Imploded cross-sections

TL;DR

This survey introduces imploded cross-sections to address non-symplectic cross-sections in Hamiltonian spaces, comparing their intersection homology with homology intersection spaces and computing related homologies.

Contribution

It provides a comprehensive overview of imploded cross-sections and their homological properties, including explicit computations for specific geometric cases.

Findings

Imploded cross-sections can be used to analyze Hamiltonian K-spaces.

Intersection homology differs from homology intersection spaces in certain examples.

Homology of intersection spaces for cones and suspensions of manifolds is explicitly computed.

Abstract

In this survey article, we describe imploded cross-sections, which were developed in order to solve the problem that the cross-section of a Hamiltonian $K$-space is usually not symplectic. In some specific examples we contrast the intersection homology of some imploded cross-sections with their homology intersection spaces. Moreover, we compute the homology of intersection spaces associated to the open cone of a simply connected, smooth, oriented manifold and the suspension of such a manifold.

Equations

$$W=\Phi^{-1}\Bigl{(}B_{\epsilon}(\alpha)\Bigr{)}\cap U$$

$$\varphi ([z_0:\cdots :z_n]{)}_{ij}=\frac{\sqrt{-1}{z}_i\overline{z_j}}{2\pi {\sum }_{k=0}^n|z_k{|}^2}.$$

$$\Phi :(u_1,\dots ,u_N)\mapsto \sum _{i=1}^N{u}_j.$$

$$M_{impl}=M/\sim ,$$

$$M_{impl}\cong (M\times (T^{\ast }K{)}_{impl}){\text{sslash}}_{0}K,$$

$$\{(z,w)\in {\mathbb{C}}^3\times {\mathbb{C}}^3\mid z\cdot w=0\}.$$

$$Y=\{(z,w)\in {\mathbb{C}}^3\times {\mathbb{C}}^3\mid z\cdot w=0,|z{|}^2+|w{|}^2=1\}.$$

$${\Phi }^{-1}({\sigma }^0)\cong K\times (0,\pi ).$$

$$M$$

$${\mathbf{k}}^{\ast }$$

$$\pi :T⟶\Sigma$$

$$\rho :T⟶[0,\infty )$$

$$L\to v.$$

$$\theta :N\setminus \Sigma \cong L\times \Sigma \times (0,1).$$

$$\overline{M}=(X\setminus \Sigma ){\cup }_{\theta }(L\times \Sigma \times [0,1)).$$

$$CT(X)=\frac{(X\setminus \Sigma ){\cup }_{\theta }(L\times \Sigma \times [0,1))}{(z,y,0)\sim (z,y^{\prime },0)}$$

$$CT(X)=\overline{M}\cong L\times [0,1).$$

$$p:\{2,3,\dots ,\}\to \{0,1,\dots \}$$

$$p(2)=0$$

$$p(k)\le p(k+1)\le p(k)+1$$

$$\dim (\sigma \cap X_{n-k})\le j-k+p(k)$$

$$f_{\ast }:H_i(L_{<k})⟶H_i(L)$$

$$L_{<k}\times \Sigma \stackrel{f\times i{d}_{\Sigma }}{\to }L\times \Sigma =\partial \overline{M}\hookrightarrow \overline{M}.$$

$$I^{\overline{p}}X=\mathrm{cone}(g)=\overline{M}{\cup }_{g}c(L_{<k}\times \Sigma ).$$

$$\tilde{H}{I}_i^{\overline{p}}(X)={\tilde{H}}_i(I^{\overline{p}}X;\mathbb{R})$$

$$H{I}_{\ast }^p(X)=\{\begin{array}{ll}{H}_j(\overline{M},\partial \overline{M})& j<k\\ H_j(\overline{M})& j>k\end{array}$$

$$\begin{diagram}$$

$${\tilde{H}}_{\ast }(\overline{M})={\tilde{H}}_{\ast }(L),$$

$${\tilde{H}}_{\ast }(\overline{M},\partial \overline{M})={\tilde{H}}_{\ast }(\overline{M}/\partial \overline{M})={\tilde{H}}_{\ast }(c^o(L))=0.$$

