Homological Stability for Spaces of Subsurfaces with Tangential Structure

TL;DR

This paper establishes homological stability for spaces of subsurfaces with various tangential structures in high-dimensional simply-connected manifolds, including new stability results for symplectic subsurfaces.

Contribution

It introduces a general criterion for homological stability of subsurface spaces with tangential structures and extends stability results to pointed embeddings and symplectic subsurfaces.

Findings

Homological stability holds for subsurfaces with certain tangential structures in manifolds of dimension ≥5.

Stability results are extended to spaces of pointedly embedded subsurfaces.

Homological stability is proved for spaces of symplectic subsurfaces.

Abstract

Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least dimension five and is simply-connected. We introduce a notion of tangential structure for subsurfaces and give a general criterion for when the space of subsurfaces with tangential structure satisfies homological stability provided that the manifold is simply-connected and has dimension $n\geq 5$. Examples of tangential structures such that the spaces of subsurface with that tangential structure satisfy homological stability are framings or spin structures of their tangent bundle, or $k$-frames of the normal bundle provided that $k\leq n-2$. Furthermore we introduce spaces of pointedly embedded subsurfaces and construct stabilization maps, as well as prove homological stability for these. This is used to prove homological stability for spaces of symplectic subsurfaces.

Figures

Equations

$$B{\text{Mod}}_g:=\mathrm{Emb}\left({\Sigma }_g,{\mathbb{R}}^{\infty }\right)/{\mathrm{Diff}}^{+}\left({\Sigma }_g\right)$$

$$\mathrm{Emb}\left({\Sigma }_{g,b},M;\delta \right)/{\mathrm{Diff}}^{+}\left({\Sigma }_{g,b}\right)\to \mathrm{Emb}\left({\Sigma }_{g^{\prime },b^{\prime }},M;\delta \right)/{\mathrm{Diff}}^{+}\left({\Sigma }_{g^{\prime },b^{\prime }}\right)$$

$$\mathrm{Gr}\left(De\right):{\Sigma }_{g,b}$$

$$x$$

$$B_2\left(M\right)$$

$${\mathrm{Emb}}^{\theta }\left({\Sigma }_{g,b},M\right):=\left\{(e,T_e)|\begin{array}{rl}& e:{\Sigma }_{g,b}\to M\text{ is an embedding and }\\ & T_e:{\Sigma }_{g,b}\to B_2\left(M\right)\text{ makes Diagram (refeqt:TangentialStructureDiagram) commute }\end{array}\right\}$$

$$-\cup P:{\mathcal{E}}_{g,b}^{\theta }\left(M;{\delta }^T\right)$$

$$W\subset M$$

$${\left(-\cup P\right)}_{\ast }:{\pi }_0\left({\mathcal{E}}_{g,b}^{\theta }\left(M;{\delta }^T\right)\right)\to {\pi }_0\left({\mathcal{E}}_{g^{\prime },b^{\prime }}^{\theta }\left(M{\cup }_{{\partial }^{0}M\times \{0\}}{\partial }^{0}M\times [0,1];{\delta }^{\prime T}\right)\right)$$

$${\left(-\cup P\right)}_{\ast }:{\pi }_0\left({\mathcal{E}}_{g,b}^{\theta }\left({\partial }^{0}M\times [-1,0];{\delta }^T\right)\right)\to {\pi }_0\left({\mathcal{E}}_{g^{\prime },b^{\prime }}^{\theta }\left({\partial }^{0}M\times [-1,1];{\delta }^{\prime T}\right)\right)$$

$${\left(-\cup P\right)}_{\ast }:{\pi }_0\left({\mathcal{E}}_{g,b}^{\theta }\left(M;{\delta }^T\right)\right)\to {\pi }_0\left({\mathcal{E}}_{g^{\prime },b^{\prime }}^{\theta }\left(M{\cup }_{{\partial }^{0}M\times \{0\}}{\partial }^{0}M\times [0,1];{\delta }^{\prime T}\right)\right)$$

$${\left(-\cup P\right)}_{\ast }:{\pi }_0\left({\mathcal{E}}_{g,b}^{\theta }\left({\partial }^{0}M\times [-1,0];{\delta }^T\right)\right)\to {\pi }_0\left({\mathcal{E}}_{g^{\prime },b^{\prime }}^{\theta }\left({\partial }^{0}M\times [-1,1];{\delta }^{\prime T}\right)\right)$$

$${\pi }_0\left({\mathcal{M}}^{{\theta }^{\prime }}\left({\Sigma }_{g,b}\right)\right)\to {\pi }_0\left({\mathcal{M}}^{{\theta }^{\prime }}\left({\Sigma }_{g,b}\right)\right)$$

$${\mathrm{Emb}}_B^{\theta }\left({\Sigma }_g,M\right):=\left\{\left(e,T_e\right)\in {\mathrm{Emb}}^{\theta }\left({\Sigma }_g,M\right)|e\left(x\right)=x_M\text{, }{T}_e\left(x\right)=B\right\}$$

$${\sigma }_g:{\mathcal{E}}_B^{\theta }\left({\Sigma }_g,M\right)\to {\mathcal{E}}_B^{\theta }\left({\Sigma }_{g+1},M\right)$$

$${\sigma }_g:H_{\ast }\left({\mathcal{S}}_{B_{\omega }}\left({\Sigma }_g,M\right)\right)\to H_{\ast }\left({\mathcal{S}}_{B_{\omega }}\left({\Sigma }_{g+1},M\right)\right)$$

$$M^k:=M{\cup }_{{\partial }^{0}M}{\partial }^{0}M\times \left[0,k\right]$$

$$\mathcal{J}:\mathrm{Emb}\left(M_1,M_2\right)\to J_{\partial }\left(M_1,M_2\right)$$

$${\mathcal{E}}_{g,b}^{+}\left(M\right)$$

$${\mathcal{J}}_{\partial }\left({\Sigma }_{g,b},M\right)$$

$${\mathcal{E}}^{+}\left(M;b\right):=\underset{g}{⨆}{\mathcal{E}}_{g,b}^{+}\left(M;\delta \right)$$

$${\mathcal{E}}_{g,b}^{+}\left(M;\delta \right)\cong \mathrm{Emb}\left({\Sigma }_{g,b},M;d\right)/{\mathrm{Diff}}_{\partial }^{+}\left({\Sigma }_{g,b}\right)$$

$$\mathrm{Gr}\left(De\right):{\Sigma }_{g,b}\to {\mathrm{Gr}}_2\left(TM\right)$$

$$B_2\left(M\right)$$

$$g^{-1}\left(\{x\}\right)\times U_x$$

$$\left(x_{X^{\prime }},x_X\right)$$

$$U_W=\left\{W^{\prime }\in {\mathcal{E}}_{g,b}^{+}\left(M;\delta \right)\mid W^{\prime }\subset N_{W,M}\text{ and }{{\pi }_{N_{W,M}}|}_{W^{\prime }}\text{ is a diffeomorphism from }{W}^{\prime }\text{ to }W\right\}$$

$$\mathrm{Emb}\left({\Sigma }_{g,b},M;d\right)$$

$${\mathrm{Emb}}^{\theta }\left({\Sigma }_{g,b},M;d^T\right)$$

$$B_2\left(M\right)$$

$$X_1$$

$$TA\oplus N\left(A,M\right)$$

Full text

Furthermore we introduce spaces of pointedly embedded subsurfaces and construct stabilization maps, as well as prove homological stability for these. This is used to prove homological stability for spaces of symplectic subsurfaces.

The author was supported by the SPP2026 ”Geometry at Infinity” of the DFG, and European Research Council Advanced Grant ”Moduli” [Grant Number 290399]

1. Introduction

The term homological stability refers to the following phenomenon in algebraic topology: Consider a sequence of spaces $X_{n}$ indexed by the natural numbers together with continuous maps $g_{n}\colon X_{n}\to X_{n+1}$, called stabilization maps. Such a sequence satisfies homological stability if $\left(g_{n}\right)_{*}\colon H_{\ast}\!\left(X_{n}\right)\to H_{\ast}\!\left(X_{n+1}\right)$ is an isomorphism for $\ast\leq{\varphi}\!\left({n}\right)$, where $\varphi\colon\mathbb{N}\to\mathbb{N}$ is a non-decreasing function such that $\lim_{n\to\infty}{\varphi}\!\left({n}\right)=\infty$. This phenomenon has been studied intensely in the context of group homology. For example it was proven in [Nak61] that the classifying spaces of the symmetric groups $BS_{n}$ together with the maps induced from the inclusion $\{0,\ldots,n\}\to\{0,\ldots,n,n+1\}$ satisfy homological stability.

Harer showed in [Har85] that homological stability holds true for the mapping class group of closed oriented surfaces of genus $g$ denoted by $\text{Mod}_{g}$. Even though in this situation we do not have stabilization maps $\text{Mod}_{g}\to\text{Mod}_{g+1}$, there are geometrically motivated maps $H_{\ast}\!\left(\text{Mod}_{g}\right)\to H_{\ast}\!\left(\text{Mod}_{g+1}\right)$ that are only defined in a range of degrees, where they turn out to be isomorphisms. Actually, homological stability for the mapping class group of surfaces is more naturally formulated for surfaces with boundary, since their mapping class groups have stabilization maps stemming from the inclusions of the surfaces. Then the homology of the mapping class group of closed surfaces can be related to the homology of the mapping class group of surfaces with boundary in a range of degrees increasing with the genus. A similar situation will repeat itself in the present text as well.

Similar results were obtained for the mapping class groups of non-orientable surfaces (see [Wah08]), the spin mapping class groups (see [Har90]) and the mapping class groups of $3$-manifolds (see [HW10]). In [RW14] it was shown, among other things, that homological stability holds for the framed mapping class group and generalized spin mapping class groups of surfaces.

Since models for the classifying space of the mapping class group resemble the spaces studied in this paper, the example of homological stability of the mapping class groups of a surface is of particular interest for this paper. Let $\Sigma_{g}$ denote a closed, orientable and connected surface of genus $g$, and ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g}\right)$ its orientation preserving diffeomorphism group. We will denote the space of smooth embeddings of $\Sigma_{g}$ into a possibly infinite-dimensional manifold $M$ by $\operatorname{Emb}\!\left(\Sigma_{g},M\right)$. A model for the classifying space $B\text{Mod}_{g}$ is given as follows:

[TABLE]

Here the diffeomorphism group of the surface acts via precomposition. Replacing $\mathbb{R}^{\infty}$ in this construction by a finite dimensional manifold $M$ yields the moduli space of submanifolds of $M$ diffeomorphic to $\Sigma_{g}$. Hence points in this quotient will be refered to as subsurfaces. This space classifies $\Sigma_{g}$-bundles together with an embedding in a trivial $M$-bundle. These spaces and their relatives will be the main focus of this paper.

Analogously to homological stability of the mapping class group of surfaces, it will be more natural to define stabilization maps for subsurfaces with boundary: Let $\Sigma_{g,b}$ denote an orientable, compact, connected surface of genus $g$ with $b$ boundary components. Suppose $M$ has non-empty boundary, then we can consider subsurfaces diffeomorphic to $\Sigma_{g,b}$ that intersect $\partial M$ in a fixed set of embedded circles $\delta$. One can define stabilization maps:

[TABLE]

These change the number of boundary components and the genus. It was shown in [CMRW17] that if $M$ is at least $5$-dimensional and simply-connected, then these stabilization maps induce isomorphism in integral homology in a range of degrees that depends on the genus $g$.

In this paper we will combine the techniques of [CMRW17] with the methods introduces in [RW16], a paper where a unified approach to homological stability for mapping class groups of surfaces with tangential structures was given, to prove homological stability for spaces of subsurfaces that are equipped with certain relative tangential structures. Furthermore we will extend these results to yield homological stability results for certain spaces of closed subsurfaces of closed manifolds. Let us proceed by introducing these relative tangential structures.

Subsurfaces with Tangential Structure:

Let $M$ denote a smooth manifold possibly with boundary, let $\operatorname{Gr}_{2}\!\left(TM\right)$ denote the Grassmannian of oriented $2$-bundles in $TM$ and note that an embedding $e\colon\Sigma_{g,b}\to M$ lifts to the Grassmannian as a map

[TABLE]

Let $B_{2}\!\left(M\right)$ denote a topological space together with a continuous Hurewicz-fibration $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$. We will call a lift $T_{e}$ of $\operatorname{Gr}\!\left(De\right)$ to $B_{2}\!\left(M\right)$ a $\theta$-structure for $e$ and refer to the pair $(e,T_{e})$ as an embedding with tangential structure. This is encapsulated in the following diagram:

[TABLE]

We will refer to $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ as a space of $\theta$-structures of subplanes of $TM$.

Example 1.1**.**

Fix a metric and an orientation on $M$, then there is an isomorphism $\operatorname{Gr}_{2}\!\left(TM\right)\cong\operatorname{Gr}_{n-2}\!\left(TM\right)$. This isomorphism corresponds to taking the normal bundle of the subsurfaces and a tangential structure corresponds to a tangential structure on the normal bundle.

Particular examples of these tangential structures are provided by the various Stiefel-manifolds associated to the tautological bundle over $\operatorname{Gr}_{n-2}\!\left(TM\right)$. In this case an embedding with tangential structure is an embedding together with a $k$-frame of its normal bundle.

Other examples are given by $B_{2}\!\left(M\right)=\operatorname{Gr}_{2}\!\left(TM\right)\times X$ where $X$ is an arbitrary topological space.

A different example will be introduced in Section 11.

Example 1.2**.**

This definition of tangential structures should be thought of as a relative tangential structure in contrast to the usual definition of a tangential structure in terms of a fibration $\theta^{\prime}\colon B\to\operatorname{Gr}_{2}\!\left(\mathbb{R}^{\infty}\right)$. Nevertheless one can recover the usual definition as a special case of the present definition by pulling back $\theta^{\prime}\colon B\to\operatorname{Gr}_{2}\!\left(\mathbb{R}^{\infty}\right)$ via a classifying map of the tautological bundle over $\operatorname{Gr}_{2}\!\left(TM\right)$ to get a tangential structure $\theta$ in our sense.

We define the space of embeddings with $\theta$-structure as follows:

[TABLE]

The definition of the topology of this space will be given in Section 2. Let $\varphi$ denote an orientation preserving diffeomorphism of $\Sigma_{g,b}$, note $\operatorname{Gr}\!\left(De\circ\varphi\right)=\operatorname{Gr}\!\left(De\right)\circ\varphi$. Therefore ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$ has a natural and free action on $\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ by precomposition.

In order to have a reasonable notion of stabilization, we will need to restrict ourselves to embeddings that intersect the boundary of $M$ in a fixed fashion:

Suppose that $\partial M$ is non-empty and that we have specified a connected component $\partial^{0}M$ of $\partial M$.

We say that two embeddings with tangential structure $e_{1},e_{2}\in\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ have the same jet along $\partial M$ if the $\infty$-jet at $\partial\Sigma_{g,b}$ of $e_{1}$ and $e_{2}$ agree, and $\left.{T_{e_{1}}}\right\rvert_{\partial\Sigma_{g,b}}=\left.{T_{e_{2}}}\right\rvert_{\partial\Sigma_{g,b}}$.

Fix an $\infty$-jet at $\partial\Sigma_{g,b}$ of embeddings with tangential structure of $\Sigma_{g,b}$ into $M$, denoted by $\delta^{T}$. Consider the space of embeddings, whose $\infty$-jet at $\partial\Sigma_{g,b}$ agrees with $\delta^{T}$. Note that the group of diffeomorphisms that fixes the $\infty$-jet of the identity at $\partial\Sigma_{g,b}$, denoted by ${\operatorname{Diff}^{+}_{\partial}}\!\left(\Sigma_{g,b}\right)$, acts freely on this subspace of the space of embeddings. We denote the quotient by ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ and call it the space of subsurfaces with $\theta$-structure, elements of this space are called subsurfaces with $\theta$-structure. The fixed boundary condition enables us to stabilize these spaces in the following way:

We can embed $\partial^{0}M\times[0,1]$ into $M$ via a collar that identifies $\partial^{0}M\times\{1\}$ with $\partial^{0}M$ via the identity and we pullback $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ to $\operatorname{Gr}_{2}\!\left(\partial^{0}M\times[0,1]\right)$. This yields a space of $\theta$-structures on subplanes in $\partial^{0}M\times[0,1]$. Let $P$ denote a subsurface with tangential structure of $\partial^{0}M\times[0,1]$ such that every connected component of $P$ has a non-empty intersection with $\partial^{0}M\times\{0\}$. If we assume that $P\cap\partial^{0}M\times\{0\}$ agrees with the image of the jet $\delta^{T}$ and that the higher jets are compatible as well, then the following map is well-defined:

[TABLE]

Here $g^{\prime},b^{\prime}$ and $\delta^{\prime T}$ depend on the topology of $P$. By identifying $M\ \cup_{\partial^{0}M\times\{0\}}\partial^{0}M\times[0,1]$ with $M$, we get stabilization maps of spaces of subsurfaces with tangential structure of $M$. We say that a tangential structure $\theta$ satisfies homological stability if $-\cup P$ induces isomorphisms in integral homology in a range of degrees that increases with $g$ and an epimorphism in the next degree i.e. the reduced homology of the mapping cone of $-\cup P$ vanishes in a range of degrees increasing with $g$.

With these notions we can paraphrase our main result, which can be understood as an adaptation of Theorem 1.2 in [RW16] to spaces of subsurfaces:

Theorem 1.3**.**

Suppose $M$ is an at least $5$-dimensional, simply-connected manifold with non-empty boundary and $\partial^{0}M$ is a codimension [math] simply-connected submanifold of $\partial M$ together with a space of $\theta$-structures on subplanes of $TM$. Suppose further that $\theta$ stabilizes on connected components, i.e.

[TABLE]

and

[TABLE]

are isomorphisms if $g$ is high enough, then $\theta$ satisfies homological stability.

See Theorem 7.3 for the most general statement including the bounds and a wider class of tangential structures. Furthermore Theorem 7.3 does not need that $\partial^{0}M$ is simply-connected. Theorem 9.1 is the more precise version of 1.3.

The following theorem (See Theorem 9.2 for a more precise statement, that also includes the bounds) yields a large number of tangential structures which satisfy the assumptions in Theorem 1.3.

Theorem 1.4**.**

In the setting of Theorem 1.3, a space of $\theta$-structures of subplanes $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ fulfils the assumptions in Theorem 1.3 i.e.

[TABLE]

and

[TABLE]

is an isomorphism if $g$ is high enough, if the homotopy fiber of $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ is simply-connected.

In particular such a tangential structure satisfies homological stability.

Examples of such spaces of tangential structures are given by $k$-framings of the normal bundle, where $k\leq n-2$. It would interesting to see wether one can use Theorem 9.1 to prove that homological stability also holds for framings of the normal bundle, similar to the situation in [RW14]. Another example, related to symplectic structures, will be introduced in Section 11.

Furthermore we prove in Theorem 9.6 that a tangential structure $\theta$ as in Example 1.2 satisfies the assumptions of Theorem 1.3, if the corresponding moduli spaces of abstract surfaces $\mathcal{M}^{\theta^{\prime}}\left(\Sigma_{g,b}\right)$ introduced in Definition 1.1 in [RW16] fulfil homological stability (see Theorem 7.1 in [RW16] for assumptions implying homological stability). In particular it is proven in [RW16], that homological stability for the abstract moduli space holds, if the stabilization maps in [RW16] yield bijections

[TABLE]

provided that $g$ is large enough.

Examples of such ”non-relative” tangential structures are given by framings of the tangent bundle, or spin structures of the subsurfaces. As was mentioned before, homological stability for these was proven in [RW14].

Since some tangential structures are more naturally defined in the context of closed subsurfaces of closed manifolds, we will prove a version of Theorem 1.3 for closed subsurfaces:

Homological Stability for Pointed Subsurfaces:

Let $M$ denote a closed manifold with space of $\theta$-structures of subplanes of $TM$, fix $x\in\Sigma_{g}$, $x_{M}\in M$ and $B\in{\theta^{-1}}\!\left({\operatorname{Gr}_{2}\!\left(T_{x_{M}}M\right)}\right)$, define

[TABLE]

As before, the group of orientation-preserving diffeomorphisms of $\Sigma_{g}$ that fix $x$, denoted by $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ admits a natural and free action on this space of embeddings by precomposition. We will denote the quotient of this action by $\mathcal{E}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$. This space is called the space of pointedly embedded $\theta$-subsurfaces of $M$. In Section 10 we will construct a stabilization map

[TABLE]

which is heuristically given by flattening the subsurfaces in a neighborhood of $x_{M}$ and then taking a connected sum with a torus in this neighborhood. We also prove that this stabilization map yields homological stability for many kinds of tangential structures (Theorem 10.17). Parts of this theorem can be summarized as follows:

Theorem 1.5**.**

Suppose $(M,x_{M})$ is an at least $5$-dimensional, simply-connected, pointed manifold and suppose further that $\theta$ is a space of tangential structures such that the homotopy fiber of $\theta$ is simply-connected, then $\sigma_{g}$ induces an isomorphism in integral homology in degrees $\ast\leq\frac{2}{3}g-1$ and an epimorphism in degrees less than $\frac{2}{3}g$.

Symplectic Subsurfaces:

As an application of this, we prove in Section 11 a homological stability result for spaces of symplectic subsurfaces: Let $(M,\omega)$ denote a simply connected symplectic manifold of dimension at least $6$ with a base point $x_{M}\in M$. We will call an embedding $e\colon\Sigma_{g}\to M$ an oriented symplectic embedding if $e^{\ast}\omega$ is a symplectic form on $\Sigma_{g}$ and $\int_{\Sigma_{g}}e^{*}\omega>0$. Fixing a symplectic $2$-plane $B_{\omega}\in\operatorname{Gr}_{2}\!\left(T_{x_{M}}M\right)$, i.e. $\left.{\omega}\right\rvert_{B_{\omega}}$ is non-degenerate, we can consider $\text{SEmb}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$, the space of oriented symplectic embeddings that map the tangent space of the base point of the surface to $B_{\omega}$. Being an oriented symplectic embedding is invariant under the action of $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ by precomposition. We denote the quotient by $\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$and call it the space of symplectic subsurfaces which are tangential to $B$.

A proposition in the context of the h-principle, namely a variation of Theorem 12.1.1 in [EM02], allows us to relate the space of symplectic subsurfaces to spaces of subsurfaces with a certain tangential structure (See Proposition 11.8). Using Theorem 1.5, one obtains:

Theorem 1.6**.**

Let $(M,\omega)$ denote a simply-connected symplectic manifold of dimension at least $6$. For every $x_{M}\in M$ and every symplectic $2$-plane $B_{\omega}$ in $T_{x_{M}}M$. There is a homomorphism of integral homology

[TABLE]

And this homomorphism induces an isomorphism for $\ast\leq\frac{2}{3}g-1$ and an epimorphism for $\ast\leq\frac{2}{3}g$.

Even though the present stabilization map exists only on the level of homology, the author is hopeful that with the presented methods, it is possible to construct an actual map $\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g},M\right)\to\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g+1},M\right)$ that realizes the map on homology (see the end of Section 11 for a more detailed discussion of this).

It was shown in [Li05] that many second homology classes can be represented by symplectic subsurfaces. Furthermore essential tools in the theory of symplectic manifolds are $J$-holomorphic curves and their moduli spaces (See [MS12] for an introduction), embedded $J$-holomorphic curves are essentially the same as symplectic subsurfaces and their moduli spaces are connected to the space of symplectic subsurfaces. We hope that the homological stability result might give some new insight into these moduli spaces.

Overview:

The first two sections describe spaces of subsurfaces with boundary, tangential structures and related concepts used throughout the paper (Section 2) and their stabilization maps (Section 3). Furthermore we show that every stabilization map can be decomposed in to simpler maps, given by stabilizing with a pair of pants or a disk) for which we will prove homological stability.

Section 4 introduces the most important tool in the proof of Theorem 1.3 and Theorem 7.3: Semi-simplicial resolutions. In Section 5 a semi-simplicial resolution of the space of subsurfaces is built and its behaviour with respect to stabilization maps is studied.

In Section 6 we introduce the notion of $k$-triviality, a concept developed in [RW16], and adapt it to spaces of subsurfaces instead of abstract surfaces. Using this notion we introduce in Section 7 the actual stability bounds and prove most of Theorem 7.3 the stronger version of Theorem 1.3. Section 8 finishes the proof by explaining how to remove boundary components.

In Section 9 we establish criteria for spaces of tangential structures to be suitable for application of Theorem 7.3 and in doing so proof among other results the strengthened version of Theorem 1.4. In Section 10, we extend the notions of spaces of subsurfaces to spaces of pointedly embedded subsurfaces and construct a stabilization map for these.

Lastly in Section 11, we apply the results of Section 9 and Section 10 to prove Theorem 1.6, the homological stability result for symplectic subsurfaces.

Most of the setting and the general proof strategy in this paper were inspired by [CMRW17], and represent an adaptation of their ideas to the newly introduced setting of tangential structures for subsurfaces. However the introduction of the concept of $k$-triviality (adapted from [RW16]) made the proof more intricate.

The results of Sections 9 to 11 are new and allow a much broader application as well as an extension of the results of [CMRW17].

Acknowledgements:

I would like to thank my PhD advisor Ursula Hamenstädt for many helpful discussions and suggestions and furthermore for introducing me to the topic of homological stability. I would also like to thank Mark Pedron for plenty of discussions about this subject and for helping me making the present text much more accessible. Frederico Cantero helped me understand parts of [CMRW17] and he helped me to put things into the correct context, for which I am very grateful. Furthermore I would like to express my gratitude for numerous corrections and hints at possible extensions of the results of this paper by an anonymous referee. Lastly I would like to thank Peter Michor for an answer on MathOverflow ([MMO]) without which I would not have been able to finish the proof of Proposition 10.4.

2. Spaces of Subsurfaces and Tangential Structures

The following section will introduce spaces of subsurfaces, tangential structures and will also lay most of the technical foundation for the rest of this paper.

2.1. Spaces of Embeddings and Embedded Submanifolds

Let $\left(M,\partial M\right)$ denote a compact smooth manifold with boundary and let $\partial^{0}M$ denote a codimension [math] submanifold (possibly with boundary) of $\partial$. We call an embedding $c\colon\partial M\times\left[0,1\right)\to M$ such that ${c}\!\left({\left(x,0\right)}\right)=x$ a collar of $M$.

We say that $M$ is collared if there is a fixed collar $c$ of $M$. Note that if $M$ is collared, then

[TABLE]

has a canonical smooth structure.

Definition 2.1**.**

Let $M_{1}$ and $M_{2}$ denote smooth collared manifolds. A smooth map $f\colon M_{1}\to M_{2}$ between collared manifolds is a smooth map that maps $\partial M_{1}$ to $\partial M_{2}$ and there exist neighborhoods of $\partial M_{1}$ and $\partial M_{2}$ such that in these neighborhoods $f$ is a product of a smooth map and the identity with respect to the collars of $M_{1}$ and $M_{2}$.

The definition of collared embedding and collared diffemorphism is completely analogous.

Since every occuring manifold and map will be assumed to be a collared manifold or a collared map we will drop the adjective.

A collared embedding from $M_{1}$ to $M_{2}$ maps the boundary of $M_{1}$ to the boundary of $M_{2}$ and the map is transverse to the boundary of $M_{2}$. If $A_{1}\subset M_{1}$ and $A_{2}\subset M_{2}$ are submanifolds, we call an embedding $e\colon\left(M_{1},A_{1}\right)\to\left(M_{2},A_{2}\right)$ an embedding of pairs if ${e^{-1}}\!\left({A_{2}}\right)=A_{1}$. We will be interested in spaces associated to the space of (collared) embeddings $\operatorname{Emb}\!\left(M_{1},M_{2}\right)$. This space is equipped with the $C^{\infty}$-topology.

Definition 2.2**.**

Let $e_{1},e_{2}\colon M_{1}\to M_{2}$ denote embeddings. We say that $e_{1}$ and $e_{2}$ have the same jet along $\partial M_{2}$ if there exists an open neighbourhood $U_{\partial M_{1}}$ of $\partial M_{1}$ such that $\left.{e_{1}}\right\rvert_{U_{\partial M_{1}}}=\left.{e_{2}}\right\rvert_{U_{\partial M_{1}}}$. This defines an equivalence relation on $\operatorname{Emb}\!\left(M_{1},M_{2}\right)$ and we denote the quotient map by

[TABLE]

We write ${\operatorname{Emb}}\!\left(M_{1},M_{2};d\right)\coloneqq{\mathcal{J}^{-1}}\!\left({d}\right)$.

From here on forth let us specialize to the case where the domain of the embedding is a compact connected oriented surface of genus $g$ with $b$ boundary components denoted by $\Sigma_{g,b}$. We denote by ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$ the group of orientation-preserving diffeomorphisms of $\Sigma_{g,b}$. This group acts freely on the space of embeddings and on $J_{\partial}$ via precomposition. We define

[TABLE]

The first space will be called the space of subsurfaces of genus $g$ and $b$ boundary components. We also define

[TABLE]

An element $W\in{\mathcal{E}^{+}_{g,b}}\!\left(M\right)$ is an unparametrized embedding or in other words a submanifold $W\subset M$ diffeomorphic to $\Sigma_{g,b}$. We will call such an unparametrized embedding a subsurface of $M$.

Since the map $\mathcal{J}$ is equivariant with respect to the ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$ -actions, we get an induced map between the quotient spaces also denoted by $\mathcal{J}$. We define ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ as ${\mathcal{J}^{-1}}\!\left({\delta}\right)$ for some $\delta\in{\mathcal{J}_{\partial}}\!\left(\Sigma_{g,b},M\right)$. Note that if $\delta=\left[d\right]$ then

[TABLE]

Here ${\operatorname{Diff}^{+}_{\partial}}\!\left(\Sigma_{g,b}\right)$ denotes the group of diffeomorphisms of $\Sigma_{g,b}$ that agree with the identity on a neighbourhood of the boundary.

As a slight abuse of notation we will also write $\delta$ for the image of some representative of $\delta\in{\mathcal{J}_{\partial}}\!\left(\Sigma_{g,b},M\right)$ in $\partial M$. We will call $\delta$ a boundary condition.

2.2. Tangential structures

Let $M$ denote a manifold possibly with boundary. There is the fiber bundle $\operatorname{Gr}_{2}\!\left(TM\right)\to M$, where $\operatorname{Gr}_{2}\!\left(TM\right)$ denotes the space of oriented $2$-planes in $TM$. For the purpose of this paper a space of $\theta$-structures of subplanes of $TM$ will denote a continuous Hurewicz-fibration $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$. For a submanifold $A$ of $M$ we have an induced map $\operatorname{Gr}_{2}\!\left(TA\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ and we will denote the pullback of $B_{2}\!\left(M\right)$ along this inclusion by $B_{2}\!\left(A\right)$.

If $W$ is an oriented subsurface, then $\operatorname{Gr}_{2}\!\left(W\right)\to W$ is a two-sheeted covering with a distinguished section. In this particular case we will identify the correctly oriented connected component of $\operatorname{Gr}_{2}\!\left(W\right)$ with $W$ and consider $B_{2}\!\left(W\right)$ as a fibration over $W$.

If we have an embedding $e\colon\Sigma_{g,b}\to M$, then we get an induced map

[TABLE]

by mapping $x\in\Sigma_{g,b}$ to ${De}\!\left({T_{x}\Sigma_{g,b}}\right)$ considered as an oriented $2$-plane. We will call this map the Grassmannian differential of $e$.

Definition 2.3**.**

Given an embedding $e\colon\Sigma_{g,b}\to M$, we call a map $T_{e}\colon\Sigma_{g,b}\to B_{2}\!\left(M\right)$ a $\theta$-structure of $e$ if the following diagram commutes:

[TABLE]

We will denote by $\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ the space of embeddings with $\theta$-structure. This space is topologized as a subspace of $\operatorname{Map}\!\left(\Sigma_{g,b},B_{2}\!\left(M\right)\right)\times\operatorname{Emb}\!\left(\Sigma_{g,b},M\right)$, where the first factor is equipped with the compact-open topology and the second factor with the $C^{\infty}$-topology. Even though elements of $\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ consist of pairs of maps, we will usually say that $e$ is an embedding with $\theta$-structure. In this case we will denote the underlying $\theta$-structure by $T_{e}$. We will say that two embeddings with $\theta$-structure $e_{1},e_{2}$ have the same $\theta$-jet along $\partial M$ if they have the same jet along $\partial M$ and $\left.{T_{e_{1}}}\right\rvert_{\partial\Sigma_{g,b}}=\left.{T_{e_{2}}}\right\rvert_{\partial\Sigma_{g,b}}$.

For a diffeomorphism $\phi\in{\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$, we have ${\operatorname{Gr}\!\left(De\right)}\!\left({x}\right)={\operatorname{Gr}\!\left(De\circ\phi^{-1}\right)}\!\left({{\phi}\!\left({x}\right)}\right)$. Hence the group ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$ acts on $\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ via precomposition, this action preserves the equivalence relation of having the same $\theta$-jet. If we restrict the embeddings to have some fixed $\theta$-jet, then we denote the corresponding space of embeddings with $\theta$-structure by $\delta^{T}$ to obtain ${\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)$ and we denote the quotient of this subspace by the action of ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g,b}\right)$ by ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$. Similarly, we obtain ${\mathcal{J}^{\theta}_{\partial}}\!\left(\Sigma_{g,b},M\right)$ and ${\mathcal{E}^{\theta}}\!\left(M;\delta^{T}\right)$. We will call the second space the space of subsurfaces with $\theta$-structure and an element in it will be called a subsurface with $\theta$-structure of $M$. If some $\theta$-jet $\delta^{T}$ of an embedding with $\theta$-structure is given, we will sometimes write $\delta$ for the image of the underlying boundary condition without $\theta$-structure.

If $\theta$ is the identity map, we will write $+$ instead of $\mathrm{Id}$. Let us recall parts of an example of the introduction:

Example 2.4**.**

Let $\gamma^{\perp}$ denote the complement of the tautological $2$-plane bundle on $\operatorname{Gr}_{2}\!\left(TM\right)$ i.e. $\pi_{\operatorname{Gr}_{2}\!\left(TM\right)}^{*}TM/\gamma$, where $\pi_{\operatorname{Gr}_{2}\!\left(TM\right)}\colon\operatorname{Gr}_{2}\!\left(TM\right)\to M$ denotes the projection and $\gamma$ denotes the tautological bundle which includes into $\pi_{\operatorname{Gr}_{2}\!\left(TM\right)}^{*}TM$. Let $S\!\left(\gamma^{\perp}\right)\coloneqq\gamma^{\perp}\setminus 0$ denote the complement of the image of the zero section of this bundle. Then $\theta\colon S\!\left(\gamma^{\perp}\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ is a Hurewicz-fibration. For an embedding $e\colon\Sigma_{g,b}\to M$ the space $B_{2}\!\left({e}\!\left({\Sigma_{g,b}}\right)\right)$ is the complement of the zero-section of the normal bundle and a $\theta$-structure corresponds to a continuous section of this bundle. In this case $\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)$ is the space of embeddings together with a continuous section of the normal bundle.

Remark 2.5**.**

In some situations it makes sense to replace $\operatorname{Map}\!\left(\Sigma_{g,b},B_{2}\!\left(M\right)\right)$ by $C^{\infty}\!\left(\Sigma_{g,b},B_{2}\!\left(M\right)\right)$ in Definition 2.3. Of course this requires some kind of smooth structure on $B_{2}\!\left(M\right)$. In Example 2.4 this would make the sections of the normal bundle smooth sections. One also has to alter the definition of having the same $\theta$-jet to include the jet of the $\theta$-structures. Everything proven in this paper works in that case as well.

Remark 2.6**.**

If we have a subsurface with a tangential structure $W$ and some isotopy $\left(\left[0,1\right],0\right)\to\left({\operatorname{Diff}^{+}_{\partial}}\!\left(M\right),\mathrm{Id}\right)$, we can move $W$ along $M$ using this isotopy to obtain $W_{t}$. Since $\theta$ is a Hurewicz-fibration we can also equip $W_{t}$ with a tangential structure and if we say nothing about tangential structures in this context, we will assume implicitly that a lift of the aforementioned path in ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is chosen. Note that the tangential structure of $W_{1}$ is not unique i.e. there might be many different tangential structures for $W_{t}$.

Lastly, using a collar $M^{k}$ and $M$ are diffeomorphic and since collars are unique up to isotopy, the associated diffeomorphism is also unique up to isotopy. This enables us to equip $M^{k}$ with a unique space of $\theta$-structures and we will always assume that $M^{k}$ has this space of $\theta$-structures. A space of $\theta$-structures on a manifold $M$ induces a space of $\theta$-structures on every submanifold and we will always assume that submanifolds are equipped with the induced structure. In particular $\partial M\times\left[0,k\right]$ is equipped with a space of $\theta$-structures.

2.3. Retractile Spaces and Fibrations

The following arguments will be used frequently throughout this paper to prove that certain maps are locally trivial fibrations and the ideas date back to [Cer61].

Definition 2.7**.**

Let $G$ denote a topological group and $X$ a $G$-space. We say that $X$ is $G$-locally retractile if every point $x\in X$ possesses an open neighborhood $U_{x}$ and a continuous map $\xi\colon\left(U_{x},x\right)\to\left(G,\mathrm{Id}\right)$, called the $G$-local retraction around $x$ , such that ${\xi}\!\left({x^{\prime}}\right)\cdot x=x^{\prime}$ for all $x\in U_{x}$. Note that this implies that $\xi$ is a homeomorphism onto its image.

The following lemma will be used several times throughout this paper.

Lemma 2.8**.**

If $X$ is $G$-locally retractile and $g\colon X^{\prime}\to X$ denotes a $G$-equivariant map, then $g$ is a locally trivial fibration.

Proof.

For $x\in X$ choose an $U_{x}$ and a $G$-local retraction $\xi\colon U_{x}\to G$ around $x$. Then

[TABLE]

gives the desired local trivialisation as is easily checked. ∎

As before let ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ denote the group of diffeomorphisms of $M$ that agree with the identity on a neighbourhood of $\partial M$. The following proposition can be found in [Cer61] (Chapter 2.2, Theorem 5).

Proposition 2.9**.**

Let $d$ denote the jet of an embedding from $A$ into $M$, then ${\operatorname{Emb}}\!\left(A,M;d\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile.

The following proposition can be found in [CMRW17] with a proof that relies on a reference to [BF81] or [Mic80]. We will give a sketch of a proof of the proposition, because similar ideas will occur later on.

Proposition 2.10**.**

The space ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile.

Proof.

Let $W\subset M$ denote an element of ${\mathcal{E}^{+}_{g,b}}\!\left(M;b\right)$ and $N_{W,M}$ a tubular neighborhood of $W$ with corresponding projection $\pi_{N_{W,M}}\colon N_{W,M}\to W$. We define

[TABLE]

which is an open subset of ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$, because the diffeomorphisms form an open subset of the maps from $W^{\prime}$ to $W$ (See Proposition 1.4.2 in Chapter 2 in [Cer61]). For $W^{\prime}$ in $U_{W}$ we can define an isotopy of the embedded subsets by pushing $W^{\prime}$ along the fiber of $N_{W,M}$ onto $W$. Furthermore we can use the isotopy extension theorem (See Theorem 6.1.1 in [Wal16]), to extend these isotopies continuously to isotopies of $M$. The resulting diffeomorphisms at time $1$ of these isotopies give us the desired ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-local retraction. ∎

The last Proposition implies the following useful observation:

Corollary 2.11**.**

The two maps

[TABLE]

are locally trivial fibrations.

Proof.

The only thing that might need clarification is the second assertion, but to prove this one only has to note that

[TABLE]

is a pullback diagram. ∎

Moreover we have the following central lemma:

Lemma 2.12**.**

The forgetful map $\pi_{\theta}\colon{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\to{\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is a Hurewicz fibration. The fiber over a surface $W$ is given by $\Gamma\!\left(B_{2}\!\left(W\right)\right)$, the space of sections of $B_{2}\!\left(W\right)$. The same holds true for the forgetful map ${\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;d^{T}\right)\to{\operatorname{Emb}}^{+}\!\left(\Sigma_{g,b},M;d^{T}\right)$

Proof.

By Theorem 42.3 and the corollary in 27.4 in [KM97] (together with the fact that the quotient of a nuclear spaces by a closed subspaces is a nuclear space) we conclude that ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is paracompact and Hausdorff, hence we conclude that a locally trivial fiber bundle over this space is actually a Hurewicz fibration by Theorem 13 in Section 2.7 of [Spa95]. Hence it suffices to prove that $\pi_{\theta}$ is a locally trivial fiber bundle. In order to do this we want to apply Lemma 2.8.

Let $\mathrm{Homeo}_{M}\!\left(B_{2}\!\left(M\right)\right)$ denote the group of homeomorphisms of $B_{2}\!\left(M\right)$ which cover a diffeomorphism of $\operatorname{Gr}_{2}\!\left(TM\right)$ induced from a diffeomorphism of $M$ that fixes a neighbourhood of the boundary i.e. the following diagram commutes and the lower map is a diffeomorphism

[TABLE]

This group acts on ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ and ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ by post-composition. Proposition 2.10 states that ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile. We will show that it is also $\mathrm{Homeo}_{M}\!\left(B_{2}\!\left(M\right)\right)$-locally retractile. For a $W\in{\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$, let $\xi\colon U_{W}\to{\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ denote a ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-local retraction around $W$. Note that ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ is locally contractible and it is paracompact (For paracompactness see Lemma 41.11 of [KM97] to conclude that ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ with the $C^{\infty}$ topology is metrizable). Hence there is a contractible, closed neighbourhood of the identity in ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$. By shrinking $U_{W}$ we can assume that ${\xi}\!\left({U_{W}}\right)$ maps to such a contractible neighbourhood of the identity.

We get a map $\operatorname{Gr}\!\left(D\mathrm{ev}\right)\colon\operatorname{Gr}_{2}\!\left(TM\right)\times U_{W}\to\operatorname{Gr}_{2}\!\left(TM\right)$ which maps $(B,W)$ to ${\operatorname{Gr}\!\left(D{\xi}\!\left({W}\right)\right)}\!\left({B}\right)$ and we can use the contractibility of $U_{W}$ together with its paracompactness and the compactness of $M$ to conclude that the pullback $\operatorname{Gr}\!\left(D\mathrm{ev}\right)^{*}B_{2}\!\left(M\right)$ is isomorphic to the pullback $\pi_{U_{W}}^{*}B_{2}\!\left(M\right)$, where $\pi_{U_{W}}\colon\operatorname{Gr}_{2}\!\left(TM\right)\times U_{W}\to\operatorname{Gr}_{2}\!\left(TM\right)$ denotes the projection. Using this isomorphism, one can construct a lift of $\xi$ to conclude that ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is $\mathrm{Homeo}_{M}\!\left(B_{2}\!\left(M\right)\right)$-locally retractile.

By Lemma 2.8 we conclude that the forgetful map $\pi_{\theta}$ is a locally trivial fiber bundle.

The specification of the fibre is evident. The proof for the second map is completely analogous. ∎

We will also need the following standard lemma from algebraic topology, whose proof is omitted:

Lemma 2.13**.**

Suppose we have the following diagram of topological spaces

[TABLE]

where $g_{1}$ is surjective. If $g_{1}$ and $g_{2}\circ g_{1}$ are fibrations, then $g_{2}$ is a fibration as well.

2.4. Tubular neighborhoods and Thickened Embeddings

If $A\subset M$ denotes a submanifold, we will write $N\!\left(A,M\right)$ for the normal bundle of $W$ i.e. $\left.{TM}\right\rvert_{A}/TA$. A tubular neighbourhood of $A$ is an embedding $e^{T}_{A,M}\colon N\!\left(A,M\right)\to M$ such that $\left.{e^{T}_{A,M}}\right\rvert_{A}$ is the identity and the composition

[TABLE]

agrees with the projection onto the second factor. If $\left(A,A^{\prime}\right)\subset\left(M,A\right)$ denotes an embedded pair, we define a tubular neighborhood of $\left(A,A^{\prime}\right)$ in $\left(M,A\right)$ to be a tubular neighborhood $e^{T}_{A,M}\colon N\!\left(A,M\right)\to M$ of $A$ such that $\left.{e^{T}_{A,M}}\right\rvert_{N\!\left(A^{\prime},A\right)}$ is a tubular neighbourhood of $A^{\prime}$ in $A$. If $A$ has a boundary we assume that every tubular neighbourhood of $A$ is actually a tubular neighbourhood of $\left(A,\partial A\right)$ in $\left(M,\partial M\right)$.

From the vector bundle $N\!\left(A,M\right)$, one can obtain a disk bundle by compactifying this fiberwise using a sphere at infinity to obtain $\overline{N\!\left(A,M\right)}$. An embedding of $\overline{N\!\left(A,M\right)}\to M$ is called a closed tubular neighborhood if its restriction to $N\!\left(A,M\right)$ is a tubular neighborhood. We denote by

[TABLE]

the subspace of closed tubular neighborhoods. Similar notation will occur if we consider tubular neighborhoods of pairs.

A proof of the following lemma is sketched in Section 2.5 of [CMRW17].

Lemma 2.14**.**

If $\left(A,A^{\prime}\right)$ is an embedded pair of compact submanifolds in $\left(M,A\right)$, then $\overline{\operatorname{Tub}}\!\left(A,M\right)$ and $\overline{\operatorname{Tub}}\!\left(\left(A,A^{\prime}\right),\left(M,A\right)\right)$ are contractible.

Definition 2.15**.**

We call an embedding, possibly equipped with a $\theta$-structure, $e\colon\Sigma_{g,b}\to M$ together with a closed tubular neighbourhood of ${e}\!\left({\Sigma_{g,b}}\right)$ a thickened embedding of $\Sigma_{g,b}$. We will write $\overline{\operatorname{TEmb}}\!\left(\Sigma_{g,b},M;d\right)$ for the space of thickened embeddings and $\overline{\operatorname{TEmb}}^{\theta}\!\left(\Sigma_{g,b},M;d^{T}\right)$ for the space of thickened embeddings with $\theta$-structure. We could furthermore replace $\Sigma_{g,b}$ by a pair $\left(\Sigma_{g,b},A^{\prime}\right)$ and $M$ by $\left(M,A\right)$ to obtain spaces of thickened embeddings of pairs.

The topology of the space of thickened embeddings is quite tricky and we refer the reader to Section 2.5 in [CMRW17] for details. The important thing to note about this space is the following proposition:

Proposition 2.16**.**

${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$* acts via post composition on $\overline{\operatorname{TEmb}}\!\left(A,M;d\right)$ and this space is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile.*

Corollary 2.17**.**

The forgetful map $\overline{\operatorname{TEmb}}\!\left(\Sigma_{g,b},M;d\right)\to{\operatorname{Emb}}\!\left(\Sigma_{g,b},M;d\right)$ is a locally trivial fibration with fiber over a basepoint $e\in{\operatorname{Emb}}\!\left(\Sigma_{g,b},M;d\right)$ given by the space of tubular neighborhoods of ${e}\!\left({\Sigma_{g,b}}\right)$.

Furthermore the forgetful map $\overline{\operatorname{TEmb}}^{\theta}\!\left(\Sigma_{g,b},M;d\right)\to{\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;d\right)$ is also a locally trivial fibration with fiber the space of tubular neighborhoods of some ${e}\!\left({\Sigma_{g,b}}\right)$.

Proof.

The first assertion follows as ${\operatorname{Emb}}\!\left(\Sigma_{g,b},M;d\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile. The second assertion follows since the following diagram is a pullback diagram:

[TABLE]

∎

We will also need the following lemma:

Lemma 2.18**.**

For every submanifold $A$ of $\Sigma_{g,b}$, the restriction maps

[TABLE]

are fibrations

The proof will use the following lemma:

Lemma 2.19**.**

Let $X$ denote a metric space and $A\subset X$ a closed subset, with a closed neighborhood $U_{A}$ such that $A$ is a strong deformation retract of $U_{A}$ via a strong deformation $\Phi(x,t)$. Then $X\times\{0\}\cup A\times[0,1]$ is a strong deformation retract of $X\times[0,1]$.

The assumptions about $X$ are quite restricting and the lemma should be true in a more general setting, but for the present context it certainly suffices.

Proof.

Define $d\colon U_{A}\to\mathbb{R}$ as

[TABLE]

As assumed above let $\Phi(-,t)$ denote a homotopy between the identity on $U_{A}$ and a retraction of the neighborhood. Then the deformation retract in the product space and the corresponding homotopy is given by the following:

[TABLE]

∎

Proof of Lemma 2.18.