$$I{\tilde{H}}_j^m((T^{\ast }SU(3){)}_{\mathrm{impl}})=\{\begin{array}{ll}\mathbb{R}& j=4\\ 0& \mathrm{otherwise}.\end{array}$$

$$c^{\circ }(X)=\frac{(0,1]\times X}{(1,x)\sim (1,x^{\prime })}.$$

Full text

1. 1 Introduction

2. 2 Imploded cross-sections

3. 2.1 Symplectic cross-section theorem

4. 2.2 Examples

5. 3 Universal imploded cross-section

6. 3.1 Introduction

7. 3.2 Universal implosion

8. 3.2.1 Universal imploded cross-section of $SU(2)$

9. 3.2.2 Universal imploded cross-section of $SU(3)$

10. 3.2.3 Quasi-Hamiltonian analogue for $SU(2)$

11. 4 Further work on symplectic implosion

12. 4.1 Hyperkähler implosion: the work of Dancer, Kirwan and Swann

13. 4.2 Derived geometry and implosion: the work of Safronov

14. 4.3 Symplectic cuts: the work of Martens and Thaddeus

15. 4.4 The work of Howard-Manon-Millson

16. 4.5 Symplectic contractions: the work of Hilgert-Manon-Martens and Lane

17. 4.6 Canonical bases and collective integrable systems: the work of Hoffman and Lane

18. 5 Topological invariants of the universal imploded cross-section

19. 5.1 Conifold transition and blow-up manifold

20. 5.2 Intersection homology and perversities

21. 5.3 Homology intersection spaces

22. 5.4 Universal imploded cross-section for $SU(3)$

23. 5.5 Intersection homology of a cone

24. 5.6 Proof of Theorem 5.13

25. A Intersection space associated to a suspension

26. A.1 Proof of Theorem A.1

1. Introduction

Let $(M,\omega)$ be a Hamiltonian $K$-manifold, where $K$ is a compact Lie group with maximal torus $T$. In other words, we assume $M$ is equipped with a symplectic structure and with the action of a group $K$ which preserves the symplectic structure and is generated by the Hamiltonian flow of a collection of functions (the moment maps for the group action). Denote the Lie algebras of $K$ and $T$ by ${\bf k}$ and ${\bf t}$ respectively. Let the moment map be denoted $\Phi:M\to{\bf k}^{*}$.

The preimage $\Phi^{-1}({\bf t}^{*})$ is in general not a manifold, and even where it is a manifold it is usually not symplectic. The purpose of the imploded cross-section construction [18] was to construct a Hamiltonian $T$ space $M_{\rm impl}$ whose symplectic quotients (with respect to the $T$ action) correspond to the symplectic quotients of $M$ with respect to its $K$ action.

In this survey article, we outline the construction of imploded cross-sections, focusing on the universal imploded cross-section, which is the imploded cross section of the cotangent bundle of $K$. The universal imploded cross-section could be seen as an analogue of the symplectic realization of ${\bf k}^{*}$. The universal imploded cross-section plays as important a role in the context of symplectic manifolds as ${\bf k}^{*}$ plays among Poisson manifolds.

The imploded cross-section can be studied using intersection cohomology. We can also study it using the homology intersection space construction [5]. We examine differences between these constructions, focusing on the universal implosion.

The layout of this article is as follow. In Section 2 we give general definitions and motivation for this construction. In Section 3 we describe the uses of the imploded cross-section. In Section 3.2 we characterize the universal imploded cross-section. In Section 4 we describe the work of Dancer, Kirwan and Swann (see for example [9]) on hyperkähler implosions. We discuss the work of Safronov [44] relating imploded cross sections with derived geometry. We also describe the work of Martens-Thaddeus [40] on the universal nonabelian symplectic cut. Next we review the work of Howard, Manon and Millson [22] about bending flows, and show how this work can be described in terms of imploded cross-sections. We outline the work of Hilgert-Manon-Martens [22] on symplectic contraction, and related work of Jeremy Lane [34]. Finally we review recent work of Hoffman and Lane on toric degenerations and integrable systems [25] which was described in [24] and [36].