Note that both statements follow from Lemma 2.8 if $\theta=\mathrm{Id}$. So consider the following lifting problem:

[TABLE]

Since the right vertical map is a fibration, we can lift $g$ to ${\operatorname{Emb}}^{+}\!\left(\Sigma_{g,b},M;d^{T}\right)$. Such a map corresponds to an embedding $e\colon\Sigma_{g,b}\times D^{N}\times[0,1]\to M\times D^{N}\times[0,1]$ which is the identity on the $D^{N}\times[0,1]$ component. Furthermore this embedding is equipped with a tangential structure on $\Sigma_{g,b}\times D^{N}\times\{0\}\cup A\times[0,1]$ which stems from $g$ and $\bar{g}$. Lifting $g$ corresponds to an extension of this tangential structure. By Lemma 2.19, $\Sigma_{g,b}\times D^{N}\times\{0\}\cup A\times[0,1]$ is a deformation retract of $\Sigma_{g,b}\times D^{N}\times[0,1]$ , hence $e^{\ast}B_{2}\!\left(M\right)$ is isomorphic to $\left.{e^{\ast}}\right\rvert_{\Sigma_{g,b}\times D^{N}\times\{0\}\cup A\times[0,1]}B_{2}\!\left(M\right)$. Composition with such an isomorphism provides the required extension of the tangential structure. The proof for the spaces of thickened embeddings is completely analogous. ∎

3. Stabilization Maps

We will be interested in two kinds of stabilization maps. Up to homotopy equivalences, they are the same, but they will play different roles in the proof and therefore will be distinguished. They are motivated by the definition of inner and outer cobordisms in [RW16].

3.1. Boundary Bordisms and Stabilization Maps

Let $M$ denote an at least $5$-dimensional manifold and $\partial^{0}M$ a codimension [math] submanifold of $\partial M$. By definition, a boundary condition $\delta^{T}\in{\mathcal{J}^{\theta}_{\partial}}\!\left(\Sigma_{g,b},M\right)$ that intersects the boundary of $\partial^{0}M$ transversely yields a submanifold with $\theta$-structure in $\partial^{0}M\times\left[0,1\right]$ by taking a product with $\left[0,1\right]$. This specifies a $\theta$-jet along $\partial^{0}M\times\{0\}$. Together with a $\theta$-jet $\bar{\delta}^{T}$ along $\partial^{0}M\times\{1\}$ and using collars one obtains boundary conditions in ${\mathcal{J}^{\theta}_{\partial}}\!\left(\Sigma_{g^{\prime},b+b^{\prime}},\partial^{0}M\times\left[0,1\right]\right)$ for any $g^{\prime}$ and $b^{\prime}$, which we will denote by $\delta^{T}\cup\bar{\delta}^{T}$. Let us denote $\bigsqcup_{g^{\prime}}{\mathcal{E}^{\theta}_{g^{\prime},b+b^{\prime}}}\!\left(\partial^{0}M\times\left[0,1\right];\delta^{T}\cup\bar{\delta}^{T}\right)$ by ${\mathcal{C}^{\theta}}\!\left(\partial^{0}M;\delta^{T},\bar{\delta}^{T}\right)$ and call it the space of boundary bordisms.

Given an element $P\in{\mathcal{C}^{\theta}}\!\left(\partial^{0}M;\delta^{T},\bar{\delta}^{T}\right)$ one obtains a continuous

[TABLE]

where $\delta^{\prime}$ consists of those boundary conditions that are not met by the boundary bordism. These maps are called stabilization maps.

We will call $P\cap\partial^{0}M\times\{0\}$ the incoming boundary of $P$ and $P\cap\partial^{0}M\times\{1\}$ the outgoing boundary of $P$. We have the following lemma, which says that $P$ decomposes into elementary pieces, for which it will be easier to prove homological stability.

Proposition 3.1**.**

Let $M$ be a manifold with space of $\theta$-structures of subplanes of $TM$. If $P$ is a boundary bordism as above then there exists $P_{1},\ldots,P_{k}$ such that:

1. (i)

$P_{i}$* is a boundary bordism sitting inside of $\partial^{0}M\times[i-1,i]$ and all of its connected components but one are homeomorphic to a cylinder.* 2. (ii)

$P_{1}\cup\ldots\cup P_{k}\simeq k\cdot P\subset\partial^{0}M\times[0,k]$* (The product means that $(x,t)\in\partial^{0}M\times[0,1]$ is mapped to $(x,kt)\in\partial^{0}M\times[0,k]$) and $\simeq$ means in this case that the underlying submanifolds are isotopic relative to their boundary and their tangential structures are homotopic* 3. (iii)

the projection onto the second coordinate of $\partial^{0}M\times[i-1,i]$ restricted to $P_{i}$ is a Morse function with at most one critical point. 4. (iv)

If any such critical point happens to be a minimum or a maximum it has to be a global maximum or minimum of $P_{1}\cup\ldots\cup P_{k}$.

Remark 3.2**.**

Since we implicitly required every connected component of $P$ to meet $\partial^{0}M\times\{0\}$ the last part of the lemma says among other things that there is no minimum.

To prove this proposition we will need to use surgery along a half-disk, a process we will explain now. We define

1. (i)

$D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}\middle|x_{2}\geq 0\text{, }\left\lVert{\left(x_{1},x_{2}\right)}\right\rVert\leq 1\right\}$ 2. (ii)

$\partial^{0}D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\partial D^{2}_{+}\middle|x_{2}=0\right\}$ 3. (iii)

$\partial^{1}D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\partial D^{2}_{+}\middle|\left\lVert{\left(x_{1},x_{2}\right)=1}\right\rVert\right\}$

Let $W\subset M$ denote a subsurface of $M$. We fix an embedded circle $c\subset\partial^{1}D^{2}_{+}\times[0,1]$ which contains $\left(\partial^{1}D^{2}_{+}\setminus(B_{\epsilon}\!\left(1\right)\cup B_{\epsilon}\!\left(-1\right))\right)\times\{0,1\}$ and connects these two connected arcs via two small circle segments. Fix once and for all an embedding $a\colon D^{2}\to D^{2}_{+}\times[0,1]$ such that ${a}\!\left({\partial D^{2}}\right)=c$ and ${a}\!\left({D^{2}\setminus\frac{1}{2}D^{2}}\right)\subset\partial^{1}D^{2}_{+}\times[0,1]$ and such that ${a}\!\left({\mathbb{R}\cdot(1,0)\cap\frac{1}{2}D^{2}}\right)$ is very close to $\partial^{0}D^{2}_{+}\times\{\frac{1}{2}\}$ (see Figure 1 to clarify the definition of $c$ and $a$). Note that ${a}\!\left({D^{2}}\right)$ cuts $\partial^{1}D^{2}_{+}\times[0,1]$ into two connected components and let $X$ denote the ”inner one” i.e. the one that does not contain $\partial^{1}D^{2}_{+}\times\{0,1\}$.

Definition 3.3**.**

Let $M$ denote a manifold as above and $W$ a subsurface of $M$. Let $e_{D^{2}_{+}}\colon(D^{2}_{+}\times[0,1],\partial^{1}D^{2}_{+}\times[0,1])\to(M,W)$ denote an embedding, then we will denote $(W\setminus{e_{D^{2}_{+}}}\!\left({X}\right))\cup{e_{D^{2}_{+}}}\!\left({{a}\!\left({D^{2}}\right)}\right)$ by $W\natural e_{D^{2}_{+}}$. The above requirements for $a$ ensure that this is a smooth manifold.

Suppose that the dimension of $M$ is at least $5$ and that $W$ is a subsurface with tangential structure. In this case $W$ and $W\natural e_{D^{2}_{+}}$ are isotopic and we can use this to give $W\natural e_{D^{2}_{+}}$ some tangential structure. Even though this is not well-defined we will denote any such subsurface with tangential structure by $W\natural e_{D^{2}_{+}}$.

Assume that the dimension of $M$ is at least $5$ and $W$ is a subsurface, given a map of a half-disk $e_{D^{2}_{+}}\colon(D^{2}_{+},\partial^{1}D^{2}_{+})\to(M,W)$, which is an embedding at every point except at $(\pm 1,0)$, where ${De_{D^{2}_{+}}}\!\left({T_{(\pm 1,0)}\partial^{1}D^{2}_{+}}\right)$ agrees with ${De_{D^{2}_{+}}}\!\left({T_{(\pm 1,0)}\partial^{0}D^{2}_{+}}\right)$. We can use the sufficiently high codimension to ensure that $e_{D^{2}_{+}}$ extends to an embedding $e_{D^{2}_{+}}^{\prime}\colon(D^{2}_{+}\times[0,1],\partial^{1}D^{2}_{+})\to(M,W)$ such that ${e_{D^{2}_{+}}}\!\left({\partial^{0}D^{2}_{+}}\right)$ agrees with ${e_{D^{2}_{+}}^{\prime}}\!\left({{a}\!\left({\mathbb{R}\cdot(1,0)\cap\frac{1}{2}D^{2}}\right)}\right)$. In this case we will still write $W\natural e_{D^{2}_{+}}$ for the surgery along this extension of $e_{D^{2}_{+}}$. Armed with this definition we can start proving the previous lemma.

Proof of Proposition 3.1.

We start by showing that there is an isotopy of $P$ such that $\pi_{[0,1]}\colon\partial^{0}M\times[0,1]\to[0,1]$ restricted to $P$ is a Morse function with the property that every critical value corresponds to a unique critical point, which proves the first three parts of the lemma. Then we proceed by explaining how to remove local minima. Using the symmetry between minima and maxima given by the flip map $(x,t)\mapsto(x,1-t)$ this also explains how to eliminate local maxima.

We have an open inclusion

[TABLE]

The isomorphism stems from the identification of maps into products as products of maps and the described map has the following form $e\mapsto(\pi_{\partial^{0}M}\circ e,\pi_{[0,1]}\circ e)$. That this map is indeed open is proven in [Cer61] Chapter 2, Section 1.2. Since Morse functions form an open and dense subset of all real functions we get that in $C^{\infty}\!\left(P,[0,1]\right)\times C^{\infty}\!\left(P,\partial^{0}M\right)$ the subset $\text{Morse}\!(P,[0,1])\times C^{\infty}\!\left(P,\partial^{0}M\right)$ is open and dense as well. Since the space of embeddings is also open there exists an embedding, such that the projection onto the second component is a Morse function, such that the embedding is arbitrarily close to the inclusion from $P$ into $\partial^{0}M$. Since the space of embeddings is locally path-connected (see Theorem 44.1 of [KM97]), we conclude that $P$ is isotopic to a subsurface such that the projection restricted to this subsurface is a Morse function. It is easy to see that we can arrange the critical points to have different values by moving them up or down a little.

For the fourth part of the lemma chose a riemannian metric $g$ on $\partial^{0}M\times[0,1]$ such that $\partial_{t}$ (the directional derivative of the interval coordinate) is orthogonal to $T_{(x,t)}\partial^{0}M\times\{t\}$ for every $(x,t)$ and assume that $\left.{\pi_{[0,1]}}\right\rvert_{P}$ is a Morse function with distinct critical values.

Before we can continue we have to fix some notation: Consider a critical point $x$ of $\left.{\pi_{[0,1]}}\right\rvert_{P}$, which is not a maximum. Consider the negative gradient flow of $\left.{\pi_{[0,1]}}\right\rvert_{P}$ in a neighborhood of $x$. Then there have to be some flow lines $\gamma$ which converge to $x$ meaning that $\lim_{T\to-\infty}{\gamma}\!\left({T}\right)=x$ but since $P$ is compact this gradient flow line has to either meet a boundary component or it is an embedding of $\mathbb{R}$ into $P$ in which case the compactness of $P$ ensures that $\lim_{T\to\infty}{\gamma}\!\left({T}\right)$ exists and has to be a critical value as well. In this case we call $x$ the starting point of $\gamma$ and the other limit point the endpoint of $\gamma$. Furthermore note that if $\gamma$ ends at a point $x$, there also has to be flow line of the gradient flow, which starts at $x$ and goes in the opposite direction as $\gamma$, by which we mean that in a Morse chart centered around $x$ $\gamma$ is given by $e^{1/2T}v$, then the other curve is given by $-e^{1/2T}v$.

Now let us proceed with the proof of the theorem. Let $p$ denote a minimum for $\left.{\pi_{[0,1]}}\right\rvert_{P}$ or in other words a [math]-handle with respect to the handle-decomposition imposed by $\left.{\pi_{[0,1]}}\right\rvert_{P}$. Since every connected component of $P$ meets $\partial^{0}M\times\{0\}$ there has to be a $1$-handle, which cancels the [math]-handle given by $p$. In other words there has to exist a gradient flow line $\gamma_{1}$ for $\left.{\pi_{[0,1]}}\right\rvert_{P}$ starting at $p$, which ends at an index $1$ critical point $q$. Following the negative gradient flow along the opposite direction defined by $\gamma_{1}$ at $q$ gives us a gradient flow line $\gamma_{2}$ that ends at some critical point $p^{\prime}$ which is lower with respect to its $t$-component than $q$. By reparametrizing and including their endpoint and starting point we can consider $\gamma_{1}$ and $\gamma_{2}$ as arcs starting and ending at critical points. We denote by $\gamma$ the concatenation of $\gamma_{1}$ and $\gamma_{2}$. By changing $P$ a little bit but fixing $\left.{\pi_{[0,1]}}\right\rvert_{P}$ we can arrange that there is no open segment of $\gamma$, where $\pi_{\partial^{0}M}S^{1}\gamma$ is constant. Fix a strictly increasing or strictly decreasing function $f\colon[0,1]\to[0,1]$ such that ${f}\!\left({0}\right)={\pi_{[0,1]}}\!\left({p^{\prime}}\right)$ and ${f}\!\left({1}\right)={\pi_{[0,1]}}\!\left({p}\right)$. Then define

[TABLE]

Using the main theorem of [Whi36], which says that embeddings are dense in the space of mappings, we can replace $H^{\prime\prime}$ by a $C^{\infty}$ close map $H^{\prime}$ such that $\bar{H}^{\prime}$ factors through an embedding of a half disc $H^{\prime}$ (this is possible since $H^{\prime\prime}$ is constant on $[0,1]\times\{0\}$ and $[0,1]\times\{1\}$) such that ${D\pi_{[0,1]}\circ H^{\prime}}\!\left({TD^{2}_{+}}\right)=<\partial_{t}>$, ${D\pi_{[0,1]}\circ H^{\prime}}\!\left({T\partial^{0}D^{2}_{+}}\right)=\lambda\partial_{t}$ for $\lambda$ a negative function or $\lambda$ a positive function depending on the height of $p$ and $p^{\prime}$, and that $\left.{H^{\prime}}\right\rvert_{\partial^{1}D^{2}_{+}}=\gamma$. We can further replace $H^{\prime}$ by $H$, using the denseness of transversal maps, such that $H$ still fulfils the conditions above and that ${H}\!\left({D^{2}_{+}}\right)\cap P=\gamma$.

Now the idea is to do surgery along $H$ to cancel pairs of critical points to get rid of minima (In fact we have to change $H$ a little bit for technical reasons, but this doesn’t change the idea). This strategy works since $P$ and $P\natural H$ are isotopic by using a deformation of $H$ to $\gamma$.

We have to consider three and a half cases: First consider the case, where $p^{\prime}$ is also a minimum. Without loss of generality we assume that $p^{\prime}$ is lower than $p$. In this case we replace $\gamma$ by the curve that ends a little bit before $p^{\prime}$ with respect to the flow time and alter $\left.{H}\right\rvert_{\partial^{0}D^{2}_{+}}$ a little bit such that it is tangential to $\gamma$ at ${H}\!\left({\pm 1,0}\right)$. We alter $\gamma$ a bit further such that it starts a little bit before $p$ in the sense that we go a short time in the opposite direction of $\gamma$ at $p$ and proceed in the same fashion as before to produce an $\bar{H}$ with the aforementioned properties. Then $P\natural\bar{H}$ is isotopic to $P$ , $\left.{\pi_{[0,1]}}\right\rvert_{P\natural\bar{H}}$ is still a Morse function with the same critical points as $\left.{\pi_{[0,1]}}\right\rvert_{P}$ except for $p$ and $q$ (this is possible by choosing a very small extension of $\bar{H}$ in the definition of surgery). In this case we have eliminated a minimum and start this process anew with another minimum.

The next case is the case where $p^{\prime}$ is an index $1$ critical point which is lower than $p$. In this case we alter $\gamma$ exactly as in the previous case and then $\left.{\pi_{[0,1]}}\right\rvert_{P\natural H}$ is still a Morse function with the same critical points as $\left.{\pi_{[0,1]}}\right\rvert_{P}$ except for $p$ and $q$. In this case we have again eliminated a minimum without producing any new critical points.

The last full case, where $p^{\prime}$ is an index $1$ critical point, which is higher than $p$, is a little bit different compared to the previous two quite similar cases and we will have to alter $H$ and $\gamma$ a little bit. In this case we let $\gamma$ start not at $p$ but at ${\gamma}\!\left({\epsilon}\right)$ for some small epsilon and choose ${H}\!\left({\partial^{0}D^{2}_{+}}\right)$ to be tangential to $\gamma$ at its starting point. Furthermore we alter $\gamma$ at its endpoint by stopping a short time before arriving at $p^{\prime}$ and then going down with respect to $\pi_{[0,1]}$ avoiding $p^{\prime}$ and points that lie on the gradient flow line which ends at $p^{\prime}$ and stopping at a point which is lower than $p^{\prime}$. As before $\left.{\pi_{[0,1]}}\right\rvert_{P\natural H}$ is a Morse function but the only critical point that could have vanished on $P\natural H$ is $q$ as can be seen by considering the derivative of $H$. Since deleting only one critical point changes the Euler characteristic of $P$, we know that this procedure has to create an index $1$ critical point somewhere, but the only place where this can occur is in a small neighborhood of ${H}\!\left({-1,0}\right)$ because at the other boundary point of $\gamma$ we can arrange $\left.{H}\right\rvert_{\partial^{0}D^{2}_{+}}$ to be tangential to $\gamma$. Another way to see this, is to note that $\left.{H}\right\rvert_{\partial^{0}D^{2}_{+}}$ becomes part of a new gradient flow line of $\left.{\pi_{[0,1]}}\right\rvert_{P\natural H}$, which has to flow down to $p$ after passing a neighborhood of ${H}\!\left({-1,0}\right)$, where it has to flow up. Because every point close to ${H}\!\left({-1,0}\right)$ is a point that can flow down along ${\text{grad}}\!\left({\left.{\pi_{[0,1]}}\right\rvert_{P\natural H}}\right)$ to a critical point that is not $p$, we conclude that the gradient flow line $\left.{H}\right\rvert_{\partial^{0}D^{2}_{+}}$ left the open unstable manifold of $p$. This procedure moves the peak of $\gamma$ below $p^{\prime}$ so that $p^{\prime}$ does not play a role anymore in the canceling of $p$.

All in all these considerations imply that we can proceed again as before but we will not run into $p^{\prime}$ again essentially reducing the number of critical points that come into question for this construction.

The last ”half” case is the case, where $\gamma$ does not run into a critical point after passing the index $1$-critical point, but rather runs into the boundary $P\cap\partial^{0}M\times\{0\}$. This case is handled completely analogous to the first two cases. ∎

This lemma allows us to split every stabilization map into a composition of maps which are given by taking the union with some $P_{i}$. The third and fourth condition limit the topology of $P_{i}$ to be either a pair of pants with incoming boundary consisting of one or two circles (a single index $1$ critical point) and corresponding outgoing boundary or a disk with incoming boundary a circle (a single index $2$ critical point). Accordingly we will write

[TABLE]

for the corresponding stabilization maps i.e. $\alpha_{g,b}$ corresponds to a map which is given by stabilization with a pair of pants with incoming boundary two circles, $\beta_{g,b}$ is a map which is given by stabilization with a pair of pants with incoming boundary a single circle and $\gamma_{g,b}$ is given by taking the union with a disk. We will say that such a map is a stabilization map of type $\alpha$,$\beta$ or $\gamma$ respectively. Even though the homotopy class of these stabilization maps depend on the isotopy type of the boundary bordism, we will suppress $P$ from the notation. We will call $P$ the corresponding boundary bordism and whenever we need it, we will clarify that it corresponds to some fixed stabilization map. In summary, one has:

Corollary 3.4**.**

Let $M$ be a manifold with a space of $\theta$-structures of subplanes of $TM$. Then every stabilization map can be written as a composition of maps of type $\alpha$, $\beta$ and $\gamma$.

3.2. Subset Bordisms

Suppose $M^{\prime\prime}$ and $M^{\prime}$ are closed (as a subset) codimension-[math]-submanifolds, possibly with corners, of $M$ such that $M^{\prime}\subset M^{\prime\prime}$. Here we do not require that the boundaries of these submanifolds lie in the boundary of $M$. We deliberately do not go into detail about manifolds with corners as in the application later on the situation will be more transparent. Suppose we have a $2$-dimensional submanifold with $\theta$-strucutre $U\subset\overline{M^{\prime\prime}\setminus M^{\prime}}$ , then we have the following map:

[TABLE]

Here $\delta^{T}$ is a boundary condition that contains (but does not necessarily equal!) $U\cap\partial M^{\prime}$ and $g^{\prime}$ and $b^{\prime}$ depend on the topology of $U$. We call $U$ a subset bordism.

Remark 3.5**.**

Note that if $\overline{M^{\prime\prime}\setminus M^{\prime}}$ is homeomorphic to a ball and either $M^{\prime\prime}$ or $M^{\prime}$ has no corners, then one can find homeomorphisms that ”smooth the corners” $\Phi^{\prime}\colon M^{\prime}\to\tilde{M}^{\prime}$ and $\Phi^{\prime\prime}\colon M^{\prime\prime}\to\tilde{M}^{\prime 1}$, where $\tilde{M}^{\prime}$ does not have corners such that

[TABLE]

where $\Phi^{\prime}_{\ast}$ and $\Phi^{\prime\prime}_{\ast}$ are induced by the aforementioned homeomorphisms and such that $\varphi$ is a stabilization map and the diagram commutes up to homotopy. In other words, in this case $-\cup U$ behaves up to homotopy like a stabilization map. In particular their homological properties (being an isomorphism or epimorphism etc.) agree.

Examples of subset bordisms will be the approximate augmentations, which will be introduced in Section 5.

4. Resolutions via Semi-Simplicial Spaces

The following section is based entirely on the fourth section of [CMRW17] and all its statements and proofs or references to proofs can be found there. Let $\Delta^{op}_{inj}$ denote the category, whose objects are non-empty finite ordinals and whose morphisms are injective order preserving maps. A semi-simplicial space is a contravariant functor $X_{\bullet}\colon\Delta^{op}_{inj}\to\textbf{Top}$, and $X_{i}$ will denote the image of $\left[i\right]\coloneqq\left\{0,\ldots,i\right\}$. We denote by $\partial_{j,i}$ the face maps that stem from the inclusion $\left\{0,\ldots,i-1\right\}\to\left\{0,\ldots i\right\}$ that misses $j\in\left\{0,\ldots i\right\}$.

A semi-simplicial space $X_{\bullet}$ together with a continuous map $\epsilon_{X_{0}}\colon X_{0}\to X$ is called a semi-simplicial space augmented over $X$ if

[TABLE]

Using the face maps and (2) one obtains one map for every level of the semi-simplicial space, the $i$-th augmentation map, $X_{i}\to X$, that we will denote by $\epsilon_{X_{i}}$. We will usually write $\epsilon_{X_{\bullet}}\colon X_{\bullet}\to X$ for the collection of all augmentation maps and call this the augmentation. If $X_{\bullet}$ and $X_{\bullet}^{\prime}$ denote two possibly augmented semi-simplicial spaces, we call a natural transformation between the functors a semi-simplicial map, if the semi-simplicial spaces are augmented we furthermore require the maps given by the natural transformation to commute with the augmentations. In that case we will denote the map by $g_{\bullet}$, the maps on the levels of the simplicial set by $g_{i}$ and the map between the topological spaces by $g$.

There is a geometric realization functor (compare [ERW19])

[TABLE]

and we call a semi-simplicial space augmented over $X$ an $n$-resolution if the induced map between the geometric realization of the semi-simplicial space and $X$ is $n$-connected. If the induced map is a weak equivalence, we will call the augmented semi-simplicial space a resolution of $X$.

Furthermore we call an augmented semi-simplicial space $\epsilon_{X_{\bullet}}\colon X_{\bullet}\to X$ an augmented topological flag complex if

1. (i)

the product map $X_{i}\to X_{0}\times_{X}\ldots\times_{X}X_{0}$, given by the product of the face maps, is an open embedding 2. (ii)

a tuple $\left(x_{0},\ldots,x_{i}\right)\in X_{0}\times_{X}\ldots\times_{X}X_{0}$ is in $X_{i}$ if and only if for each $0\leq j<j^{\prime}\leq i$ we have $\left(x_{j},x_{j}^{\prime}\right)\in X_{0}\times_{X}X_{0}$ lies in $X_{1}$.

The following three lemmas will be crucial for the proofs of the following sections and can be found in Section 4 of [CMRW17].

Lemma 4.1**.**

Let $\epsilon_{X_{\bullet}}\colon X_{\bullet}\to X$ be an augmented topological flag complex. Suppose that

1. (i)

$\epsilon_{X_{0}}\colon X_{0}\to X$* has local sections that is $\epsilon_{X_{0}}$ is surjective and for each $x_{0}\in X_{0}$ such that ${\epsilon_{X_{0}}}\!\left({x_{0}}\right)=x\in X$ there is a neighbourhood $U_{x}$ of $x$ and a map $s\colon U_{x}\to X_{0}$ such that ${\epsilon_{X_{0}}\circ s}\!\left({x^{\prime}}\right)=x^{\prime}$ for all $x^{\prime}\in U_{x}$ and ${s}\!\left({x}\right)=x_{0}$.* 2. (ii)

given any finite collection $\left\{x_{0}^{1},\ldots,x_{0}^{n}\right\}\subset X_{0}$ in a single fiber of $\epsilon_{X_{0}}$ over some $x\in X$, there is an $x_{0}^{\infty}\in\mathrm{Fib}_{x}\!\left(\epsilon_{X_{0}}\right)$ such that each $\left(x_{0}^{1},x_{0}^{\infty}\right)$ is a $1$-simplex

then $\left|{\epsilon_{X_{\bullet}}}\right|\colon\left|{X_{\bullet}}\right|\to X$ is a weak equivalence.

Lemma 4.2**.**

Let $\epsilon_{X_{\bullet}}\colon X_{\bullet}\to X$ denote an augmented semi-simplicial space. If each $\epsilon_{X_{i}}\colon X_{i}\to X$ is a fibration and $\mathrm{Fib}_{x}\!\left(\epsilon_{X_{i}}\right)$ denotes its fiber at $x\in X$, then the realization of the semi-simplicial space $\mathrm{Fib}_{x}\!\left(\epsilon_{X_{i}}\right)$ is weakly homotopy equivalent to the homotopy fiber of $\left|{\epsilon_{X_{\bullet}}}\right|$ at $x$.

Definition 4.3**.**

If $g\colon X_{1}\to X_{2}$ denotes a continuous map, we will write $C\!\left(g\right)$ for the pair $\left(M_{g},X_{1}\right)$, where $M_{g}$ denotes the mapping cylinder of $g$.

The following lemma hides an occurring spectral sequence argument in the proof. The proof of the lemma can be found in Criterion 4.4. in [CMRW17] and it is an abstraction of the first part of the proof of Theorem 9.3 in [RW16].

Lemma 4.4**.**

Let $g_{\bullet}\colon X_{\bullet}\to X_{\bullet}^{\prime}$ be a map of augmented semi-simplicial spaces such that $X_{\bullet}\to X$ is an $(n-1)$-resolution and $X_{\bullet}^{\prime}\to X^{\prime}$ is an $n$-resolution.

Suppose there is a sequence of path connected based spaces $(B_{i},b_{i})$ and maps $p_{i}\colon X_{i}^{\prime}\to B_{i}$, and form the map

[TABLE]

induced by the composition with $g_{i}$.

{\mathrm{HoFib}_{b_{i}}\!\left(p_{i}\circ g_{i}\right)}$${\mathrm{HoFib}_{b_{i}}\!\left(p_{i}\right)}$${X_{i}}$${X_{i}^{\prime}}$${B_{i}}$${X}$${X^{\prime}}$$\scriptstyle{g_{i}}$$\scriptstyle{g_{i}}$$\scriptstyle{g}

Suppose that there is a $c\leq n+1$ such that

[TABLE]

Then the map induced in homology by the composition of the inclusion of the fiber and the augmentation map

[TABLE]

is an epimorphism in degrees $q\leq c$.

If in addition $H_{c}\!\left(C\!\left(g_{0}\right)\right)=0$, then $H_{q}\!\left(C\!\left(g\right)\right)=0$ in degrees $q\leq c$.

5. Resolutions of the Space of Subsurfaces

The goal of this section is to establish a resolution of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ and to understand how this resolution behaves with respect to stabilization maps. This will enable us in Section 7 to relate spaces of subsurfaces and homological properties of maps between them to homological properties of maps between spaces of subsurfaces with smaller genera.

5.1. Constructing a Resolution for the Space of Subsurfaces

Recall the following notation introduced in Section 3:

1. (i)

$D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}\middle|x_{2}\geq 0\text{, }\left\lVert{\left(x_{1},x_{2}\right)}\right\rVert\leq 1\right\}$ 2. (ii)

$\partial^{0}D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\partial D^{2}_{+}\middle|x_{2}=0\right\}$ 3. (iii)

$\partial^{1}D^{2}_{+}\coloneqq\left\{\left(x_{1},x_{2}\right)\in\partial D^{2}_{+}\middle|\left\lVert{\left(x_{1},x_{2}\right)=1}\right\rVert\right\}$

From here on forth let us fix an at least $5$-dimensional simply-connected manifold $M$ together with a non-empty codimension [math] submanifold of the boundary denoted by $\partial^{0}M$ and a space of tangential structures $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$. Furthermore let $\delta^{T}$ denote some fixed boundary condition for subsurfaces with tangential structure in $M$. Lastly, let $\ell$ denote a codimension [math] ball in $\partial^{0}M$ that intersects $\delta$ in two intervals $\ell_{0}$ and $\ell_{1}$, which we label and orientate according to the orientation of $\delta$ once and for all.

To define a resolution of the space of subsurfaces, we will need the following definition:

Definition 5.1**.**

Given $W\in{\mathcal{E}^{\theta}}\!\left(M;\delta^{T}\right)$, we call an embedding

[TABLE]

which maps $\partial^{0}D^{2}_{+}$ to $\ell$ and $\left(1,0\right)$ to $\ell_{0}$ and $\left(-1,0\right)$ to $\ell_{1}$ an arc in $W$ with embedded boundary isotopy. We will call a thickening of an arc in $W$ with embedded boundary isotopy $\left(a,\mathbf{a}\right)$ a thickened arc in $W$ with embedded boundary isotopy if the image of $\mathbf{a}$ restricted to the normal bundle of $\partial^{0}D^{2}_{+}$ lies in $\ell$. In this case we will call the image of $\left.{a}\right\rvert_{\partial^{1}D^{2}_{+}}$ the underlying arc of $a$ and the image of of $\mathbf{a}$ restricted to the normal bundle of ${a}\!\left({\partial^{1}D^{2}_{+}}\right)$ the thickened underlying arc of $a$ and we will write $\mathbf{a}_{W}$ for it. For notational reasons we will usually just write $a$ for the thickened arc in $W$ with embedded boundary isotopy even tough the correct notation would include $\mathbf{a}$.

An embedding (without the presence of a subsurface) $a\colon D^{2}_{+}\to M$ such that ${a}\!\left({\partial^{1}D^{2}_{+}}\right)\subset M\setminus\partial M$, ${a}\!\left({\partial^{0}D^{2}_{+}}\right)\subset\ell$ and ${a}\!\left({\left((-1)^{k},0\right)}\right)\in\ell_{k}$ , will be called an arc with embedded boundary isotopy i.e. we drop $W$ from the previous notation. Similarly for a thickened arc with embedded boundary isotopy.

Definition 5.2**.**

Let $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{\bullet}$ denote the following semi-simplicial space: The space of $i$-simplices consists of tuples $\left(W,\left(a^{0},\mathbf{a}^{0}\right),\ldots,\left(a^{i},\mathbf{a}^{i}\right)\right)$ such that:

1. (i)

$W\in{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ is a surface with tangential structure in $M$. 2. (ii)

All the $\left(a^{j},\mathbf{a}^{j}\right)$ are thickened arcs in $W$ with embedded boundary isotopy. 3. (iii)

The images of all $\mathbf{a}^{j}$ are disjoint. 4. (iv)

$W$ without all underlying thickened arcs is connected, i.e. the arc system consisting of the underlying arcs is coconnected. 5. (v)

The starting and endpoints of the underlying arcs are ordered from [math] to $i$ in $\ell_{0}$ and ordered from $i$ to [math] in $\ell_{1}$ (Note that this makes sense as $\ell_{k}$ is oriented). In this case, we will say that the arc system consisting of the underlying arcs is ordered.

The $j$-th face map forgets the $j$-th embedding and we topologize the set of $i$-simplices as a subspace of

[TABLE]

where in $\overline{\operatorname{TEmb}}\!\left((D^{2}_{+})^{i+1},M;\ell\right)$ the boundary condition means just that $\partial^{0}D^{2}_{+}$ maps to $\ell$. This semi-simplicial space possesses an augmentation map $\epsilon_{\bullet}$ to ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ which forgets the thickened arcs with embedded boundary isotopy.

Notation**.**

Sometimes we want to distinguish between the cases, where the intersection of $\delta$ and $\ell$ meets a single connected component of $\delta$ or two different components. To emphasize this we will sometimes write $\mathcal{O}^{\prime\theta}_{g,\delta}\!\left(M,\delta^{T};\ell\right)_{\bullet}$ for the single component case and $\mathcal{O}^{\prime\prime\theta}_{g,\delta}\!\left(M,\delta^{T};\ell\right)_{\bullet}$ for the different components case.

The general proof strategy of the following proposition was communicated to me by Frederico Cantero as a proposed fix to some errors in the proof of Proposition 5.3 in [CMRW17].

Proposition 5.3**.**

Suppose that $M$ is an at least $5$-dimensional and simply-connected manifold, then $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{\bullet}$ is a $\left(g-1\right)$-resolution of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ i.e.

[TABLE]

is $\left(g-1\right)$-connected or in other words the homotopy fiber of this map is $\left(g-2\right)$-connected.

Proof.

Since ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta^{T}\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile and the augmentation maps

[TABLE]

are equivariant with respect to the natural action of ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ via post composition, Lemma 2.8 implies that the augmentation maps are locally trivial fibrations. Using Lemma 4.2 we get that the homotopy fiber $\mathrm{Fib}_{W}\!\left(\left|{\epsilon_{\bullet}}\right|\right)$ is weakly homotopy equivalent to $\left|{\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)}\right|$. The $i$-simplices of $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$ are disjoint, ordered thickened arcs in $W$ with embedded boundary isotopy.

Fix $x_{k}\in\ell_{k}$ and charts $U_{x_{k}}$ centered at $x_{k}$. We say that an embedding $e\colon\left[0,1\right]\to W$ meets $x_{k}$ in a radial fashion if its image in $U_{x_{k}}$ is a straight ray meeting [math].

Let $X\!\left(W;x_{0},x_{1}\right)_{\bullet}$ denote the following semi-simplicial set: Its set of zero simplices consists of embeddings of an arc into $W$ that meets $x_{0}\in\ell_{0}$ and $x_{1}\in\ell_{1}$ in a radial fashion together with a tubular neighborhood of said arc and we furthermore require these arcs to be non-isotopic to a part of the boundary. The set of $i$-simplices is given by collections of [math]-simplices such that the complement of the arcs is connected, and such that the tubular neighborhoods are disjoint except for their intersection with $U_{x_{k}}$, lastly we require that the ordering of the arcs at $x_{k}$ with respect to the angle of the corresponding ray to be order-preserving at $x_{0}$ and order-reversing at $x_{1}$.

There is a map $g\colon\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)\to X\!\left(W,x_{0},x_{1}\right)_{\bullet}$ given by sending an element in the fiber to the underlying thickened arcs and then adding a collar of $W$ to $W$ and then joining the arcs to $x_{k}$ in a controlled way to produce an element in $X\!\left(W,x_{0},x_{1}\right)_{\bullet}$.

Let $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)^{\delta}$ denote the semi-simplicial set (not space) $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$ i.e. the set of $i$-simplices is given by the underlying set of $\mathrm{Fib}_{W}\!\left(\epsilon_{i}\right)$. Now we want to apply Theorem A.7 of [Kup13], which says that, if

1. (i)

$X\!\left(W,x_{0},x_{1}\right)_{\bullet}$ is weakly Cohen-Macauley of dimension $(g-1)$ i.e. it is $(g-2)$-connected and the link of every $p$-simplex is $(g-2-p-2)$-connected. 2. (ii)

$\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$ is Hausdorff and an ordered flag space (See [Kup13] for the definition of ordered flag space) 3. (iii)

$\left|{g}\right|\colon\left|{\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)^{\delta}}\right|\to X\!\left(W,x_{0},x_{1}\right)_{\bullet}$ is simplexwise injective i.e. for every $p$-simplex $\left\{a_{0},\ldots,a_{p}\right\}$ with $p\geq 1$ we have ${g}\!\left({a_{i}}\right)\neq{g}\!\left({a_{j}}\right)$ for all $i\neq j$. 4. (iv)

For all finite collections $\left\{a_{1},\ldots,a_{j}\right\}\subset\mathrm{Fib}_{W}\!\left(\epsilon_{0}\right)$ and $e_{0}\in X\!\left(W,x_{0},x_{1}\right)_{\bullet}$ such that $\left(e_{0},{g}\!\left({a_{i}}\right)\right)$ is a $1$-simplex in $X\!\left(W,x_{0},x_{1}\right)_{\bullet}$ then there exists an element $a_{0}\in\mathrm{Fib}_{W}\!\left(\epsilon_{0}\right)$ such that ${g}\!\left({a_{0}}\right)=e_{0}$ and such that $\left(a_{0},a_{i}\right)$ is a $1$-simplex in $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$.

then $\left|{\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)}\right|$ is $(g-2)$-connected.

It is an easy observation that the second and third condition hold. For the first condition, note that Theorem 2.9 in [Nar15] proves that $\left|{X\!\left(W,x_{0},x_{1}\right)_{\bullet}}\right|$ is $(g-2)$-connected, so we only have to prove that the link of a $p$-simplex is $(g-2-p-2)$-connected. Given a $p$-simplex $\sigma_{p}$, the link of this simplex consists of all arc systems whose union with the arc system of $\sigma_{p}$ is still a coconnected arc system. Consider the closed complement of the thickened arcs in $\sigma_{p}$ denoted by ${W}\!\left({\sigma_{p}}\right)$ (where this means, that we take the complement of the tubular neighborhoods, then take the closure in $W$ and then resolve the singularities at $x_{k}$). Evidently every simplex in the link of $\sigma_{p}$ consists of arcs that lie in ${W}\!\left({\sigma_{p}}\right)$. Furthermore the points $x_{k}$ correspond to $p+2$ points in ${W}\!\left({\sigma_{p}}\right)$ and each of these points corresponds to an interval between two consecutive arcs in $\sigma_{p}$ in the ordering of arcs connecting the points $x_{k}$. We will denote the subspace of the link of $\sigma_{p}$ consisting of those arcs that are bigger than all arcs in $\sigma_{p}$ in the ordering of the arcs connecting $x_{k}$ by $Y_{\bullet}$. Unbending corners, $Y_{\bullet}$ is isomorphic to $X\!\left({W}\!\left({\sigma_{p}}\right),x^{\prime}_{0},x^{\prime}_{1}\right)_{\bullet}$ which is at least $(g-p-3)$-connected by Theorem 2.9 in [Nar15] and a simple Euler characteristic computation.

We will show, using the techniques of Section 2.1 (i.e. bad and good simplices) in [HV17], that the link has the same connectivity as its subspace $Y_{\bullet}$. We call a simplex in the link of $\sigma_{p}$ bad if all its arcs are smaller than some arc in $\sigma_{p}$. These certainly satisfy the conditions for a set of bad simplices in [HV17]. Furthermore given a bad simplex $\sigma_{k}^{\prime}$, then $G_{\sigma_{k}^{\prime}}$ is given by $X\!\left({W}\!\left({\sigma_{p}\cup\sigma_{k}^{\prime}}\right),x^{\prime}_{0},x^{\prime}_{1}\right)_{\bullet}$, which is $(g-p-3-k-1)$-connected. Thus by (b) of Corollary 2.2 in [HV17] the link is at least $(g-2-p-1)$-connected.

For the fourth condition, note that since $\left(e_{0},{g}\!\left({a_{i}}\right)\right)$ is a $1$-simplex, in particular $e_{0}$ and the underlying arc of ${g}\!\left({a_{i}}\right)$ are coconnected and ordered correctly, we can find a thickened embedded arc $\bar{a}_{0}$ in $W$ such that ${g}\!\left({\bar{a}_{0}}\right)=e_{0}$ (this expression makes sense, since $g$ only takes the underlying thickened arcs of the elements in $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$) and such that the pairs consisting of $\bar{a}_{0}$ and the thickened underlying arcs of $a_{i}$ form an ordered and coconnected arc system. Using that $M$ is simply-connected and the main result of [Hae61] we can find an extension of $\bar{a}_{0}$ to an arc in $M$ with embedded boundary isotopy. Since the dimension of $M$ is at least $5$ we conclude that a small perturbation of this embedding yields an embedding disjoint from all the other $a_{i}$ by transversality. Adding a sufficently small tubular neighborhood finishes the proof of the fourth assumption and therefore implies the desired claim about the connectivity of $\mathrm{Fib}_{W}\!\left(\epsilon_{\bullet}\right)$.

To finish the proof, just observe that the following diagram is a pullback diagram, where the lower map denotes the map forgetting the tangential structure.

{\left|{\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{\bullet}}\right|}$${\left|{\mathcal{O}^{+}_{g,b}\!\left(M,\delta^{T};\ell\right)_{\bullet}}\right|}$${{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)}$${{\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)}

∎

We will need to establish some more notation.

Definition 5.4**.**

A thickened strip with embedded boundary isotopy consists of the following data:

1. (i)

A thickened arc with embedded boundary isotopy $a\colon D^{2}_{+}\to M$ 2. (ii)

A subbundle $L$ of $N\!\left({a}\!\left({\partial^{1}D^{2}_{+}}\right),M\right)$

such that the restriction $\left.{\mathbf{a}}\right\rvert_{\left.{L}\right\rvert_{(1,0)}}$ lies in $\ell_{0}$ and $\left.{\mathbf{a}}\right\rvert_{\left.{L}\right\rvert_{(-1,0)}}$ lies in $\ell_{1}$. We will denote the restriction of $\mathbf{a}$ to $L$ by $\mathbf{a}_{L}$. If we add a tangential structure for the image of $\mathbf{a}_{L}$, which agrees with the tangential structure specified by $\delta^{T}$, wherever this makes sense, we will call this a thickened strip with tangential structure and embedded boundary isotopy. We will call the image of $\mathbf{a}_{L}$ the strip of $(a,L)$.

Notation**.**

Similar as before we will usually suppress $\mathbf{a}$ and $\mathbf{a}_{L}$ from this notation and only write $a$ for the tuple $(a,\mathbf{a},L)$.

Definition 5.5**.**

Let $A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ denote the set of tuples $\left(a^{0},\ldots,a^{i}\right)$ such that all the $a^{k}$ are thickened strips with tangential structures and embedded boundary isotopies such that all the images of $\mathbf{a}^{k}$ are disjoint and the starting and endpoints of the underlying arcs lie in $\ell_{0}$ and $\ell_{1}$ and they are ordered from [math] to $i$ in $\ell_{0}$ and ordered from $i$ to [math] in $\ell_{1}$.

We topologize this as a subset of $\overline{\operatorname{TEmb}}\!\left(D^{2}_{+}\times\left[i\right],M;\ell\right)\times\operatorname{Emb}^{\theta}\!\left(I\times I\times\left[i\right],M\right)$, where the thickened arcs with embedded boundary isotopies correspond to elements in the first factor and the strips and their tangential structure to the second factor.

There is a continuous map $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}\to A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ which forgets the surface but keeps the tubular neighbourhood in the surface and the tangential structure on it.

Lemma 5.6**.**

The restriction map $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}\to A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ is a Serre fibration and the fiber over a point $a=\left(a_{0},\ldots,a_{i}\right)$ can be identified with ${\mathcal{E}^{\theta}_{g-i-1,b+i+1}}\!\left({M}\!\left({a}\right);{\delta}\!\left({a}\right)^{T}\right)$ if we are considering $\mathcal{O}^{\prime\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$. If we are considering $\mathcal{O}^{\prime\prime\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ instead, then the fiber is given by ${\mathcal{E}^{\theta}_{g-i,b+i-1}}\!\left({M}\!\left({a}\right);{\delta}\!\left({a}\right)^{T}\right)$.

Here we define ${M}\!\left({a}\right)=\overline{M\setminus\cup_{j}\mathbf{a}_{j}}$ and the boundary condition ${\delta}\!\left({a}\right)^{T}$ is given by the boundary of the image of the strips together with the intersection of the boundary condition $\delta$ and the submanifold ${M}\!\left({a}\right)$.

Remark 5.7**.**

Note that ${M}\!\left({a}\right)$ is a manifold with corners and ${\delta}\!\left({a}\right)^{T}$ is a boundary condition for a manifold with corners, but we can circumvent this by fixing a homeomorphism from ${M}\!\left({a}\right)$ to $M$ that is a diffeomorphism at all points except the corner points. Heuristically such a homeomorphisms pushes the dent which came from removing $a$ to the outside.

Note that the boundary condition ${\delta}\!\left({a}\right)^{T}$ maps the corners of ${\Sigma_{g,\delta}}\!\left({a}\right)$, i.e. the surface with the corresponding arcs removed, to the corners of ${M}\!\left({a}\right)$. Therefore postcomposing a subsurface with the aforementioned homeomorphism yields an embedded subsurface without corners in a manifold without corners. Using this we can treat the occurring spaces of subsurfaces with corners of manifolds with corners up to homeomorphism like ordinary spaces of subsurfaces.

Proof.

We will construct the following diagram and prove that the maps labelled fib are fibrations to prove the claim:

[TABLE]

Fix a set $\sigma$ of $i+1$ disjoint arcs in $\Sigma_{g,b}$ starting at ${d^{-1}}\!\left({\ell_{0}}\right)$ and ending at ${d^{-1}}\!\left({\ell_{1}}\right)$, and fulfilling the ordering condition of Definition 5.2, and such that their complement is connected. Here $d$ is a parametrization of $\delta$. Then, the top map is the restriction map, which was proven to be a fibration in Lemma 2.18. The right vertical map forgets the boundary isotopy. Note that this is a fibration as the following diagram is a pullback and the right hand map is a fibration by Lemma 2.8:

[TABLE]

The definition of the space $O^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ is as follows: We define $\overline{\mathcal{O}}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ just like the arc resolution, but without the boundary isotopies (i.e. subsurfaces with arc systems). We define $O^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ via the following diagram in which the space in the top left corner of both squares are defined as pullbacks:

[TABLE]

Note that ${\operatorname{Diff}^{+}_{\sigma}}\!\left(\Sigma_{g,b}\right)$ the group of diffeomorphisms that fix the boundary and $\sigma$ acts on the space of embeddings in the diagram and the left most map is the quotient map associated to this action. It is a fibration by Lemma 2.8 and the fact that $\overline{\mathcal{O}}^{+}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile. Since the left vertical map is a fibration, all vertical maps in the diagram are fibrations. Furthermore the top square in (3) is also easily seen to be a pullback diagram. Hence all maps in that square are fibrations. Since all maps in (3), except for the diagonal one, are fibrations Lemma 2.13 implies that the diagonal map is a fibration as well.

Lastly we have to determine the fiber over $a$. Note that removing $i+1$ strips from a surface increases its Euler characteristic by $i+1$, since it corresponds to taking out $i+1$ one-cells. To calculate the genus and the number of boundary components of the fiber it is enough to specify its number of boundary components. We will take out the $i+1$ strips consecutively and there are two cases we have to distinguish: Either all boundary points lie in the same connected component of the boundary of the surface (Case 1) or they all lie in different connected components (Case 2). But note that removing the first arc in Case 2, reduces the calculation to Case 1 for the rest of the arcs. But in the first case we see that taking out an arc increases the number of boundary components by one and the requirement for the ordering of the arcs ensures that the consecutive arcs all connect the same connected component of the boundary.

All in all we conclude that in Case 2 the number of boundary components changes to $b+i-1$ and in the first case it changes to $b+i+1$. Using the Euler-characteristic formula for surfaces, we can compute the corresponding genus to get the above specifications of the fiber. ∎

Definition 5.8**.**

For some $\left(a^{0},\ldots,a^{i}\right)\in A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ we call the composition of the inclusion of the fiber in Lemma 5.6 with the augmentation of $\epsilon_{i}\colon\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}\to{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ the $i$-approximate augmentation of the resolution $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ over the $i$-simplex $\left(a^{0},\ldots,a^{i}\right)$.