In Section 5 we review intersection homology and describe the intersection homology of a universal imploded cross-section of $SU(3)$ (joint work of the first author with Nan-Kuo Ho [23]). In subsection 5.1 we contrast intersection homology (IH) with the homology intersection spaces (HI) of Banagl and Hunsicker [5]. In subsection 5.2 we outline intersection homology. In subsection 5.3 we outline the construction of homology intersection spaces. In subsection 5.4 we outline the construction of the universal imploded cross-section of $SU(3)$, while in subsection 5.5 we describe the intersection homology of a cone. In subsection 5.6 we give a computation of the HI of the universal imploded cross-section studied above. Finally, in the Appendix A we compute the HI of suspensions, by methods similar to those from subsection 5.6.

The authors thank Markus Banagl for guidance on Section 5 of the paper, and thank Reyer Sjamaar for suggestions regarding Section 4. They would also like to thank Ben Hoffman and Jeremy Lane for providing the detailed description of their work that appears in Section 4.6 below.

2. Imploded cross-sections

In this section and the next, we follow [18].

We fix $K$ to be a compact connected Lie group. Let $(M,\omega)$ be a Hamiltonian $K$-manifold with equivariant moment map $\Upphi:M\rightarrow{\bf k}$. Here ${\bf k}$ denotes the Lie algebra of the group $K$. Moreover, we assume that $T$ is a maximal torus of $K$ and ${\bf t}_{+}^{*}$ is is a chosen fundamental Weyl chamber in ${\bf t}^{*}$. Here ${\bf t}$ denotes the Lie algebra of the maximal torus $T$ of $K$.

Symplectic manifolds with Hamiltonian $K$ actions are parametrizations of systems equipped with a symmetry group $K$. It is desirable to divide out by the symmetry group to obtain a simpler system. However, in general the new system will not be symplectic.

If [math] is a regular value of the moment map, and $K$ acts freely on $\Phi^{-1}(0)$, then the symplectic quotient $\Phi^{-1}(0)/K$ is a smooth manifold. If [math] is a regular value of the moment map, then $K$ acts with finite stabilizers at all points of $\Phi^{-1}(0)$ and $\Phi^{-1}(0)/K$ is an orbifold, in other words a topological space that is locally homeomorphic to the quotient space of an open subset of a smooth manifold by a finite group action.

If we instead want a space that parametrizes not the symplectic quotient at [math], but rather symplectic quotients at other orbits in ${\bf k}^{*}$, then it makes sense to look at $\Phi^{-1}(\mathcal{O}_{\lambda})/K$ for a general orbit $\mathcal{O}_{\lambda}$ of the coadjoint action in ${\bf k}$. If we take the preimage of one such orbit of the coadjoint action under the moment map and then take the quotient by the action of $K$, in general we recover an orbifold, or in good situations a smooth manifold. For example, for $K=SO(3)$, the Lie algebra ${\bf k}^{*}$ is identified with ${\bf R}^{3}$ and the coadjoint action of $K$ on it is the rotation action. In this case, the orbits $\mathcal{O}_{\lambda}$ are 2-spheres with center [math] through the points $0\neq\lambda\in{\bf t}^{*}$.

2.1. Symplectic cross-section theorem

In general there is an open subset $U$ of ${\bf t}^{*}$ for which $\Phi^{-1}(U)/T$ is foliated by the symplectic quotients $\Phi^{-1}(\lambda)/T$ for $\lambda\in U$. Moreover, for each $\lambda\in{\bf t}^{*}$, we have $\Phi^{-1}(\lambda)/T\cong\Phi^{-1}(\mathcal{O_{\lambda}})/K.$ The symplectic cross-section theorem of Guillemin and Sternberg ([19], Section 41) states the following.

Let $M$ be a Hamiltonian $K$-manifold, where $K$ is a compact connected Lie group. Let $T$ be a maximal torus of $K$ with Lie algebra ${\bf t}$. Denote the moment map for the $K$ action on $M$ by $\Phi:M\to{\bf k}^{*}$.

Let $\alpha$ be a point in ${\bf t}^{*}$. Let $p$ be a point in $M$ with $\alpha=\Phi(p)$. Let $B_{\epsilon}(\alpha)$ be a ball of radius $\epsilon$ in ${\bf t}^{*}$ around $\alpha$. By [19] Theorem 26.7, there is a $T$-invariant neighbourhood $U$ of $\Phi^{-1}(\alpha)$ in $M$ such that

[TABLE]

is a symplectic submanifold of $M$, where $B_{\epsilon}(\alpha)$ is a ball in ${\bf k}^{*}$ with center $\alpha$ and radius $\epsilon$. Then $W$ is $T$-invariant and the action of $T$ on $W$ is Hamiltonian and the moment map is just the restriction of $\Phi:W\to{\bf t}^{*}$. The space $W$ is called a slice for the $G$ action at $p$.

Every $K$-orbit in $M$ that intersects $W$ intersects it in a single $T$-orbit ([19], Proposition 41.2)

Finally $W$ can be reconstructed from $\alpha$ and the isotropy representation of $T$ on $TW_{p}$, as a Hamiltonian $T$ space ([19], Theorem 41.2).

2.2. Examples

Example 2.1**.**

1. (1)

The space ${\bf C}P^{1}$, equivalently $S^{2}$, is a coadjoint orbit of the rotation group $SO(3).$ The moment map is the inclusion map into ${\bf R}^{3}$. We may choose a maximal torus of $SO(3)$ so that the dual of the Lie algebra of the maximal torus is identified with the vertical axis $\{(0,0,s)|s\in{\bf R}\}$ in the dual of Lie algebra of $SO(3)$, which is identified with ${\bf R}^{3}$. The preimage of the Lie algebra of ${\bf t}^{*}$ under $\Phi$ consists of two points, the north and south poles. 2. (2)

The group $U(n+1)$ acts on the space ${\bf C}P^{n}$ by right multiplication. The moment map for this action is

[TABLE]

The preimage of ${\bf t}^{*}$ is the subspace where the image of the moment map is a diagonal matrix with entries in $\sqrt{-1}{\bf R}$.

This means $z_{i}\bar{z_{j}}=0$ if $i\neq j$, and hence that at least one of $z_{i}$ or $z_{j}$ is [math]. This is only possible if all but one of $z_{i}$ are [math]. These points of ${\bf C}P^{n}$ are the fixed points of the action of the maximal torus by right multiplication. In this case the preimage of ${\bf t}^{*}$ under the moment map consists of isolated points. 3. (3)

We now consider the product of a collection of spheres. Let $K=SU(2)$ act diagonally on a space $M$ which is the product of $N$ copies of the sphere ${\bf C}P^{1}$. The moment map is the sum

[TABLE]

Here, each $u_{j}$ is regarded as a point of ${\bf k}^{*}\cong{\bf R}^{3}$ (so the sum $\sum_{j}u_{j}$ makes sense). Each $u_{j}$ satisfies $|u_{j}|=1$ for each $j$ since each $u_{j}$ is a member of $S^{2}$.

In this case the condition that the moment map takes values in ${\bf t}^{*}$ is that the sum $\sum_{j}u_{j}$ is in ${\bf t}^{*}$, in other words that this sum is on a chosen axis in ${\bf R}^{3}$, for example the vertical axis.

Notice that the space of products of spheres has a subspace of $\Phi^{-1}({\bf t}^{*})$ for which the restriction of the symplectic form from the product of spheres is not everywhere symplectic. This restriction is of course closed, but it may be degenerate. Fix a point $x\in\Phi^{-1}({\bf t}^{*})\subset M$.