5.2. Stabilizing the arc resolution

We want to extend the maps $\alpha_{g,b}$ and $\beta_{g,b}$ to the aforementioned resolution as shown below:

[TABLE]

and similarly for $\beta_{g,b}$. Let $P$ denote a subsurface in $\partial^{0}M\times I$ used in the definition of a stabilization map. We define $\bar{\ell}=\ell\times\{1\}$ and $\bar{\ell_{i}}=\ell_{i}\times\{1\}$ for $i\in\{0,1\}$ and we assume without loss of generality that $P\cap\left(\ell\times I\right)=\left(\ell\cap\delta\right)\times I$, in particular $\bar{\ell}\cap\bar{\delta}=\left(\ell\cap\delta\right)\times\{1\}$, where $\bar{\delta}^{T}$ denotes the boundary condition of the image of the stabilization map. (Here we isotope $P$ not relative the boundary to get that $\ell\times I$ is contained in $P$).

Define $\tilde{a}$ for $a\in A^{\theta}\!\left(M;\delta^{T},\ell\right)_{0}$ as ${a}\!\left({\partial^{0}D^{2}_{+}}\right)\times I$. This allows us to extend the stabilization maps to $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}$ as follows

[TABLE]

and we write $\bar{a}$ for $a\cup\tilde{a}$. This yields the dashed lifts. Since they commute with the face maps and with the augmentation maps we get a map of augmented semi-simplicial spaces

[TABLE]

which is augmented over $\alpha_{g,b}$. If $\bar{\ell}$ intersects two different components of the new boundary condition, then we obtain analogously

[TABLE]

augmented over $\beta_{g,b}$ .

All of these considerations imply the following corollary:

Corollary 5.9**.**

The semi-simplicial pair $C\!\left(\alpha_{g,b}^{\bullet}\right)$ together with the natural augmentation map to $C\!\left(\alpha_{g,b}\right)$ is a $g$-resolution i.e. the map between pairs is $g$-connected.

The semi-simplicial pair $C\!\left(\beta_{g,b}^{\bullet}\right)$ together with the natural augmentation map to $C\!\left(\beta_{g,b}\right)$ is a $(g-1)$-resolution i.e. the map between pairs is $(g-1)$-connected.

There is a commutative square

[TABLE]

using collars one sees that the lower map is a homotopy equivalence. By commutativity of the above square we get a map between the fibers over the points $a$ and $\bar{a}$

[TABLE]

If $P$ denotes the bordism defining the map $\alpha_{g,b}$ in question, then this map is given by taking the union with ${P}\!\left({a}\right)\coloneqq\overline{P\setminus\cup_{i}a_{i}}$. These are again manifolds with corners, but this is quite easily fixed using similar techniques as in Remark 3.5 and Remark 5.7.

This map between the fibers is quite easily seen to be of the form $\beta_{g-i,b+i-1}$. As the map $A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}\to A^{\theta}\!\left(M^{1};\bar{\delta}^{T},\bar{\ell}\right)_{i}$ is a homotopy equivalence, we conclude that the space of subsurfaces ${\mathcal{E}^{\theta}_{g-i,b+i-1}}\!\left({M}\!\left({a}\right);{\delta^{T}}\!\left({a}\right)\right)$ is homotopy equivalent to the homotopy fiber of the composition of the map $\mathcal{O}^{\theta}_{g,b}\!\left(M,\delta^{T};\ell\right)_{i}\to A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ with the aforementioned homotopy equivalence. Moreover we have shown that the map between the fibers is a stabilization map of the form $\beta_{g-i,b+i-1}$. This is expressed in the following diagram:

[TABLE]

Repeating the same procedure for maps of the form $\beta_{g,b}$, we obtain the following corollary:

Corollary 5.10**.**

The induced map between the homotopy fibers of

[TABLE]

is given by $\beta_{g-i,b+i-1}$ and analogously the induced map between the homotopy fibers of

[TABLE]

is given by $\alpha_{g-i-1,b+i+1}$.

Definition 5.11**.**

We call the map $C\!\left(\beta_{g-i,b+i-1}\right)\to C\!\left(\alpha_{g,b}\right)$ given by the composition of the inclusion of the fiber in the previous corollary into $C\!\left(\alpha_{g,b}^{i}\right)$ with the projection onto $C\!\left(\alpha_{g,b}\right)$ the relative $i$-approximate augmentation of the resolution $\left(\alpha_{g,b}^{\bullet}\right)$. Analogously we call $C\!\left(\alpha_{g-i-1,b+i+1}\right)\to C\!\left(\beta_{g,b}\right)$ defined as in the previous case the relative $i$-approximate augmentation of the resolution $\left(\beta_{g,b}^{\bullet}\right)$. If we want to emphasize the point $a\in A^{\theta}\!\left(M;\delta^{T},\ell\right)_{i}$ over which the fiber lives, then we call it the relative $i$-approximate augmentation over $a$.

6. $k$-Triviality and $\pi_{0}$-stabilization

The following notions are inspired by the corresponding definitions in Section 6 of [RW16] and will serve as the main requirements of a tangential structure in order to fulfil homological stability. The following definition is an adaptation of Definition 6.1 in [RW16].

Definition 6.1**.**

We say a subset bordism $U\subset\overline{M\setminus M^{\prime}}$ absorbs a boundary bordism $P\subset\partial^{0}M$ if $U\cap\partial^{0}M\times[0,1]\subset P$ and there exists a $Z\subset\left(M\setminus M^{\prime}\right)^{1}\cup\partial^{0}M\times[1,2]$ , where $\partial^{0}M^{\prime}=\partial^{0}M\cap M^{\prime}$ (note that $M^{\prime 1}$ can be embedded into $M^{2}$ and $Z$ is a subset of the difference of these two) such that

[TABLE]

the following diagram commutes up to homotopy, where the lower maps labeled by $\simeq$ are given gluing on a cylinder.

Remark 6.2**.**

While this seems hard to check it is actually just a question about the path components of the occuring bordism spaces that define the spaces. This will be further emphasized in the proof of Proposition 6.9.

The following definition is the analogue of the first part of Definition 6.2 in [RW16] of test pairs of height $k$:

Definition 6.3**.**

Given a boundary condition $\delta$, a sequence of pairs of disjoint embedded intervals $\ell_{0}^{i}\subset\delta$ and $\ell_{1}^{i}\subset\delta$ for $1\leq i\leq k$, one can glue [math]-handles along $\ell_{0}^{i}$ and $\ell_{1}^{i}$ to $\delta$. Doing this for the first $i-1$ pairs of intervals yields an (abstract) $1$-manifold with $k-i$ embedded pairs of intervals which we denote by $\delta^{i}$.

We call such a sequence together with two more intervals $\bar{\ell_{0}}$ and $\bar{\ell_{1}}$ genus maximizing of length $k$ for maps of type $\alpha$ and boundary condition $\delta$ if:

1. (i)

All occurring embedded intervals are disjoint and $\ell_{0}^{i}\subset\bar{\ell_{0}}$ and $\ell_{1}^{i}\subset\bar{\ell_{1}}$ 2. (ii)

$\ell_{0}^{1}$ and $\ell_{1}^{1}$ lie in different connected components of $\delta$ 3. (iii)

$\ell_{0}^{i}$ and $\ell_{1}^{i}$ lie in the same connected component of $\delta^{i}$ if $i$ is even and in different components if $i$ is odd.

Such a sequence is genus maximizing of length $k$ for maps of type $\beta$ and boundary condition $\delta$ if:

1. (i)

All occurring embedded intervals are disjoint and $\ell_{0}^{i}\subset\bar{\ell_{0}}$ and $\ell_{1}^{i}\subset\bar{\ell_{1}}$ 2. (ii)

$\ell_{0}^{1}$ and $\ell_{1}^{1}$ lie in the same connected components of $\delta$ 3. (iii)

$\ell_{0}^{i}$ and $\ell_{1}^{i}$ lie in the same connected component of $\delta^{i}$ if $i$ is odd and in different components if $i$ is even.

Furthermore given a genus maximizing sequence of length $k$, we call a sequence of codimension [math] balls $\ell^{i}$ all contained in a ball $\overline{\ell}$ such that $\ell^{i}\cap\delta=\ell_{0}^{i}\cup\ell_{1}^{i}$ and $\overline{\ell}\cap\delta=\overline{\ell_{0}}\cup\overline{\ell_{1}}$ together with a sequence of $a^{i}\in A^{\theta}\!\left({M}\!\left({a^{1},\ldots,a^{i-1}}\right);(\delta^{i-1})^{T},\ell^{i}\right)_{0}$ for $1\leq i\leq k$ compatible with $(\ell_{0}^{i},\ell_{1}^{i})$. We will usually suppress $\ell$ from this notation. In this case $\delta^{i}$ is naturally a boundary condition and not just an abstract manifold.

Remark 6.4**.**

Note that it is always possible to chose $\overline{\ell}$ and $\ell^{i}$ which intersect $\delta$ as required. If $\partial M$ is simply-connected, then this ball is unique up to isotopy.

We will be interested in tangential structures for which certain sequences of approximate augmentations of the resolutions introduced in Section 5 absorb stabilization maps of type $\alpha$ and $\beta$.

Definition 6.5**.**

Given a genus maximizing sequence of length $k$ with compatible $a^{i}$, we call the composition of all the [math]-th approximate augmentations corresponding to the strips $a^{i}\in A^{\theta}\!\left({M}\!\left({\bigcup_{j\in\{1,\ldots,i\}}a^{j}}\right);(\delta^{i})^{T},\ell^{i}\right)_{0}$ the associated augmentation composition of length $k$ and denote it by

[TABLE]

where

[TABLE]

Note that the augmentation compositions are subset bordisms.

If we have a stabilization map defined via $P$, where the defining boundary bordism $P$ fulfils $P\cap\left(\left(\bar{\ell_{0}}\cup\bar{\ell_{1}}\right)\times[0,1]\right)=\left(\bar{\ell_{0}}\cup\bar{\ell_{1}}\right)\times[0,1]$, in this case we call $-\cup P$ compatible with the genus maximizing sequence. Then $-\cup P$ lifts to a map

[TABLE]

where $g^{\prime}$ and $b^{\prime}$ are as before, and if $P$ is of type $\alpha$ or $\beta$ and the sequence is genus maximizing for maps of same type, then $-\cup{P}\!\left({\bigcup_{j\in\{1,\ldots,k\}}a^{j}}\right)$ is a map of the same type as before if $k$ is even, and of the other type if $k$ is odd. We call this map a ** relative augmentation compositions of length $k$ ** and denote it by $\mathfrak{a}^{k}_{g,b}$ if $-\cup P$ is of type alpha, and $\mathfrak{b}^{k}_{g,b}$ in the other case.

Remark 6.6**.**

Note that by the dimension restriction on $M$, and $\partial M$ being simply-connected, we can find an isotopy that changes a $P$ into one that is compatible with the genus maximizing sequence.

Again this definition is inspired by the second part of Definition 6.2 in [RW16].

Definition 6.7**.**

We call a space of $\theta$-structures $k$-trivial if for every genus maximizing sequence of length $k$ and a compatible sequence $a^{i}$, and every compatible stabilization map of type $\alpha$ or $\beta$, the boundary bordism $P$ is absorbed by the subset bordism defining the relative augmentation compositions of length $k$ associated to the genus maximizing sequence.

In Section 9 we will prove that it suffices to have stabilisation of connected components in order to obtain $k$-triviality for some $k$. To state this we need the following definition, which is the analogue of Definition 6.3 in [RW16]:

Definition 6.8**.**

We say a tangential structure $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ $\pi_{0}$-stabilizes at genus $g$ if all stabilization maps

[TABLE]

induce bijections on $\pi_{0}$ for all $g^{\prime}\geq g$ and they induce a surjection on $\pi_{0}$ for all $g^{\prime}\geq g-1$.

We say a tangential structure $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ $\pi_{0}$-stabilizes at the boundary at genus $g$ if all stabilization maps:

[TABLE]

induce bijections on $\pi_{0}$ for all $g^{\prime}\geq g$ and they induce a surjection on $\pi_{0}$ for all $g^{\prime}\geq g-1$.

The rather abstract property of $k$-triviality is already implied by $\pi_{0}$-stabilization. This is encapsulated in the following proposition, which will be proven in Section 9 and is an adaptation of Proposition 6.5 in [RW16].

Proposition 6.9**.**

Suppose $M$ is an at least $5$-dimensional manifold which is simply connected and $\partial^{0}M$ is a simply-connected codimension [math] submanifold of $\partial M$. Suppose further that $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ $\pi_{0}$-stabilizes at the boundary at $g$, then it is $2g+1$-trivial.

7. Proving Homological Stability

In order to prove homological stability we will need the stability bounds. The following definition stems from Definition 6.9 in [RW16].

Definition 7.1**.**

Given two natural numbers $k\geq 1$ and $m\geq 0$, we define the following recursive functions $A,B,SA,SB\colon\mathbb{Z}\to\mathbb{Z}$ as follows:

1. (i)

[TABLE]

for $g\leq m-2$ and

[TABLE] 2. (ii)

We define

[TABLE] 3. (iii)

If $k=1$, we define:

[TABLE] 4. (iv)

If $k=2l$ with $l>0$, we define:

[TABLE] 5. (v)

If $k=2l+1$ with $l>0$, we define:

[TABLE]

Remark 7.2**.**

If $k=1$ and $m=0$, then ${A}\!\left({g}\right)={SA}\!\left({g}\right)=\left\lfloor\frac{1}{3}(2g+1)\right\rfloor$ and ${B}\!\left({g}\right)={SB}\!\left({g}\right)=\left\lfloor\frac{2}{3}g\right\rfloor$ gives the aforementioned functions.

The goal of the rest of this section is to prove the most of the first two parts of the following theorem. The proof will be an adaptation of the proof of the main theorem of [CMRW17] to include $k$-triviality:

Theorem 7.3**.**

Suppose $M$ is an at least $5$-dimensional, simply-connected manifold with non-empty boundary and $\partial^{0}M$ is a codimension [math] submanifold of $\partial M$. Suppose further that $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ is a $k$-trivial space of $\theta$-structures on subplanes of $TM$ which $\pi_{0}$-stabilizes at $m$, then

• •

the homology of $C\!\left(\alpha_{g,b}\right)$ vanishes in degrees $\ast\leq{A}\!\left({g}\right)$

• •

the homology of $C\!\left(\beta_{g,b}\right)$ vanishes in degrees $\ast\leq{B}\!\left({g}\right)$. If one of the newly created boundary components is contractible in $\partial^{0}M$ and the tangential structure on that boundary component vanishes in $\pi_{1}\!\left(B_{2}\!\left(M\right)\right)$, then $\beta_{g,b}$ is a monomorphism in all degrees

• •

the homology of $C\!\left(\gamma_{g,b}\right)$ vanishes in degrees $\ast\leq{B}\!\left({g}\right)$. If $b>1$, then $\gamma_{g,b}$ induces an epimorphism in all degrees

where $A$ and $B$ are defined as in Definition 7.1 and $C\!\left(-\right)$ denotes the mapping pair as in Definition 4.3.

The proof will proceed via induction. More precisely we will use induction for the following two statements:

• •

$A_{h}$: For all $g\leq h$ and for all $\alpha_{g,b}$, the reduced homology of $C\!\left(\alpha_{g,b}\right)$ vanishes in degrees $\ast\leq{A}\!\left({g}\right)$

• •

$B_{h}$: For all $g\leq h$ and for all $\beta_{g,b}$, the reduced homology of $C\!\left(\beta_{g,b}\right)$ vanishes in degrees $\ast\leq{B}\!\left({g}\right)$

Furthermore we will also need the following two auxiliary statements:

• •

$SA_{h}$: For all $g\leq h$ and all strips with embedded boundary isotopy, the relative [math]-approximate augmentation induces a surjection $H_{\ast}\!\left(C\!\left(\beta_{g,b-1}\right)\right)\to H_{\ast}\!\left(C\!\left(\alpha_{g,b}\right)\right)$ in homology in degrees $\ast\leq{SA}\!\left({g}\right)$.

• •

$SB_{h}$: For all $g\leq h$ and and all strip with embedded boundary isotopy, the relative [math]-approximate augmentation induces a surjection $H_{\ast}\!\left(C\!\left(\alpha_{g-1,b+1}\right)\right)\to H_{\ast}\!\left(C\!\left(\beta_{g,b}\right)\right)$ in degrees $\ast\leq{SB}\!\left({g}\right)$.

Note that the start of the induction is provided by the assumption that the tangential structure $\pi_{0}$-stabilizes at genus $m$.

We will also denote the mapping cone of a map $g$ by $\bar{C}\!\left(g\right)$, if $g$ is an inclusion $A\to X$ we will write $\bar{C}\!\left(X,A\right)$ instead. The following technical lemma will be needed in the proof. It is Lemma 7.5 in [CMRW17].

Lemma 7.4**.**

Suppose we have a map of pairs

[TABLE]

such that there exists a map $t\colon X\to A^{\prime}$ making the bottom triangle commute up to a homotopy $H$. Then the induced maps between mapping cones $(f,g)\colon\bar{C}(X,A)\to\bar{C}(X^{\prime},A^{\prime})$ factors as $\bar{C}(X,A)\xrightarrow{p}\bar{C}A\cup_{i}\bar{C}X\xrightarrow{h}\bar{C}(X^{\prime},A^{\prime})$, where $p$ comes from the Puppe sequence of the pair $(X,A)$.

Additionaly if there is a homotopy $G$, which makes the bottom triangle commute, then the composite $\bar{C}A\cup_{i}\bar{C}X\xrightarrow{h}\bar{C}(X^{\prime},A^{\prime})\xrightarrow{p^{\prime}}\bar{C}A^{\prime}\cup_{j}\bar{C}X^{\prime}$ is nullhomotopic.

We have the following proposition of convoluted implications finishing the proof of Theorem 7.3:

Proposition 7.5**.**

With the assumptions of Theorem 7.3 we have the following implications:

1. (i)

$B_{h}$* implies $SA_{h}$.* 2. (ii)

$A_{h-1}$* implies $SB_{h}$* 3. (iii)

If $k=1$, then $SA_{h}$ and $A_{h-1}$ imply $A_{h}$

If $k>1$ then $SA_{h}$, $SB_{h}$ and \begin{cases}B_{h-l}\text{ if k=2l}\\ A_{h-l-1}\text{ ifk=2l+1 }\end{cases} imply $A_{h}$ 4. (iv)

If $k=1$, then $SB_{h}$ and $B_{h-1}$ imply $B_{h}$

If $k>1$ then $SB_{h}$, $SA_{h-1}$ and \begin{cases}A_{h-l-1}\text{ if k=2l}\\ B_{h-l-1}\text{ ifk=2l+1 }\end{cases} imply $B_{h}$

Proof of (i) and (ii).

We will prove (i), the other part is completely analogous. By corollary 5.10 we get the following diagram:

[TABLE]

Furthermore the augmentation $\mathcal{O}^{\prime\prime\theta}_{h,b}\!\left(M,\delta^{T};\ell\right)_{\bullet}\to{\mathcal{E}^{\theta}_{h,b}}\!\left(M;\delta^{T}\right)$ is $(h-2)$-connected by Proposition 5.3. Since ${M}\!\left({a}\right)$ is homeomorphic to $M$, $B_{h}$ implies that $H_{q}\!\left(\beta_{h-j,b+j-1}\right)=0$ for $q\leq{B}\!\left({h-j}\right)$. This gives us all the ingredients to apply Lemma 4.4 with $c={SA}\!\left({h}\right)$, using that ${SA}\!\left({h}\right)\geq{B}\!\left({g-i}\right)+i$. The lemma implies that the induced map $H_{q}\!\left(C\!\left(\beta_{h,b-1}\right)\right)\to H_{q}\!\left(C\!\left(\alpha_{h,b}\right)\right)$ is an epimorphism for $q\leq{SA}\!\left({h}\right)$, which is exactly the statement of $SA_{h}$. ∎

Proof of (iii) and (iv).

We will prove (iii) for $k=2l$ here. The proof for odd $k$ is completely analogous and we will indicate the necessary minor changes for the proof of (iv).

Pick a genus maximizing sequence $(\ell_{j}^{i})_{1,\ldots,k+1}$ of length $k+1$ for maps of type $\alpha$ with a compatible sequence $(a^{i})_{1,\ldots,k+1}$ of arcs with embedded boundary isotopy such that $(a^{i})_{2,\ldots,k,1}$ is also a genus maximizing sequence of length $k+1$. To get such a sequence, choose for example $k+1$ disjoint intervals in the two components of $\delta$ and then label them positively oriented from $1$ to $k+1$, this gives the desired genus maximizing sequence (For (iv) choose $2k+2$ disjoint intervals in the single component of $\delta$ and label them as follows: $\ell_{0}^{k+1},\ldots\ell_{0}^{1},\ell_{1}^{k+1},\ldots\ell_{1}^{1}$; in this case $\ell_{0}^{1}$ and $\ell_{1}^{1}$ switch if one moves $a^{1}$ to the end of the sequence).

Now form the following diagram, where we denote $a^{1}\cup\ldots\cup a^{k+1}$ by $d$, $\bar{a}^{1}\cup\ldots\cup\bar{a}^{k+1}$ by $\bar{d}$, $a^{2}\cup\ldots\cup a^{k+1}$ by $c$, and $\bar{c}$ analogously:

[TABLE]

Here the rows are given by the corresponding Puppe sequences, note that the first and last line agree. The vertical maps are augmentation compositions and we have indicated in brackets which arcs with boundary isotopy are used there.

Since $\theta$ is $k$-trivial we can find a $Z\subset c\cup\partial^{0}{M}\!\left({a^{1}}\right)\times[1,2]$ providing a diagonal map for the lower left square. Note that $Z\cup(\bar{a}^{1}\setminus a^{1})\subset c\cup\partial^{0}M\times[1,2]$ gives a bordism that provides a diagonal map for the second square. This implies that the diagonal map of the lower square postcomposed with $\sigma^{1}_{h,b}(a^{1})$ agrees up to homotopy with the composition of $\sigma^{1}_{h-l,b}(\bar{a}^{1})$ and the diagonal map of the second square. Using this observation and Lemma 7.4, one obtains the following:

[TABLE]

This implies that $h$ induces the zero morphism in homology in those degrees, where $\Sigma\sigma^{1}_{h-l,b}(a^{1})$ induces a surjective map in homology. Note that by replacing the occurring manifolds with corners by manifolds with boundary, $\sigma^{1}_{h-l,b}(a^{1})$ becomes equivalent to a map of type $\beta_{h-l,b}$, hence is surjective in degrees ${B}\!\left({h-l}\right)$. Hence $h\circ p$, which is homotopic to $\bar{C}\!\left(\alpha_{h-l,b}\right)\to\bar{C}\!\left(\alpha_{h,b}\right)$, the relative augmentation composition on mapping cones instead of mapping pairs, is the null homomorphism in those degrees as well. Since this map is defined as a composition of relative [math]-approximate augmentations (on cones not pairs), which are epimorphisms in a range of degrees (depending on the genus occurring in the approximate augmentations) by $SA_{h}$ and $SB_{h}$, one has $H_{\ast}\!\left(C\!\left(\alpha_{h,b}\right)\right)$ vanishes in all degrees $\ast\leq\min\{{SA}\!\left({h-l+1}\right),{SB}\!\left({h-l+1}\right),{B}\!\left({h-l}\right)\}$. This finishes the proof as ${A}\!\left({h}\right)$ is by definition smaller or equal than the left hand side. ∎

8. Homological Stability for Capping off Boundary Components

All that is left to do to finish the proof of Theorem 7.3 is to prove the homological stability statement for maps of type $\gamma$ and prove that $\beta_{g,b}$ induces a monomorphism in integral homology provided that one of the newly created boundary components is contractible and the tangential structure on it extends to a disk. To do this we will first observe that homological stability for maps of type $\gamma_{g,b}$ where $b>1$ is an easy corollary of homological stability of maps of type $\beta_{g,b}$ and then we will establish everything needed to use Lemma 4.4 again to relate the homological stability for maps of type $\gamma_{g,1}$ to maps of type $\gamma_{g,b}$ for $b>1$.

So consider a subsurface $P\subset\partial^{0}M\times[0,1]$, which defines a map of type $\beta_{g,b}$. Suppose that one of the boundary components of the outgoing boundary of the pair of pants component of $P$ is contractible in $\partial^{0}M\times[0,1]$. Fix a contraction and, using the main result of [Hae61], realize it up to homotopy as an embedding $(D^{2},\partial D^{2})\to(\partial^{0}M\times[0,1],\partial^{0}M\times\{1\})$.

By flipping this embedded contraction (i.e. flipping the interval direction) and equipping it with a tangential structure extending the one on the boundary given by $P$, yields a bordism defining a map of type $\gamma_{g,b}$ and by construction $\gamma_{g,b+1}\circ\beta_{g,b}$ is a union with a cylinder and hence the composition is a homotopy equivalence and induces an isomorphism in homology. This implies in particular that $\beta_{g,b}$ induces a monomorphism in integral homology in all degrees.