Taking a basis for the tangent space to $\Phi^{-1}({\bf k}^{*})$ at $x$, denote by $\iota:\Phi^{-1}({\bf t}^{*})\to\Phi^{-1}({\bf k}^{*})$ the inclusion map, and its adjoint $\pi:=\iota^{*}$ is the projection from the tangent space to $\Phi^{-1}({\bf k}^{*})$ at $x$ to the tangent space to $\Phi^{-1}({\bf t}^{*})$ at $x$ using the chosen invariant inner product. Denote by $A_{x}$ the matrix that represents the symplectic form in this basis. Then the condition that the restriction of the symplectic form is degenerate at $x$ is that the matrix $\pi\circ A_{x}\circ\iota$ is degenerate, in other words that the determinant of $\pi\circ A_{x}\circ\iota$ is [math]. This means that one particular minor of this matrix is [math]. This minor is the determinant of a square submatrix of real codimension $\dim(K)-\dim(T)$ $:=\ell$. In appropriate coordinates on the tangent space to $M$ at $x$, it is the determinant of the first $2N-\ell$ rows and the first $2N-\ell$ columns of the matrix.

Example 2.2** (Actions on coadjoint orbits).**

[35]**

Jeremy Lane has studied the action of $SU(n-1)$ on a coadjoint orbit of $SU(n)$ where the action arisis from the inclusion of $SU(n-1)$ in $SU(n)$ and the coadjoint action of $SU(n)$ on the orbit. He finds that there are loci where the symplectic form on the orbit becomes degenerate when restricting to the dual of the Lie algebra of the maximal torus of $SU(n-1)$. For example, Lane finds that for $n=3$, the symplectic cross-section of such an orbit is a Lagrangian submanifold of dimension $3$.

When taking a symplectic quotient at a value of the moment map which is not a regular value, the symplectic quotient is a stratified symplectic space [45]. In other words, the symplectic quotient is not a smooth manifold, but decomposes into strata each of which is a smooth manifold and has a symplectic structure. The preimage of the Lie algebra of the maximal torus under the moment map (called the symplectic cross-section) is also a stratified space, but the strata are not necessarily symplectic. The imploded cross-section is designed to repair this so that each stratum of the preimage of the maximal torus under the moment map has a symplectic structure.

Define a relation $\sim$ on $\Upphi^{-1}({{\bf t}}_{+}^{*})$ as follows:

Definition 2.3**.**

*Let $K$ act on ${\bf k}^{*}$ by the coadjoint action. Then $m_{1}\sim m_{2}$ if there exists $k\in[K_{\Upphi(m_{1})},K_{\Upphi(m_{2})}]$ such that $k\cdot m_{1}=m_{2}$, where $k\cdot m$ denotes the image of the action of $k$ on $m$. *

It turns out that this defines an equivalence relation on $\Upphi^{-1}({{\bf t}^{*}})$. Indeed, by equivariance of the moment map $\Upphi$, $m_{1}\sim m_{2}$ implies that $K_{\Upphi(m_{1})}=K_{\Upphi(m_{2})}$ and therefore this equivalence relation is transitive.

Definition 2.4**.**

The symplectic implosion $M$, denoted by $M_{impl}$, is defined as

[TABLE]

where $\sim$ is the above equivalence relation. This space is equipped with the quotient space topology.

One can lift the left action of $K$ on itself to a Hamiltonian action on the cotangent bundle $T^{*}K$. The implosion of the cotangent bundle, $(T^{*}K)_{impl}$, is called the universal imploded cross-section of $K$. The following theorem explains why this space is called ``universal":

Theorem 2.5**.**

([18], Theorem 4.9) For any Hamiltonian $K$-manifold $M$, there exists an isomorphism

[TABLE]

where $\sslash_{0}$ denotes the symplectic quotient and the symplectic quotient is with respect to the diagonal action of $K$.

The above theorem tells us that the imploded cross-section of $M$ is the same as the symplectic quotient at [math] of the product of $M$ with the imploded cross-section of the cotangent bundle of $K$ (in other words the universal imploded cross-section of $K$).