Similarly given a disk defining a map of type $\gamma_{g,b}$ such that there exists another component of $\delta$ in the same connected component of $\partial^{0}M$, we can find a pair of pants defining a map $\beta_{g,b-1}$ such that $\gamma_{g,b}\circ\beta_{g,b-1}$ is a homotopy equivalence. We have already shown that $\beta_{g,b}$ induces an isomorphism in homology in degrees $\leq{B}\!\left({g}\right)-1$ and an epimorphism in the next degree. Moreover in this case $\beta_{g,b-1}$ induces a monomorphism in all degrees from which we conclude that it actually induces an isomorphism in all degrees $\leq{B}\!\left({g}\right)$. This implies that $\gamma_{g,b}$, as a left inverse for this map, induces an isomorphism in these degrees as well. All in all this gives:

Lemma 8.1**.**

Suppose $M$ fulfils the assumptions of Theorem 7.3: If one of the new boundary components of the bordism that defines a map of type $\beta_{g,b}$ is contractible in $\partial^{0}M$, then $\beta_{g,b}$ induces a monomorphism in homology in all degrees .

Similarly if there exists another boundary component of $\delta$ in the same connected component of $\partial^{0}M$, where the disk component of the bordism defining $\gamma_{g,b}$ lies, then $\gamma_{g,b}$ induces an epimorphism in homology in all degrees and furthermore an isomorphism in degrees $\leq{B}\!\left({g}\right)+1$.

Therefore we only have to concern ourselves with the case where there is no more connected component of $\delta$ in the same connected component as the disk component of the bordism defining $\gamma_{g,b}$. We will tackle this case using a certain resolution, which lets us relate $C\!\left(\gamma_{g,b}\right)$ to $C\!\left(\gamma_{g,b+j}\right)$. This section is very similar to Section 5 albeit it is much easier and does not require any inductive arguments.

Let $\ell$ denote a codimension [math] ball that is in the same connected component of $\partial^{0}M$ as the disk component of the bordism defining some fixed map of type $\gamma_{g,b}$ and that is disjoint from $\delta$.

Definition 8.2**.**

Fix an embedded subsurface $W\in{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$. We call an embedding

[TABLE]

such that ${y}\!\left({0}\right)\in\ell$ and ${y}\!\left({1}\right)\in M\setminus\partial M$ a boundary path of $W$. If we add a tubular neighborhood of $y$ in the pair $(M,W)$ denoted by $\mathbf{y}=(\mathbf{y}_{M},\mathbf{y}_{W})$ such that $\mathbf{y}\cap\partial M\subset\ell$, then we will call the pair $(y,\mathbf{y})$ a thickened boundary path of $W$.

If we only consider embeddings as above $y\colon([0,1],0)\to(M,\partial M)$ without a particular subsurface, we will drop the $W$ from the notation i.e. we will call these just boundary paths.

We call a boundary path together with a $2$-plane $B$ in the normal bundle at ${y}\!\left({\frac{1}{2}}\right)$ a thickened boundary path. A thickened boundary path with tangential structure is a thickenend boundary path with a tangential structure on ${\mathbf{y}}\!\left({B}\right)$.

With this notation at hand we can proceed to define another resolution of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$.

Definition 8.3**.**

Let $\mathcal{Q}^{\theta}_{g,b}\!\left(M;\delta^{T},\ell\right)_{\bullet}$ denote the semi-simplicial space, whose $i$-simplices are tuples $(W,(y^{0},\mathbf{y}^{0}),\ldots,(y^{i},\mathbf{y}^{i}))$, such that:

1. (i)

$W\in{\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ 2. (ii)

$(y^{j},\mathbf{y}^{j})$ is a thickened boundary path of $W$. 3. (iii)

The neighbourhoods $\mathbf{y}^{0},\ldots,\mathbf{y}^{i}$ are pairwise disjoint.

The $j$-th face map forgets the $j$-th boundary path and we topologize this as a subspace of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\times\overline{\operatorname{TEmb}}\!\left([0,1]\times X_{i},M;q\right)$. There is also an augmentation map $\epsilon_{\bullet}$ to ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$, which forgets the boundary paths.

Lemma 8.4**.**

If $M$ is connected and has dimension at least $3$, then the semi-simplicial space $\mathcal{Q}^{\theta}_{g,b}\!\left(M;\delta^{T},\ell\right)_{\bullet}$ is a resolution of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$.

Proof.

There is the following commutative square:

[TABLE]

By Lemma 2.8 ${\mathcal{E}^{+}_{g,b}}\!\left(M;\delta\right)$ is ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$-locally retractile. The group ${\operatorname{Diff}^{+}_{\partial}}\!\left(M\right)$ acts on $\mathcal{Q}^{+}_{g,b}\!\left(M;\delta,\ell\right)_{\bullet}$ and the lower map is equivariant, hence the bottom map is a Hurewicz-fibration. Furthermore it is obvious that the square is a pullback square, from which we conclude that the top map is also a Hurewicz-fibration.

It is easy to see that $\mathcal{Q}^{+}_{g,b}\!\left(M;\delta,\ell\right)_{\bullet}$ is a topological flag complex. We want to apply Lemma 4.1 to show that it is a resolution. Since the augmentation is a fiber bundle (and the fiber is non-empty as $M$ is connected) this flag complex has local sections. All that is left to show to apply the lemma is that for $\{y^{1},\ldots y^{n}\}\subset\mathrm{Fib}_{W}\!\left(\epsilon_{0}\right)$, there exists an $y^{\infty}$ such that $(y^{i},y^{\infty})$ is a $1$-simplex for every $i$. Since $M$ is at least $3$-dimensional we conclude that $M\setminus(\mathbf{y}^{0}\cup\ldots\cup\mathbf{y}^{n})$ is still path-connected, hence we can find a boundary path in this complement. ∎

Definition 8.5**.**

Let $R^{\theta}\!\left(M,\ell\right)_{i}$ denote the set of tuples $(\mathbf{y}^{0},\ldots,\mathbf{y}^{i})$, where $\mathbf{y}^{j}$ are thickened boundary paths with tangential structure such that all $\mathbf{y}^{j}$ are disjoint. We topologize this as a subspace of $\overline{\operatorname{TEmb}}\!\left([0,1]\times X_{i},M;q\right)\times\operatorname{Emb}\!\left(D^{2}\times X_{i},M\right)$.

There is a map from $\mathcal{Q}^{\theta}_{g,b}\!\left(M;\delta^{T},\ell\right)_{\bullet}$ to $R^{\theta}\!\left(M,\ell\right)_{i}$, which forgets the surface, but keeps the tubular neighbourhood in the surface and the corresponding tangential structure on it.

Lemma 8.6**.**

For a point $y=(y_{0},\ldots,y_{i})\in R^{\theta}\!\left(M,\ell\right)_{i}$ there is a homotopy fiber sequence:

[TABLE]

where ${M}\!\left({y}\right)$ denotes $\overline{M\setminus\left(\bigcup_{j}\mathbf{y}_{j}\right)}$ and ${\delta^{T}}\!\left({\mathbf{y}}\right)$ denotes $\delta^{T}\cup\left(\bigcup_{j}\partial\mathbf{y}_{j}^{\text{Disk}}\right)$where $\mathbf{y}_{j}^{\text{Disk}}$ denotes the embedding with $\theta$-structure of $D^{2}$ into $M$ associated to the two-plane in the definition of thickened boundary path with tangential structure.

The proof of this lemma is completely analogous to the proof of Lemma 5.6 and is therefore omitted.

The issues here that arise from the occurring manifolds with corners are dealt with in the exact same fashion as in the discussion after Lemma 5.6.

Similar to Section 5, we want to understand how this resolution behaves under stabilization maps. Let us take a look at the following diagram, in which we want to constructed the dashed arrow:

[TABLE]

Let us denote the surface defining $\gamma_{g,b}$ by $P$. Replace $P$ by a different but isotopic bordism if necessary to ensure that $(\ell\times[0,1])\cap P=\emptyset$. We will write $\bar{\ell}$ for $\ell\times[0,1]$. For a $y\in R^{\theta}\!\left(M,\ell\right)_{i}$ we define $\tilde{y}^{j}={y^{j}}\!\left({0}\right)\times[0,1]$ and $\bar{y}^{j}=y^{j}\cup\tilde{y}^{j}$. The dashed arrow is now given by mapping $W$ to $W\cup P$ and $y^{j}$ to $\bar{y}^{j}$. These maps commute with the face and augmentation maps and so they define a map of augmented semi-simplicial spaces, which we denote by $\gamma_{g,b}^{\bullet}$.

There is a map $R^{\theta}\!\left(M,\ell\right)_{i}\to R^{\theta}\!\left(M^{1},\bar{\ell}\right)_{i}$ defined by $y\mapsto\bar{y}$. This map is a homotopy equivalence. We have the following diagram:

[TABLE]

The commutativity of the right square gives us a map between the fibers and it is easy to see that the left vertical arrow is given by $\gamma_{g,b+i+1}$. Putting this together yields:

Corollary 8.7**.**

The induced map between the homotopy fibers of

[TABLE]

is given by $\gamma_{g,b+i+1}$.

Finally with all these tools at hand we are able to finish the proof of Theorem 7.3 by concluding:

Proposition 8.8**.**

Suppose $M$ fulfils the requirements of Theorem 7.3, then

[TABLE]

for $\ast\leq{B}\!\left({g}\right)$.

Proof.

We want to apply Lemma 4.4. The role of $B_{i}$ will be played by $R^{\theta}\!\left(M,\ell\right)_{i}$ and $\mathcal{Q}^{\theta}_{g,b}\!\left(M;\delta^{T},\ell\right)_{\bullet}$ will be the two resolutions, $\gamma_{g,b}^{\bullet}$ will be the map between them. Corollary 8.7 specified the occurring homotopy fibers.

We choose $n=\infty$ and $c={B}\!\left({g}\right)+1$. By Lemma 8.1 and 8.7 we conclude that the homology of the reduced mapping cone of the map between the fibers equals zero in the desired range i.e.

[TABLE]

for $\ast\leq{B}\!\left({g}\right)$, which finishes the proof of the proposition. ∎

9. Tangential Structures that fulfil Homological Stability

The goal of this section is to establish criteria for when a space of tangential structures satisfies $k$-triviality, which was the key requirement for a tangential structure to fulfil homological stability, and prove Proposition 6.9, which will be fundamental in the proofs of these criteria. This will yield more easily applicable versions of Theorem 7.3.

9.1. Stability of Connected Components implies

$k$-triviality

Proof of Proposition 6.9.

We will prove $k$-triviality for maps of type $\alpha$, the case of maps of type $\beta$ is completely analogous. Let $\left(\ell_{0}^{i},\ell_{1}^{i}\right)$ together with $\bar{\ell_{0}}$ and $\bar{\ell_{1}}$ denote a genus maximizing sequence of length $2g+1$ for a map of type $\alpha$ and compatible $a^{i}$ as in the definition of $k$-triviality. Let $P$ denote a stabilization bordism such that $P\cap\left(\left(\bar{\ell_{0}}\cup\bar{\ell_{1}}\right)\times[0,1]\right)=\left(\bar{\ell_{0}}\cup\bar{\ell_{1}}\right)\times[0,1]$. Furthermore let us denote $\bigcup_{i}a_{L}^{i}$ by $a_{L}$ and $\bigcup_{i}\bar{a}_{L}^{i}$ by $\bar{a}_{L}$ and $\bigcup_{i}a^{i}$ by $a$ and analogously for $\bar{a}$. We want to prove that $a_{L}$ absorbs $P$ or in other words there exists a $Z\subset\bar{a}\cup\partial^{0}M\times[1,2]$ such that

[TABLE]

commutes up to homotopy.

Let us denote $\overline{\partial a_{L}\setminus\delta}$ by $\partial^{0}a_{L}$ and $\overline{\partial\overline{a}_{L}\setminus\delta^{\prime}}$ by $\partial^{0}\overline{a}_{L}$. We define

[TABLE]

We have the following commutative diagram:

[TABLE]

Note that the occuring spaces in this diagram are manifolds with corners, this can be fixed by the usual smoothing the angle techniques explained in Remark 5.7 and Remark 5.7, furthermore note that in this diagram $-\cup{P}\!\left({a}\right)$ is a complicated subset bordism and not a stabilization bordism.

Note that the non-cylcindrical parts of the occuring bordisms in the first diagram occur as elements of the spaces in this second diagram. For example the bordism for the composition of the two arrows on the right of the first diagram is given by $\bar{a}_{L}\cup\delta^{\prime T}\times[1,3]$ which is an element of the lower left space. Analogously the bordism of the composition on the left hand side $a_{L}\cup\delta^{T}\times[0,2]$ is an element of the top right space. That the genus and the number of boundary conditions is as stated follows from the requirement that the sequence of intervals is genus maximizing and that $P$ contains $\left(\bar{\ell_{0}}\cup\bar{\ell_{1}}\right)\times[0,1]$.

$Z$ will be an element of the top left space and the compositions with $-\cup Z$ in the first diagram correspond to composition in the second diagram. Furthermore note that the left map is $\pi_{0}$-surjective and the right hand is $\pi_{0}$-injective. Hence chose an element $Z$ in ${\mathcal{E}^{\theta}_{g-1,4}}\!\left(\bar{a}\cup\partial^{0}M\times[1,2];a\right)$ such that $Z\cup P+2$ lies in the same connected component as $\bar{a_{L}}\cup\delta^{\prime T}\times[1,3]$. Then the lower triangle in the first diagram commutes up to homotopy by construction.

Since the second diagram is commutative the image of $Z\cup{P}\!\left({\bar{a}}\right)$ under the right map is isotopic to $\bar{a_{L}}\cup\delta^{\prime T}\times[1,3]\cup{P}\!\left({a}\right)$ which in turn is isotopic to $a_{L}\cup\delta^{T}\times[0,2]\cup P+2$, since this is the image of $a_{L}\cup\delta^{T}\times[0,2]$ under the right hand map, which is $\pi_{0}$-injective, we conclude that the upper triangle in the first diagram commutes as well. ∎

As a corollary to Proposition 6.9 and Theorem 7.3 one obtains:

Theorem 9.1**.**

Suppose $M$ is an at least $5$-dimensional, simply-connected manifold with non-empty boundary and $\partial^{0}M$ is a codimension [math] simply-connected submanifold of $\partial M$ with a space of $\theta$-structures of subplanes of $TM$. Suppose further that $\theta$ $\pi_{0}$-stabilizes at $m_{1}$ and $\pi_{0}$-stabilizes at the boundary at $m_{2}$, then

1. (i)

The homology of $C\!\left(\alpha_{g,b}\right)$ vanishes in degrees $\ast\leq{A}\!\left({g}\right)$ 2. (ii)

The homology of $C\!\left(\beta_{g,b}\right)$ vanishes in degrees $\ast\leq{B}\!\left({g}\right)$ 3. (iii)

The homology of $C\!\left(\gamma_{g,b}\right)$ vanishes in degrees $\ast\leq{B}\!\left({g}\right)+1$

Where $k$ in the definition of $A$ and $B$ (Definition 7.1) equals $2m_{2}+1$.

9.2. Criterions for Stabilization of Connected Components

The goal of this subsection is to prove the following theorem:

Theorem 9.2**.**

Suppose that $M$ is a simply connected manifold of dimension at least $5$ such that $\partial^{0}M$ is simply-connected as well and $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ is a space of $\theta$-structures of subplanes of $TM$ where $\mathrm{Fib}_{B}\!\left(\theta\right)$ is simply-connected, then

1. (i)

The homology of $C\!\left(\alpha_{g,b}\right)$ vanishes in degrees $\ast\leq\left\lfloor\frac{1}{3}(2g+1)\right\rfloor$ 2. (ii)

The homology of $C\!\left(\beta_{g,b}\right)$ vanishes in degrees $\ast\leq\left\lfloor\frac{2}{3}g\right\rfloor$ 3. (iii)

The homology of $C\!\left(\gamma_{g,b}\right)$ vanishes in degrees $\ast\leq\left\lfloor\frac{2}{3}g+1\right\rfloor$

The proof will use Theorem 9.1 and Remark 7.2.

As was explained in the introduction, examples of spaces of $\theta$-structures with simply-connected fiber are given by $k$ framings of the normal bundle of the subsurfaces, where $k\leq n-2$.

In order to apply Theorem 9.1, we need a better understanding of the space of connected components of ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$if the fiber of the space of tangential structures is simply-connected.

Lemma 9.3**.**

Suppose that $M$ is simply-connected and of dimension at least $5$, and that the fiber $\mathrm{Fib}_{B}\!\left(\theta\right)$ is simply connected, then $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$ carries a canonical group structure and there is an action of this group on $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\right)$, given by connected sum, such that it is an affine $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$-set.

Proof.

The proof is divided into three steps. We will first construct the group structure on $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$ by relating this space to the second homotopy groups of $B_{2}\!\left(M\right)$ and $\operatorname{Gr}_{2}\!\left(TM\right)$, then we will relate this group to $\pi_{0}\!\left({\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)\right)$ using the simply-connectedness of $M$. Lastly, we will see that the identifications we proved along the way behave very well with respect to the action of the diffeomorphism group of the surface to get an understanding of $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\right)$.

For the first step, consider the following tower of fibrations:

[TABLE]

Let us calculate $\pi_{2}\!\left(\operatorname{Gr}_{2}\!\left(TM\right)\right)$. Consider the following diagram, which stems from the fact that the fibration $\operatorname{Gr}_{2}\!\left(TM\right)\to M$ is the pullback of $\operatorname{Gr}_{2}\!\left(\gamma\right)$, where $\gamma$ denotes the tautological bundle of some $\operatorname{Gr}_{n}\!\left(\mathbb{R}^{N}\right)$

[TABLE]

Hence the upper long exact sequence breaks down to the following:

[TABLE]

Since $M$ is simply-connected, we have an isomorphism $\pi_{2}\!\left(M\right)\simeq\pi_{0}\!\left(\operatorname{Map}\!\left(S^{2},M\right)\right)$ and this group is isomorphic to $\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)$ by the main result of [Hae61]. Therefore the split in the aforementioned diagram is given by taking an embedding as a representative and then applying $\operatorname{Gr}\!\left(D-\right)$. From here on forth we consider $\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)$ as a subgroup of $\pi_{2}\!\left(\operatorname{Gr}_{2}\!\left(TM\right)\right)$.

Consider the the following long exact sequence:

[TABLE]

We claim that ${\left(\theta\right)_{*}^{-1}}\!\left({\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)}\right)$ is isomorphic to $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$. There is a canonical map $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)\to{\left(\theta\right)_{*}^{-1}}\!\left({\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)}\right)$, which is evidently surjective. For injectivity of this map suppose two elements $\left[e\right],\left[e^{\prime}\right]\in\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$ map to the same element in ${\left(\theta\right)_{*}^{-1}}\!\left({\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)}\right)$. Hence there is a homotopy $H$ from $T_{e}$ to $T_{e^{\prime}}$ as maps into $B_{2}\!\left(M\right)$. The main theorem of [Hae61] implies that $\theta\circ H$ is actually homotopic via $\bar{H}$ relative to $e$ and $e^{\prime}$ to an isotopy. We have the following commutative diagram, where the dashed arrows exists since $\theta$ is a fibration:

[TABLE]

The indicated lift of $\bar{H}$, when evaluated at $\left(-,1\right)$, gives us the desired isotopy from $\left[e\right]$ to $\left[e^{\prime}\right]$ as embeddings with $\theta$-structure. All in all this implies that $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$ has a group structure, which stems from the isomorphism to ${\left(\theta\right)_{*}^{-1}}\!\left({\pi_{2}\!\left(\operatorname{Gr}_{2}\!\left(TM\right)\right)}\right)$. For notational reasons, let us denote $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$ by $\pi^{\theta}$.

For the second step, consider the cofibration $\Sigma^{1}_{g,b}\to\Sigma_{g,b}$, where $\Sigma^{1}_{g,b}$ denotes a small tubular neighbourhood of the $1$-skeleton of $\Sigma_{g,b}$ to which only a single $2$-cell is attached via $\sigma$. Since the inclusion $\Sigma^{1}_{g,b}\to\Sigma_{g,b}$ is a cofibration, $\operatorname{Map}\!\left(\Sigma_{g,b},M\right)\to\operatorname{Map}\!\left(\Sigma^{1}_{g,b},M\right)$ is a fibration, with fiber over an embedding $e$ given by $\operatorname{Map}\!\left(\left(D^{2},\partial D^{2}\right),\left(M,\mathfrak{d}\right)\right)$, where $\mathfrak{d}$ is the image of the boundary of the single two cell, denoted by $D^{2}$, glued to $\Sigma^{1}_{g,b}$under $e$. Again by [Hae61] and transversality, $\pi_{0}\!\left(\operatorname{Map}\!\left(\left(D^{2},\partial D^{2}\right),\left(M,\mathfrak{d}\right)\right)\right)$ is isomorphic to $\pi_{0}\!\left({\operatorname{Emb}}\!\left(D^{2},M\setminus{e}\!\left({\Sigma^{1}_{g,b}}\right);\mathfrak{d}\right)\right)$. Similar considerations as before show that the a priori pointed set $\pi_{0}\!\left({\operatorname{Emb}}\!\left(D^{2},M\setminus{e}\!\left({\Sigma^{1}_{g,b}}\right);\mathfrak{d}\right)\right)$ can be considered as a subgroup of $\pi_{2}\!\left(\operatorname{Gr}_{2}\!\left(TM\right),\mathfrak{d}\right)$, where we consider $\mathfrak{d}$ (or its image to be more precise) as a subset of $\operatorname{Gr}_{2}\!\left(TM\right)$ via the Grassmannian differential.

Since $\mathrm{Fib}_{B}\!\left(\theta\right)$ is simply-connected, there exists an up to homotopy unique $\theta$-structure on $\mathfrak{d}$, which we denote by $\mathfrak{d}^{T}$.

Let us introduce the shorthand notation $\pi^{\theta}_{\text{rel}}$ for ${\left(\theta\right)_{*}^{-1}}\!\left({\pi_{0}\!\left({\operatorname{Emb}}\!\left(D^{2},M\setminus{e}\!\left({\Sigma^{1}_{g,b}}\right);\mathfrak{d}\right)\right)}\right)$, here $\theta\colon\left(B_{2}\!\left(M\right),\mathfrak{d}^{T}\right)\to\left(\operatorname{Gr}_{2}\!\left(TM\right),\mathfrak{d}\right)$is considered as a map of pairs. We claim that $\pi^{\theta}$ considered as a subgroup of $\pi_{2}\!\left(B_{2}\!\left(M\right)\right)$ acts transitively and freely on $\pi^{\theta}_{\text{rel}}$. In order to prove this claim, consider the long exact sequence of the pair $\left(B_{2}\!\left(M\right),\mathfrak{d}^{T}\right)$:

[TABLE]

That the action is free follows immediately from this long exact sequence. To prove that this action is transitive consider the following diagram given by considering long exact sequences of pairs and the corresponding maps between them:

[TABLE]

The commutativity of the lower left square gives us that $\pi_{0}\!\left(\operatorname{Emb}\!\left(S^{2},M\right)\right)$ acts freely and transitively on $\pi_{0}\!\left({\operatorname{Emb}}\!\left(D^{2},M\setminus{e}\!\left({\Sigma^{1}_{g,b}}\right);\mathfrak{d}\right)\right)$and taking preimages with respect to $\theta$ does not change the transitivity of the action.

Furthermore the same arguments as for $\pi^{\theta}$ before show that $\pi^{\theta}_{\text{rel}}$ is isomorphic to the pointed set $\pi_{0}\!\left({\operatorname{Emb}}^{\theta}\!\left(D^{2},M\setminus{e}\!\left({\Sigma^{1}_{g,b}}\right);\mathfrak{d}^{T}\right)\right)$.

We claim that $\pi^{\theta}_{\text{rel}}$ is isomorphic to $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(\Sigma_{g,b},M\right)\right)$. Consider the long exact sequence in homology of the pair $\left(B_{2}\!\left(M\right),\delta^{T}\right)$(As before $\delta$ is the boundary condition for the spaces of subsurfaces) :

[TABLE]

Note that $B_{2}\!\left(M\right)$ is simply-connected since $\mathrm{Fib}_{B}\!\left(\theta\right)$ and $\operatorname{Gr}_{2}\!\left(TM\right)$ are simply-connected. Likewise the pair $\left(B_{2}\!\left(M\right),\mathfrak{d}^{T}\right)$ is simply-connected. This yields the claimed isomorphisms by the Hurewicz-Theorem. The long exact sequence implies that $\pi^{\theta}\subset\pi_{2}\!\left(B_{2}\!\left(M\right)\right)\cong H_{2}\!\left(B_{2}\!\left(M\right)\right)$ acts freely on ${\partial^{\prime-1}}\!\left({[\delta^{T}]}\right)$. We have a map from $\pi^{\theta}_{\text{rel}}$ to ${\partial^{\prime-1}}\!\left({[\delta^{T}]}\right)$ by extending an embedding by $\left.{e}\right\rvert_{\Sigma^{1}_{g,b}}$ and considering the image of the fundamental class of $\Sigma_{g,b}$. By definition this map factorizes through $\pi_{0}\!\left({\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)\right)$ and it is equivariant with respect to the $\pi^{\theta}$-action on $H_{2}\!\left(B_{2}\!\left(M\right)\right)$ and ${\partial^{\prime-1}}\!\left({[\delta^{T}]}\right)$. This implies that the map from $\pi^{\theta}_{\text{rel}}$ to $\pi_{0}\!\left({\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)\right)$ is injective.

To see that this map is surjective consider the fibration (proven to be a fibration in Lemma 2.18) ${\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)\to{\operatorname{Emb}}^{\theta}\!\left(\Sigma^{1}_{g,b},M;\delta^{T}\right)$ and consider its long exact sequence:

[TABLE]

Hence the map is surjective, as $\pi_{0}\!\left({\operatorname{Emb}}^{+}\!\left(\Sigma^{1}_{g,b},M;\delta\right)\right)$ has only one element by the main result of [Hae61], and the simply-connectedness of $\mathrm{Fib}_{B}\!\left(\theta\right)$.

Let us proceed with the last step. We have proven so far that the $\pi^{\theta}$-equivariant map from $\pi^{\theta}_{\text{rel}}$ to $\pi_{0}\!\left({\operatorname{Emb}}^{\theta}\!\left(\Sigma_{g,b},M;\delta^{T}\right)\right)$ is an isomorphism and that the map from there to ${\partial^{\prime-1}}\!\left({[\delta^{T}]}\right)$ is an injection. But this map factorizes through $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\right)$, since an orientation preserving diffeomorphism does not change the image of the fundamental class. Therefore we conclude that $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\right)$ is an affine $\pi^{\theta}$-set. ∎

From this we obtain almost immediately the following corollary.

Corollary 9.4**.**

If $M$ is simply-connected and of dimension at least $5$, $\partial^{0}M$ is simply-connected as well and the fiber of $\theta$ is simply-connected, then $\theta$ $\pi_{0}$-stabilizes at [math] and $\pi_{0}$-stabilizes at the boundary at [math] as well.

Proof.

The stabilization maps are given by gluing in surfaces, which induce $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$-equivariant maps between $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)\right)$ and $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g+1,b-1}}\!\left(M;\delta^{T}\right)\right)$ or $\pi_{0}\!\left({\mathcal{E}^{\theta}_{g,b+1}}\!\left(M;\delta^{T}\right)\right)$. Since both of these spaces are affine $\pi_{0}\!\left(\operatorname{Emb}^{\theta}\!\left(S^{2},M\right)\right)$-spaces, the map is a bijection. Repeating the same argument for $\partial^{0}M\times[0,1]$ proves the claim. ∎

Applying this corollary to $M$ together with Theorem 9.1 for $m_{1}=m_{2}=0$, one obtains Theorem 9.2.

9.3. Non-relative Tangential Structures

As was explained in the introduction, the notion of tangential structure in this paper is a generalization of the usual notion to include relative tangential structures. This subsection will deal with non-relative tangential structures.

There is a (up to homotopy) canonical map $\operatorname{Gr}_{2}\!\left(TM\right)\to\operatorname{Gr}_{2}\!\left(\mathbb{R}^{\infty}\right)\simeq BSO\left(2\right)$ and recall that in the common definition of tangential structure one considers a fibration $\theta\colon B_{2}\!\left(\operatorname{Gr}_{2}\!\left(\mathbb{R}^{\infty}\right)\right)\to\operatorname{Gr}_{2}\!\left(\mathbb{R}^{\infty}\right)$. In this case one obtains classifying spaces for surface bundles equipped with a $\theta$-structure, denoted by $\mathcal{M}^{\theta}\left(\Sigma_{g,b}\right)$ (we are a bit sloppy here, since there are also boundary conditions involved). We will not define these classifying spaces here and the reader is advised to consult [RW16] in particular Definition 1.1 in that paper.

Definition 9.5**.**

We say that a tangential structure $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ (in the sense of this paper) is a non-relative tangential structure stemming from $\theta^{\prime}$ if there exists a tangential structure (in the common sense)

[TABLE]

such that the following diagram is a pullback diagram:

[TABLE]

The following theorem states that for the question of whether a non-relative tangential structure fulfils homological stability, it does not matter, whether one considers it as tangential structures of abstract surfaces or subsurfaces of a (finite-dimensional) manifold.

Theorem 9.6**.**

Suppose $M$ is a simply-connected at least $5$-dimensional manifold and that $\partial^{0}M$ is simply-connected. Let $\theta\colon B_{2}\!\left(TM\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ denote a non-relative tangential structure stemming from $\theta^{\prime}$ such that $\mathcal{M}^{\theta^{\prime}}\left(\Sigma_{g,b}\right)$ fulfils homological stability as stated in Theorem 7.1 in [RW16] (Note that the definitions of $F$ and $G$ in the theorem, agree with our definition of $A$ and $B$).

Then $\theta$ $\pi_{0}$-stabilizes at genus $g$, and also $\pi_{0}$-stabilizes at the boundary at genus $g$, where $g$ is the first degree in which the first homology of $C\!\left(\alpha_{g,b}\right)$, of $C\!\left(\beta_{g,b}\right)$, and of $\ MappingPair{\gamma_{g,b}}$ vanishes, here these stabilization maps are stabilization maps of abstract surfaces in the sense of Subsection 1.3 in [RW16]. Hence $\theta$ is $2g+1$-trivial and ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ fulfills homological stability as stated in Theorem 7.3.

In particular $\theta$ fulfils homological stability if $\theta^{\prime}$ stabilizes on connected components of $\mathcal{M}^{\theta^{\prime}}\left(\Sigma_{g,b}\right)$

Proof.

By Theorem 9.1 it suffices to prove that ${\mathcal{E}^{\theta}_{g,b}}\!\left(M;\delta^{T}\right)$ stabilizes on connected components. There is the following pullback diagram, where the horizontal maps stems from a fixed embedding of $M$ into $\mathbb{R}^{\infty}$,

[TABLE]

Here $\mathcal{M}^{+}\left(\Sigma_{g,b}\right)$ is the classifying space of oriented surface bundles. Let us denote the fiber of the vertical maps of this pullback diagram by $F_{g,b}$. Let us prove stabilization of connected components for a stabilization map of type $\alpha_{g,b}$, the case of another stabilization map is treated analogously.

One obtains the following diagram, where the rows are exact sequences and the vertical maps are steming from stabilization maps $\alpha_{g,b}$:

[TABLE]

From this we can we conclude that if $\left(\alpha_{g,b}\right)_{*}\colon H_{1}\!\left(\mathcal{M}^{\theta^{\prime}}\left(\Sigma_{g,b}\right)\right)\to H_{1}\!\left(\mathcal{M}^{\theta^{\prime}}\left(\Sigma_{g+1,b-1}\right)\right)$ and $\left(\alpha_{g,b}\right)_{*}\colon H_{0}\!\left(\mathcal{M}^{+}\left(\Sigma_{g,b}\right)\right)\to H_{0}\!\left(\mathcal{M}^{+}\left(\Sigma_{g+1,b-1}\right)\right)$ are isomorphisms, then the induced map between the fibers $\left(\alpha_{g,b}\right)_{*}\colon H_{0}\!\left(F_{g,b}\right)\to H_{0}\!\left(F_{g+1,b-1}\right)$ is an isomorphism as well.

Hence we obtain:

[TABLE]

Repeating the same argument for $\partial^{0}M\times[0,1]$, we conclude that $\theta$ $\pi_{0}$-stabilizes and $\pi_{0}$-stabilizes at the boundary. The rest of the proof follows from Theorem 9.1. ∎

As was explained in the introduction, examples of such tangential structures are given by framings of the tangent bundle of the subsurface or spin structures on the subsurfaces (See [RW16] for the corresponding statement about the moduli spaces of abstract surfaces with these tangential structures).

10. Spaces of Pointed Subsurfaces and their Stabilization

While stabilizing at the boundary, as we have done so far, is the most intuitive way to stabilize subsurfaces, it is not suitable for many kinds of tangential structures. In particular it is not well suited for symplectic subsurfaces, which we will consider in Section 11. In order to deal with this, we will introduce spaces of pointedly embedded subsurfaces and construct stabilization maps between them that are heuristically given by taking the connected sum with a fixed torus and then prove homological stability for these stabilization maps.

10.1. Spaces of Pointedly Embedded Subsurfaces

Let $M$ denote a closed smooth $n$-manifold, where $n\geq 5$ and let us abbreviate $\Sigma_{g,0}$ by $\Sigma_{g}$. Fix $x\in\Sigma_{g}$, $x_{M}\in M$ and an oriented $2$-plane with tangential structure $B\in{\theta^{-1}}\!\left({\operatorname{Gr}_{2}\!\left(T_{x_{M}}M\right)}\right)$, by abuse of notation we will also write $B$ for the underlying $2$-plane. We will consider the space

[TABLE]

equipped with the same topology as in the non-pointed case. Let $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ denote the subgroup of ${\operatorname{Diff}^{+}}\!\left(\Sigma_{g}\right)$ that fixes $x$. This group acts on $\operatorname{Emb}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$ freely via precomposition and we denote the quotient by $\mathcal{E}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$. Elements in this space will be called pointed subsurfaces of $M$

Remark 10.1**.**

In this situation, one could replace $M$ by a manifold with boundary and $\Sigma_{g}$ by $\Sigma_{g,b}$ and include some boundary condition $\delta^{T}$ into this notation. Since all proofs presented here, will work verbatim for the pointed case with boundary, the results hold in that case as well.

As before the following lemma will play a central role.

Lemma 10.2**.**

The forgetful map $\pi_{\theta}\colon\mathcal{E}^{\theta}_{B}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ is a Hurewicz fibration..

The proof is verbatim the same as the proof of Lemma 2.12. One only has to replace the occuring homeomorphism and diffeomorphism groups by their pointed analogues and note that $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ is $\operatorname{Diff}_{x_{M}}\!\left(M\right)$-locally retractile, where this is the group of diffeomorphisms fixing $T_{x_{M}}M$ pointwise.

10.2. Stabilization Maps for Pointedly Embedded Subsurfaces

The goal of this subsection is to construct stabilization maps

[TABLE]

As the first and most complicated step, we will show that there are homotopy equivalent spaces of subsurfaces where the subsurfaces meet a neighbourhood of $x_{M}$ in a prescribed way. From there it will be quite easy to define the stabilization maps.

Let $V$ denote a coordinate chart centered around $x_{M}$ diffeomorphic to $\mathbb{R}^{n}$ such that the fixed tangent plane $B$ corresponds to $\mathbb{R}^{2}$ in $\mathbb{R}^{n}$, where this inclusion is given by specifying the last $n-2$ coordinates to be zero. Fix once and for all $0<R\in\mathbb{R}$ and we write $D^{2}_{R}\coloneqq\mathbb{R}^{2}\cap B_{R}\!\left(0\right)$.

Definition 10.3**.**

$\mathcal{E}^{\theta}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$ is the subspace of $\mathcal{E}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$ consisting of those pointed subsurfaces such that their intersection with the open ball of a radius $R$ in $V$ equals $D^{2}_{R}$, and such that the tangential structure is constantly $B$ on $B_{\frac{R}{2}}\!\left(0\right)$ with respect to some fixed trivialization of $B_{2}\!\left(D^{2}_{R}\right)$.

We will need the following two results to define pointed stabilization maps. The first proposition will deal with the case without tangential structure and its proof will occupy most of this section, the subsequent corollary will extend this to the case with tangential structures.

Proposition 10.4**.**

There is a coordinate chart centered at $x_{M}$ such that the inclusion

[TABLE]

is a homotopy equivalence with homotopy inverse denoted by $\Psi_{\text{flat}}$.

Remark 10.5**.**

It is far easier to prove that this inclusion is a weak homotopy equivalence, but we will need a homotopy inverse to the inclusion (i.e. maps that flatten the subsurfaces) to construct the stabilization maps.

Corollary 10.6**.**

The inclusion $\mathcal{E}^{\theta}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$ is a homotopy equivalence with homotopy inverse denoted by $\Psi_{\text{flat}}$ as well.

Proof.

We define $\mathcal{E}^{\theta}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)^{\ast}$ as ${\pi_{\theta}^{-1}}\!\left({\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)}\right)$, where

[TABLE]

denotes the projection. Note that the following diagram is a pullback diagram:

[TABLE]

It is a standard fact that a pullback of a homotopy equivalence along a Hurewicz-fibration produces an induced map, which is also a homotopy equivalence (see for example Proposition 2.3 in [Fra]). This fact together with Lemma 10.2 (which implies that the right arrow is a fibration) implies that the top map is a homotopy equivalence. Furthermore the homotopy inverse is a map over the homotopy inverse of the base map.

So all that is left to do is trivializing the tangential structure on the disk but this is easily done by using a fixed trivialization of $B_{2}\!\left(D^{2}_{R}\right)\cong D^{2}_{R}\times{\theta^{-1}}\!\left({\operatorname{Gr}_{2}\!\left(T_{0}D^{2}_{R}\right)}\right)$ and just continuously extend the tangential structure of $B$ to $D^{2}_{\frac{R}{2}}$. ∎

Proof of Proposition 10.4

In order to proof Proposition 10.4, we will define two subspaces of $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$, which both contain $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$, and fit in the following diagram

[TABLE]

The hooked arrows are inclusions and the $\Psi_{i}$ represent their respective homotopy inverses. We will denote $\Psi_{3}\circ\Psi_{2}\circ\Psi_{1}$ by $\Psi_{\text{flat}}$ Let us define these spaces:

Definition 10.7**.**

Let $\pi_{2}\colon\mathbb{R}^{n}\to\mathbb{R}^{2}$ denote the projection onto the first two coordinates i.e. the map that sends $(x_{1},\ldots,x_{n})$ to $(x_{1},x_{2})$. For $r\in(0,\infty)$, we define

[TABLE]

The action of $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ via precomposition on $\operatorname{Emb}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$ restricts to a free action on the subspace $\operatorname{Emb}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ and we denote the quotient by $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$. For a $W\in\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ we will write $W_{r,x}$ for the image of $U_{x}$ in $M$.

Let $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ denote the subspace of $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ of such subsurfaces $W$ such that $D^{2}_{R}\subset W\cap B_{R}\!\left(0\right)$

Remark 10.8**.**

Note that the subtle difference between $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ and $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$ is that subsurfaces in the latter intersect $B_{R}\!\left(0\right)$ exactly in $D^{2}_{R}$, while surfaces in the former only have to contain $D^{2}_{R}$, but there can be more parts of these surfaces that intersect $B_{R}\!\left(0\right)$.

After the following three auxillary lemmas we will prove that all the aforementioned inclusions are indeed homotopy equivalences by constructing their inverse and the necessary homotopies. This will finish the proof of Proposition 10.4.

Lemma 10.9**.**

Let $\iota\colon X^{\prime}\to X^{\prime\prime}$ denote an inclusion and let $\Psi_{\iota}\colon X^{\prime\prime}\to X^{\prime}\subset X^{\prime\prime}$ be a map such that there exists a homotopy $H\colon X^{\prime\prime}\times[0,1]\to X^{\prime\prime}$ such that ${H}\!\left({-,0}\right)=\mathrm{Id}_{X^{\prime\prime}}$ and ${H}\!\left({-,1}\right)=\Psi_{\iota}$ and ${H}\!\left({X^{\prime}\times[0,1]}\right)\subset X^{\prime}$, then $\iota$ is a homotopy equivalence with homotopy inverse $\Psi_{\iota}$.

Proof.

$H$ is already a homotopy between $\iota\circ\Psi_{\iota}$ and the identity of $X^{\prime\prime}$, therefore we only need to construct a homotopy between $\Psi_{\iota}\circ\iota$ and the identity on $X^{\prime}$. But $\left.{\mathrm{Id}_{X^{\prime\prime}}}\right\rvert_{X^{\prime}}=\mathrm{Id}_{X^{\prime}}$, therefore the necessary homotopy is again given by $H$. ∎

This lemma will be used to prove that the $\Psi_{i}$ are homotopy inverse to the aforementioned inclusions. The next rather technical lemma will be used towards the end of the proof of Proposition 10.4 to show that a certain map is continuous.

Lemma 10.10**.**

Suppose we have a diagram of the following form:

[TABLE]

where $X_{2}$ is compact and ${g_{\mathrm{inf}}}\!\left({x_{1}}\right)=\inf_{x_{2}\in X_{2}}{g}\!\left({x_{1},x_{2}}\right)$. Then $g_{\mathrm{inf}}$ is continuous.

Proof.

Since intervals of the form $(C,\infty]$ and $(-\infty,C)$ form a subbasis of the topology of $\mathbb{R}\cup\{\infty\}$ it suffices to show that their preimages are open. We will first show that ${g_{\mathrm{inf}}^{-1}}\!\left({(C,\infty]}\right)$ is open for every $C$. For this one only has to note that ${g_{\mathrm{inf}}^{-1}}\!\left({(C,\infty]}\right)={\pi_{1}}\!\left({\left({g^{-1}}\!\left({(-\infty,c]}\right)\right)^{c}}\right)$ which is open since $g$ is continuous and $\pi_{1}$ is closed since $X_{2}$ is compact.

Now we will proof that ${g_{\mathrm{inf}}^{-1}}\!\left({(-\infty,C]}\right)$ is open, observe that ${g_{\mathrm{inf}}^{-1}}\!\left({(-\infty,C]}\right)={\pi_{1}}\!\left({{g^{-1}}\!\left({(-\infty,C)}\right)}\right)$. This is open since $\pi_{1}$ is an open map. ∎

Lastly we need the following lemma before we can start the proof of Proposition 10.4.

Lemma 10.11**.**

$\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$* is an open subset of $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ for every $r\in(0,\infty)$ and if $\epsilon_{k}$ denotes a sequence of positive real numbers that converges to zero, then*

[TABLE]

Proof.

By the inverse function theorem, for every $W\in\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ there is an $r>0$ such that $\left.{\pi_{2}}\right\rvert_{{\pi_{2}^{-1}}\!\left({\overline{B_{r}\!\left(0\right)}\cap W}\right)}$ restricted to the connected component of ${\pi_{2}^{-1}}\!\left({\overline{B_{r}\!\left(0\right)}\cap W}\right)$ containing $x_{M}$ is a diffeomorphism. This implies that every $W\in\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ is contained in some $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ for $r$ small enough.

To finish the proof we have to show that $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ is an open subset of $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$, which is equivalent to its preimage in $\operatorname{Emb}^{\theta}_{B}\!\left(\Sigma_{g},M\right)$ being open. Take some embedding $e$ in this set. There is some closed neighborhood $U_{x}$ of $x$ in $\Sigma_{g}$ such that $\left.{\pi_{2}}\right\rvert_{{e}\!\left({U_{x}}\right)}$ is a diffeomorphism onto $\overline{B_{r}\!\left(0\right)}$. Furthermore this implies that the differential of $\pi_{2}\circ e$ is an isomorphism on these points. Since the isomorphisms form an open subset of all linear maps we conclude that there is actually an open neighborhood $U_{x}^{\prime}$ of $U_{x}$ such that on the closure of this neighborhood the differential of $\pi_{2}\circ e$ is still an isomorphism. Since $\left.{\pi_{2}\circ e}\right\rvert_{\overline{U_{x}^{\prime}}}$ is proper because $\overline{U_{x}^{\prime}}$ is compact and a local diffeomorphism onto $\mathbb{R}^{2}$ by the requirement that the differential is an isomorphism, we conclude that it is a covering of its image. Since a closed disk can only cover a closed disk via a diffeomorphism we see that this map is actually a diffeomorphism onto some closed disk in $\mathbb{R}^{2}$ containing $\overline{B_{r}\!\left(0\right)}$. Let $2\varepsilon$ denote $\min_{x^{\prime}\in\partial U_{x}^{\prime}}\left\lVert{{\pi_{2}\circ e}\!\left({x^{\prime}}\right)}\right\rVert-r>0$. Consider the following neighbourhood of $e$:

[TABLE]

With our previous considerations we conclude that $U_{e}$ lies in the preimage of $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ in the embedding space $\operatorname{Emb}^{+}_{B}\!\left(\Sigma_{g},M\right)$. As it is by definition an open neighbourhood in the $C^{1}$ topology and the $C^{\infty}$ topology is finer than all the $C^{k}$ topologies we conclude that it is in fact an open neighborhood of $e$. ∎

Lemma 10.12**.**

If $V$ sits inside another coordinate chart $V^{\prime}$ and the coordinate change is given by $(x_{1},\ldots,x_{n})\mapsto({\mathrm{arctan}}\!\left({x_{1}}\right),\ldots,{\mathrm{arctan}}\!\left({x_{n}}\right))$, then the inclusion $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ is a homotopy equivalence for any $r\in(0,\infty)$ with homotopy inverse $\Psi_{1}$.

Proof.

By Theorem 42.3 and the corollary in 27.4 in [KM97] (together with the fact that the quotient of a nuclear spaces by a closed subspaces is a nuclear space) we conclude that $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ is paracompact and Hausdorff and by Lemma 27.8 in the same source, we conclude that it is actually metrizable (This line of argument was communicated to me in [MMO]). Therefore we can find a continuous function $\varphi_{\text{radius}}\colon\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)\to(0,r]$, which is $r$ on $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$ and for every $W$ it is smaller or equal than the maximal $r^{\prime}$ such that $\mathcal{E}^{+}_{B,R^{\prime}}\!\left(\Sigma_{g},M\right)$ contains $W$ (This uses the fact that $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$ is closed in $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$).

Let us now construct the homotopy inverse to the inclusion from $\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$. Let $\psi_{1}\colon(0,\infty)\to{\operatorname{Diff}^{+}_{c}}\!\left(M\right)$ denote the following continuous map into the compactly supported diffeomorphisms: Fix a smooth monotone function $\varphi\colon\mathbb{R}\to[0,1]$ such that $\varphi\equiv 0$ on an open neighborhood of $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ and $\varphi\equiv 1$ outside of $(-\pi,\pi)$. We define $\psi_{1}$ via the following formula:

[TABLE]

Since this is the identity outside the cube $(-\pi,\pi)^{n}$ we can consider this as a compactly supported diffeomorphism of $M$ by extending it via the identity. Let $\Psi_{1}$ denote

[TABLE]

This maps $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ into $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$. Furthermore this gives us a homotopy equivalence between $\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ and $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ since $\varphi_{\text{radius}}$ can be homotoped to be the map that is constantly $r$ via linear interpolation and such a homotopy induces maps $\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)\times[0,1]\to\mathcal{E}^{+}_{B}\!\left(\Sigma_{g},M\right)$ as in Lemma 10.9. Hence $\Psi_{1}$ is indeed a homotopy inverse to the inclusion. ∎

Lemma 10.13**.**

If $V$ sits inside another coordinate chart $V^{\prime}$ and the coordinate change is given by $(x_{1},\ldots,x_{n})\mapsto({\mathrm{arctan}}\!\left({x_{1}}\right),\ldots,{\mathrm{arctan}}\!\left({x_{n}}\right))$, then the inclusion $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B,r}\!\left(\Sigma_{g},M\right)$ is a homotopy equivalence with homotopy inverse denoted by $\Psi_{2}$.

Proof.

Fix a smooth function $\varphi_{1}\colon\mathbb{R}^{2}\to\mathbb{R}$ that is invariant under rotation and $1$ on $\overline{B_{R}\!\left(0\right)}$ and [math] outside of $B_{2R}\!\left(0\right)$. We define a map

[TABLE]

as follows: Note that $\pi_{2}$ extends to $V^{\prime}$. For an $x^{\prime}\in B_{2R}\!\left(0\right)$ we will denote the corresponding point in $W_{2R,x_{M}}$ by ${W}\!\left({x^{\prime}}\right)$. Fix a smooth monotone function $\varphi_{2}\colon[[0,\infty)\to\mathbb{R}$ such that ${\varphi_{2}^{-1}}\!\left({1}\right)=[0,\sqrt{n}\frac{\pi}{4}]$ and ${\varphi_{2}^{-1}}\!\left({0}\right)=(2\sqrt{n}\pi,\infty)$. Now we define

[TABLE]

Note that ${W}\!\left({x^{\prime}}\right)$ is only defined when $\left\lVert{{\pi_{2}}\!\left({x^{\prime}}\right)}\right\rVert\leq 2R$ but in that case ${\varphi_{1}}\!\left({{\pi_{2}}\!\left({x^{\prime}}\right)}\right)=0$. The map $\psi_{1}$ is well-defined i.e. the image is indeed a diffeomorphism, since a map

[TABLE]

for $v$ some vector and a positive, monotonously decreasing and bounded function $g$, is a diffeomorphism if $\max\left\lVert{g}\right\rVert\left\lVert{v}\right\rVert<1$ because this ensure the derivative to be an isomorphism which implies that the map is locally injective. Furthermore such a map extends to a degree $1$ map from $S^{2n-1}$ into itself and therefore it also has to be surjective. Hence it is a diffeomorphism, if the derivative of $g$ is small enough, but we can chose $g$ a posteriori with a small enough derivative.

Lastly we have to argue that $\psi_{2}$ is a continuous map as can be seen since it can be written in a neighborhood of $W$ in terms of the image of $W$ and the sections of the ”normal” bundle of $W$ (where we require the ”normal” bundle to have $\{0\}\times\mathbb{R}^{n-2}$ as fiber over $W_{2R,x_{M}}$), which is a neighborhood of $W$ in $\mathcal{E}^{+}_{B,2R}\!\left(\Sigma_{g},M\right)$.

We define ${H}\!\left({W,t}\right)={{\psi_{2}}\!\left({W,t}\right)}\!\left({W}\right)$. Note that $\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)$ includes into $\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)$ and with a similar construction as in Lemma 10.12 we can see that this inclusion is actually a homotopy equivalence with homotopy inverse, which is given by postcomposing with a diffeomorphism of $M$, denoted by $\Psi_{R,2R}$. Note that $H$ fixes ${\Psi_{R,2R}}\!\left({\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)}\right)$ and hence by Lemma 10.9, we conclude that the inclusion $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)$ followed by $\Psi_{R,2R}$ is a homotopy equivalence with homotopy inverse ${H}\!\left({-,1}\right)$. We get the following diagram:

{\mathcal{E}^{+}_{B,2R}\!\left(\Sigma_{g},M\right)}$${\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)}$${\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)}$$\scriptstyle{{H}\!\left({-,1}\right)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\Psi_{R,2R}}

This diagram shows that ${H}\!\left({-,1}\right)\circ\Psi_{R,2R}$ is a homotopy inverse to the inclusion of $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ into $\mathcal{E}^{+}_{B,R}\!\left(\Sigma_{g},M\right)$ and we denote it by $\Psi_{2}$. ∎

Lastly we have the following lemma which finishes the proof of Proposition 10.4.

Lemma 10.14**.**

The inclusion $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ is a homotopy equivalence with homotopy inverse denoted by $\Psi_{3}$.

Proof.

Consider the function $\varphi\colon\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)\to\mathbb{R}$ that sends $W$ to $\inf_{x^{\prime}\in W\setminus D^{2}_{R}}\left\lVert{x^{\prime}}\right\rVert$. Note that $\varphi$ is bigger than zero for every surface and bounded by $R$. Assume for now that this function is continuous. We define $\psi_{3}\colon[0,R]\times[0,1]\to{\operatorname{Diff}^{+}_{c}}\!\left(\mathbb{R}^{n}\right)$ as follows: ${\psi_{3}}\!\left({t_{1},t_{2}}\right)$ is the diffeomorphism that is the identity outside the ball of radius $2R$ and the identity for $t_{2}=0$, and ${\psi_{3}}\!\left({t_{1},t_{2}}\right)$ sends the ball of radius $t_{1}$ bijectively onto the ball of radius $Rt_{2}+(1-t_{2})t_{1}$ and finally only acts on the norm component if an element in $\mathbb{R}^{n}$ is represented in polar coordinates. We then define

[TABLE]

Then ${H}\!\left({-,1}\right)$ is a homotopy inverse to the inclusion of $\mathcal{E}^{+}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)$ into $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ by Lemma 10.9. We denote it by $\Psi_{3}$.

So the only thing left to prove is that $\varphi$ is continuous. Let $\operatorname{Emb}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ denote the subspace of those embeddings in $\operatorname{Emb}^{+}_{B}\!\left(\Sigma_{g},M\right)$ whose $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$-equivalence class lie in $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ and we denote by $\pi_{\operatorname{Emb}}$ the projection of this space to $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$. Let $\mathrm{ev}\colon\operatorname{Map}\!\left(X_{1},X_{2}\right)\times X_{1}\to X_{2}$ denote the canonical evaluation map. Let $\bar{A}={\mathrm{ev}^{-1}}\!\left({\left(D^{2}_{R}\right)^{c}}\right)\subset\operatorname{Emb}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)\times\Sigma_{g}$. $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ acts on $\operatorname{Emb}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ as well as $\bar{A}$ (Here we let $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ from the left on the $\Sigma_{g}$ component of $\bar{A}$). We denote the quotient of this action of $\bar{A}$ by $A$.

The projection onto the first factor from $\bar{A}$ to $\operatorname{Emb}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ descends to a map from $A$ to $\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ which we denote by $\pi_{A}$ and we claim that this map is a fiber bundle. Let $W\in\mathcal{E}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)$ denote an embedded subsurface and $N_{W,M}$ a tubular neighborhood of $W$ with corresponding projection $\pi_{W}$. Consider the following set

[TABLE]

which is open because $N_{W,M}$ is open and the space of diffeomorphisms is an open subset of $\operatorname{Map}\!\left(W^{\prime},W\right)$ (see Section 1.2 of the second chapter of [Cer61]). We will proof that $\pi_{A}$ is a locally trivial fibration. Note that

[TABLE]

are inverses of another. Here $e_{W}$ is any representative of $W$. Therefore we conclude that $\pi_{A}$ is a locally trivial fibration, whose fiber is given by $\Sigma_{g}$ with an open disc removed.

So we have the following diagram:

[TABLE]

where $\mathrm{ev}$ denotes again the evaluation map, which descends from $\operatorname{Emb}^{+}_{B,\text{almost}}\!\left(\Sigma_{g},M\right)\times\Sigma_{g}$ to $A$ and $\left\lVert{-}\right\rVert$ denotes the norm mapping in the chart $V$ extended by $\infty$ to the complement of $V$. We give $\mathbb{R}\cup\{\infty\}$ the topology coming from open sets in $\mathbb{R}$ and neighborhoods of $\infty$ have the form $(C,\infty]$. Note that ${\varphi}\!\left({W}\right)$ is given by taking the infimum over the composition of the horizontal maps for all elements in ${\pi_{A}}\!\left({W}\right)$ and this always lands in $(0,R]$. Since $\pi_{A}$ is locally a trivial fibration, Lemma 10.10 shows that $\varphi$ is locally continuous and therefore continuous. ∎

This finishes the proof of Proposition 10.4.

Remark 10.15**.**

At no point in the proof did we use that the submanifolds were two-dimensional. The corollary would also hold true for higher dimensional submanifolds.

To proceed with the construction of the stabilization maps we need the following corollary of our previous considerations:

Corollary 10.16**.**

There is a homotopy equivalence

[TABLE]

where $B_{\frac{R}{2}}\!\left(x_{M}\right)$ is the preimage of $B_{\frac{R}{2}}\!\left(x_{M}\right)\subset V$ in $M$ and $\delta^{T}$ is the preimage of the intersection of $B$ with the sphere of radius $\frac{R}{2}$ in $V$ with the tangential structure stemming from the constant one. We denote its homotopy inverse by $\Psi_{+B}$

Proof.

Removing $B_{\frac{R}{2}}\!\left(x_{M}\right)$ yields a map $\Phi\colon\mathcal{E}^{\theta}_{B,\text{flat}}\!\left(\Sigma_{g},M\right)\to{\mathcal{E}^{\theta}_{g,1}}\!\left(\overline{M\setminus B_{\frac{R}{2}}\!\left(x_{M}\right)};\delta^{T}\right)$ its image is the subspace of ${\mathcal{E}^{\theta}_{g,1}}\!\left(\overline{M\setminus B_{\frac{R}{2}}\!\left(x_{M}\right)};\delta^{T}\right)$ of those subsurfaces that intersect $V\setminus B_{\frac{R}{2}}\!\left(0\right)$ like $B\cap V\setminus B_{\frac{R}{2}}\!\left(0\right)$, and we will denote it by ${\mathcal{E}^{\theta}_{g,1}}\!\left(\overline{M\setminus B_{\frac{R}{2}}\!\left(x_{M}\right)};\delta^{T}\right)^{\text{flat}}$. Evidently $\Phi$ is a homeomorphism onto its image. Furthermore the inclusion ${\mathcal{E}^{\theta}_{g,1}}\!\left(\overline{M\setminus B_{\frac{R}{2}}\!\left(x_{M}\right)};\delta^{T}\right)^{\text{flat}}\to{\mathcal{E}^{\theta}_{g,1}}\!\left(\overline{M\setminus B_{\frac{R}{2}}\!\left(x_{M}\right)};\delta^{T}\right)$ is a homotopy equivalence, as can be seen using collars. ∎

We define the pointed stabilization maps $\sigma_{g}$ via the composition indicated in the following diagram:

[TABLE]

Note that we required the boundary condition of the lower right space of subsurfaces to agree with the boundary condition of the top right space.

In the pointed case, we say that a space of $\theta$-structures $\pi_{0}$-stabilizes at $g$ if $\sigma_{g^{\prime}}$ induces an isomorphism on $\pi_{0}$ provided that $g^{\prime}\geq g$ and a surjection for all $g^{\prime}\geq g-1$. Since $\sigma_{g}$ is defined in terms of homotopy equivalences (Corollary 10.6 and Corollary 10.16) and stabilization maps of type $\alpha$ and $\beta$, one immediately obtains (using Theorem 9.2 for the second part):

Theorem 10.17**.**

Suppose $(M,x_{M})$ is an at least $5$-dimensional, simply-connected, pointed manifold with a $k$-trivial space of $\theta$-structures of subplanes of $TM$ $\theta\colon B_{2}\!\left(M\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ which $\pi_{0}$ stabilizes at $m$, then the homology of $C\!\left(\sigma_{g}\right)$ vanishes in degrees $\ast\leq\min\left\{{A}\!\left({g}\right),{B}\!\left({g}\right)\right\}$ where $A$ and $B$ are defined as in Definition 7.1.

In particular if $\mathrm{Fib}_{B}\!\left(\theta\right)$ is simply-connected, then the homology of $C\!\left(\sigma_{g}\right)$ vanishes in degrees $\ast\leq\left\lfloor\frac{2}{3}g\right\rfloor$.

11. Homological Stability for Symplectic Subsurfaces

In this last section we will explain how to use Theorem 10.17 to prove homological stability for pointed symplectic subsurfaces of a given closed simply-connected symplectic manifold $(M,\omega)$ (i.e. $\omega$ is a point-wise non-degenerate $2$-form on $M$) of dimension $2n\geq 6$. The proof will use a classical theorem, that is proven via the h-principle, to relate symplectic subsurfaces to tangential structures as they were defined in the previous parts of this paper.

Let $\pi_{\operatorname{Gr}}\colon\operatorname{Gr}_{2}\!\left(TM\right)\to M$ denote the Grassmannian of oriented two-planes in $TM$ and $\Sigma_{g}$ an oriented, connected, closed surface of genus $g$. Fix $x\in\Sigma_{g}$, $x_{M}\in M$ and $B_{\omega}\in\operatorname{Gr}_{2}\!\left(T_{x_{M}}M\right)$ such that $\left.{\omega}\right\rvert_{B_{\omega}}$ is non-degenerate and the orientation of $B_{\omega}$ agrees with the orientation induced by $\left.{\omega}\right\rvert_{B_{\omega}}$.

Definition 11.1**.**

We say an embedding $e\colon\Sigma_{g}\to M$ is an oriented, pointed symplectic embedding if

1. (i)

${e}\!\left({x}\right)=x_{M}$ and ${De}\!\left({T_{x}\Sigma_{g}}\right)=B_{\omega}$ 2. (ii)

$\left.{\omega}\right\rvert_{{De}\!\left({T\Sigma_{g}}\right)}$ is non-degenerate. 3. (iii)

$\int_{\Sigma_{g}}e^{*}\omega>0$

We equip the set of symplectic embeddings with the $C^{\infty}$-topology and we denote this space by $\text{SEmb}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$.

Similarly we say that a smooth map $F\colon\Sigma_{g}\to\operatorname{Gr}_{2}\!\left(TM\right)$ is a formal solution to the oriented symplectic embedding problem if

1. (i)

$\pi_{\operatorname{Gr}}\circ F$ is an embedding such that ${(\pi_{\operatorname{Gr}}\circ F)}\!\left({x}\right)=x_{M}$ and ${D(\pi_{\operatorname{Gr}}\circ F)}\!\left({T_{x}\Sigma_{g}}\right)={F}\!\left({x}\right)=B_{\omega}$ 2. (ii)

$\left.{\omega}\right\rvert_{{F}\!\left({x^{\prime}}\right)}$ is non-degenerate for every point $x^{\prime}\in\Sigma_{g}$ 3. (iii)

$\int_{\Sigma_{g}}(\pi_{\operatorname{Gr}}\circ F)^{*}\omega>0$

Note that the last condition ensures that the orientations of $\Sigma_{g}$ induced by $\omega$ and by the orientation of $\Sigma_{g}$ agree.

Remark 11.2**.**

For an oriented, pointed symplectic embedding $e\colon\Sigma_{g}\to M$, the Grassmannian differential $\operatorname{Gr}\!\left(De\right)\colon\Sigma_{g}\to\operatorname{Gr}_{2}\!\left(TM\right)$ is a formal solution to the oriented symplectic embedding problem.

Definition 11.3**.**

We call a continuous map $H\colon\Sigma_{g}\times[0,1]\to\operatorname{Gr}_{2}\!\left(TM\right)$ a solution homotopy if

1. (i)

There exists an embedding $e\colon\Sigma_{g}\to M$ such that ${H}\!\left({-,0}\right)$ agrees with $\operatorname{Gr}\!\left(De\right)$, ${e}\!\left({x}\right)=x_{M}$ and ${De}\!\left({T_{x}M}\right)=B_{\omega}$ 2. (ii)

The following diagram commutes:

[TABLE] 3. (iii)

For all $t\in[0,1]$, we have ${H}\!\left({x,t}\right)=B_{\omega}$ 4. (iv)

${H}\!\left({-,1}\right)$ is a formal solution to the oriented symplectic embedding problem

The set of solution homotopies is a subspace of $\operatorname{Emb}\!\left(\Sigma_{g},M\right)\times\operatorname{Map}\!\left(\Sigma_{g},\operatorname{Map}\!\left([0,1],\operatorname{Gr}_{2}\!\left(TM\right)\right)\right)$ and hence inherits a topology, where the first factor is equipped with the $C^{\infty}$ topology and the second one with the compact-open topology.

Note that there is an inclusion from the space of symplectic embeddings into the space of solution homotopies by sending an embedding to the constant solution homotopy over that embedding. The proof of the following theorem can be found in Section 12 of [EM02].

Theorem 11.4**.**

If $(M,\omega)$ is an at least six-dimensional symplectic manifold, then the inclusion of the symplectic embeddings into the space of solution homotopies is a weak equivalence.

11.1. Interpreting Solution Homotopies as Tangential Structures:

Inspired by Theorem 11.4, we want to construct a space of $\theta$-structures of subplanes of $TM$ such that the space of subsurfaces with that tangential structures is the space of solution homotopies.

We call an element $B_{\omega}\in\operatorname{Gr}_{2}\!\left(TM\right)$ a symplectic $2$-plane if $\left.{\omega}\right\rvert_{B_{\omega}}$ is non-degenerate and we denote by $\operatorname{Gr}^{\omega}_{2}\!\left(TM\right)$ the (open) subspace of symplectic two-planes of $TM$. Analogously we define $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)$. For a $B_{\omega}^{\prime}\in\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ we will denote the homotopy fiber of the inclusion $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ by $\mathcal{P}_{2}\!\left(M\right)_{B_{\omega}^{\prime}}$. More explicitly this is the space of paths in $\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ starting at $B_{\omega}^{\prime}$ and ending in $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)$. If we allow $B_{\omega}^{\prime}$ to vary, we obtain the mapping path space $\mathcal{P}_{2}\!\left(M\right)_{x^{\prime}}$ of the inclusion $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$, i.e.

[TABLE]

This space is a subset of the following space, which is equipped with the compact open topology

[TABLE]

If we trivialize $\operatorname{Gr}_{2}\!\left(TM\right)\to M$ via a Darboux chart defined on $U\subset M$, we conclude that the inclusion $\operatorname{Gr}^{\omega}_{2}\!\left(TM\right)\to\operatorname{Gr}_{2}\!\left(TM\right)$ is locally equivalent to $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\times U\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)\times U$. This implies that

[TABLE]

is locally equivalent to $\mathcal{P}_{2}\!\left(M\right)_{x^{\prime}}\times U\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)\times U$, which implies that this map is a Hurewicz-fibration as it is the product of two Hurewicz-fibrations, one being the projection of the mapping path space $\mathcal{P}_{2}\!\left(M\right)_{x^{\prime}}\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ and the other one being the identity. Using Theorem 13 in Chapter 2.7 of [Spa95], we conclude that it is indeed a Hurewicz-fibration as it is locally a Hurewicz-fibration and the basespace $\operatorname{Gr}_{2}\!\left(TM\right)$ is paracompact.

Lemma 11.5**.**

The fiber of

[TABLE]

is simply-connected.

Proof.

The claim that $\theta_{\omega}$ is indeed a fibration and hence a space of $\theta_{\omega}$-structures of subplanes was just shown. Using Proposition 6.9 and Corollary 9.4 it suffices to show that the fiber of $\theta_{\omega}$ is simply-connected:

The fiber over an oriented 2-plane $B_{\omega}^{\prime}$ in $T_{x^{\prime}}M$ is given by the set of paths in $\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ from $B_{\omega}^{\prime}$ to $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)$. As we have mentioned before, this is the homotopy fiber of $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$. Choose some almost complex structure $J$ on $TM$ compatible with the symplectic form $\omega$. Let $V^{\mathbb{R}}_{2}(T_{x^{\prime}}M)$ denote the compact $2$-Stiefel manifold i.e. the space of real (orthogonal) $2$-frames of $T_{x^{\prime}}M$ and let $V^{symp}_{2}(T_{x^{\prime}}M)$ denote the space of $2$-frames $(v_{1},v_{2})$ such that ${\omega}\!\left({v_{1},v_{2}}\right)=1$. Furthermore define $V^{\mathbb{C}}_{1}(T_{x^{\prime}}M)$ as the compact Stiefel manifold of complex $1$-frames (i.e. just non-zero vectors). Then we have an inclusion

[TABLE]

and we claim that this is a homotopy equivalence with homotopy inverse given by the projection on the first vector. Indeed the homotopy is given by

[TABLE]

which is easily seen to be well-defined. Therefore we get the following maps between fiber sequences:

{S^{1}\simeq Sp(2,\mathbb{R})}$${SO(2)\cong S^{1}}$${V^{\mathbb{C}}_{1}(T_{x^{\prime}}M)\simeq V^{symp}_{2}(T_{x^{\prime}}M)}$${V^{\mathbb{R}}_{2}(T_{x^{\prime}}M)}$${\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)}$${\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)}

Note that $V^{\mathbb{C}}_{1}(T_{x^{\prime}}M)\cong S^{2n-1}$ and $V^{\mathbb{R}}_{2}(T_{x^{\prime}}M)\cong STS^{2n-1}$ (the unit sphere bundle of the tangent bundle of $S^{2n-1}$). Since both of these spaces are $(2n-3)$-connected and the top horizontal map is a homotopy equivalence, we conclude that the map from $\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\to\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right)$ induces an isomorphism on $\pi_{i}$ for $i\leq 3$ or in other words $\pi_{i}\!\left(\operatorname{Gr}_{2}\!\left(T_{x^{\prime}}M\right),\operatorname{Gr}^{\omega}_{2}\!\left(T_{x^{\prime}}M\right)\right)$ for $i\leq 2$, which implies that the homotopy fiber is two-connected, since

[TABLE]

∎

11.2. Spaces of Symplectic Subsurfaces and Homological Stability:

The group $\operatorname{Diff}_{x}\!\left(\Sigma_{g}\right)$ acts freely on $\text{SEmb}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$ via precomposition, since being a symplectic embedding is independent of the parametrization. We denote the quotient by this group action by $\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$.

Elements of this space are referred to as pointed symplectic subsurfaces. By a slight abuse of notation, let $B_{\omega}$ also denote the path in $\mathcal{P}_{2}\!\left(M\right)$ that is constantly $B_{\omega}$. The inclusion of the space of symplectic embeddings into the space of solution homotopies descends to an inclusion

[TABLE]

Since Condition (iii) in Definition 11.1 implies that $\omega$ evaluates positively on the fundamental class of pointed symplectic subsurfaces, this inclusion can not be $\pi_{0}$-surjective. In order to use this inclusion later on we will alter its image. We define:

[TABLE]

With these definition the following corollary follows immediately:

Corollary 11.6**.**

$\operatorname{Emb}^{\theta_{\omega}}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$* is the space of solution homotopies.*

Observe that the stabilization map of $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$ defined in Section 10 restricts to $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$ namely

[TABLE]

since for every $W\in\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$ we have $[W]=[{\sigma_{g}}\!\left({W}\right)]\in H_{2}\!\left(M\right)$ and if $[\omega]$ denotes the real cohomology class corresponding to $\omega$, then $\int_{W}\omega>0$ is equivalent to ${[\omega]}\!\left({[W]_{\mathbb{R}}}\right)>0$, where $[W]_{\mathbb{R}}$ denotes the real fundamental class. Since $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$ is a union of connected components of $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$ and $(\sigma_{g})_{*}\colon\pi_{0}\!\left(\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)\right)\to\pi_{0}\!\left(\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g+1},M\right)\right)$ is an isomorphism, Theorem 10.17 implies that homological stability holds for $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$ as well i.e.

[TABLE]

is an isomorphism for $\ast\leq\frac{2}{3}g-1$ and an epimorphism in the next degree.

Remark 11.7**.**

The same arguments show that even in the general case, homological stability still holds for all kinds of homological constraints we could put on $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g+1},M\right)$ as long as these constraints depend only on $H_{2}\!\left(M\right)$.

The following proposition allows us to relate homological stability for the space of solution homotopies to $\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g},M\right)$.

Proposition 11.8**.**

$\mathcal{S}_{B_{\omega}}\!\left(\Sigma_{g},M\right)\to\mathcal{E}^{\theta_{\omega}}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$* is a weak equivalence.*

Proof.

We have the following map between fibrations:

[TABLE]

Note that the lower right map is a fibration by 10.2, the lower map on the left side is a fibration as a restriction of this fibration and the horizontal arrow in the middle is a weak equivalence by Theorem 11.4. Therefore we conclude from the $5$-Lemma and the long exact sequence of homotopy groups for a fibration that the inclusion is in fact a weak equivalence. ∎

This proposition together with Theorem 10.17 and the previously discussed fact that $\mathcal{E}^{\theta}_{B_{\omega}}\!\left(\Sigma_{g},M\right)^{\omega}$ also fulfils homological stability immediately implies:

Theorem 11.9**.**

Let $(M,\omega)$ denote a simply-connected symplectic manifold of dimension at least $6$. For every $x_{M}\in M$ and every symplectic $2$-plane $B_{\omega}$ in $T_{x_{M}}M$. There is a homomorphism of integral homology

[TABLE]

And this homomorphism induces an isomorphism for $\ast\leq\frac{2}{3}g-1$ and an epimorphism for $\ast\leq\frac{2}{3}g$.

The main problem here is that the present text does not provide an actual stabilization map for symplectic subsurfaces in the spirit of the pointed stabilization maps $\sigma_{g}$, but only an abstract one that uses the inverses of the aforementioned weak equivalence and therefore only exists on the level of homology.

Nevertheless the author is confident that the construction in Section 10 can be adapted to work in the symplectic setting. One would have to be more careful in choosing the deformations so that the embedded subsurfaces stay symplectic.

Furthermore there are nice symplectic embeddings of tori with a hole into $\mathbb{R}^{2n}$, which should provide the necessary tori in $B_{R}\!\left(0\right)$ to provide an actual stabilization map in the symplectic case, which would be defined like $\sigma_{g}$ i.e. it would be defined by taking the connected sum with one of these symplectic tori and map symplectic subsurfaces to symplectic subsurfaces.