3. Universal imploded cross-section

3.1. Introduction

Let $K$ be a compact connected Lie group. The symplectic quotient of a symplectic manifold equipped with a Hamiltonian $K$ action is a stratified symplectic space (in other words each stratum is equipped with a symplectic structure [45]), but the preimage of the dual of the Lie algebra of the maximal torus $T$ under the moment map is not necessarily symplectic. The imploded cross-section of such a manifold has the property that the intersection of each stratum with the preimage of the Lie algebra of the dual of the maximal torus under the moment map is symplectic.

3.2. Universal implosion

The universal imploded cross-section is the imploded cross-section of the cotangent bundle of a Lie group. It has the property that the symplectic quotient at [math] of the product of a Hamiltonian $K$-manifold $M$ with the universal imploded cross-section of $T^{*}K$ is the imploded cross-section of $M$. The definition of the universal imploded cross-section was stated in [18].

3.2.1. Universal imploded cross-section of $SU(2)$

The universal imploded cross-section of $K=SU(2)$ is a copy of ${\bf C}^{2}$, as shown in Example 4.7 of [18]. One way to describe this correspondence is that the cotangent bundle of $K$ may be identified with $K\times{\bf k}^{*}$, with the moment map given by projection on ${\bf k}^{*}$. Hence the preimage of ${\bf t}^{*}$ is $K\times{\bf t}^{*}$. The implosion is obtained by collapsing the fiber above the identity element $e$ by the action of the commutator subgroup $[K,K]$. But $[K,K]=K$ so this action collapses $K\times\{0\}$ to a point. Hence the imploded cross-section is $K\times{\bf R}/\sim$. Here the equivalence relation identifies $(x,0)\sim(y,0)$. If $s\neq 0$ and $t\neq 0$, then $(x,t)\sim(y,s)$ implies $t=s$ and $x=y$. This identifies the imploded cross-section as the cone on $S^{3}$, in other words ${\bf C}^{2}$.

3.2.2. Universal imploded cross-section of $SU(3)$

As described in Example 6.16 in [18], the universal imploded cross-section of $SU(3)$ has a structure of an irreducible affine complex variety which is given by

[TABLE]

This space has an isolated singularity at $(0,0)$. It turns out that this space is homeomorphic to the open cone over the compact connected Riemannian manifold

[TABLE]

We show in [23] (Theorem 4.2) that the space $Y$ decomposes as the union of two spaces $W$ and $X$, where the disjoint union of two copies of $S^{5}$ is a deformation retraction of $W$, while $SU(3)$ is a deformation retraction of $X$ and the disjoint union of two copies of $SU(3)$ is a deformation retraction of $W\cap X$. A Mayer-Vietoris sequence enables us to compute the homology of $Y$, which completes the computation of the intersection homology of $X$.

3.2.3. Quasi-Hamiltonian analogue for $SU(2)$

Quasi-Hamiltonian $K$-spaces were introduced by Alekseev, Malkin and Meinrenken [1]. A quasi-Hamiltonian $K$-space is a manifold equipped with a $K$-action and equipped with a 2-form $\omega$. The form $\omega$ is neither closed nor nondegenerate, but satisfies three key properties with respect to the $K$ action ([1], Definition 2.2) which are analogous to certain properties of a Hamiltonian $K$-manifold (namely that it is equipped with a 2-form which is closed and nondegenerate, and there is an equivariant moment map $\mu:M\to{\bf k}^{*}$).

The space $K\times K$ is the quasi-Hamiltonian analogue of the cotangent bundle $T^{*}K$, and has a group valued moment map (the commutator map $\Phi:K\times K\to K$). It is then possible to define an imploded cross-section for quasi-Hamiltonian $K$-spaces in a manner analogous to the definition of imploded cross-sections for Hamiltonian $K$-spaces.

We now specialize to $K=SU(2)$ for the remainder of this section. Under this definition, the imploded cross-section of the quasi-Hamiltonian space $K\times K$ (the double for the group $K$) is isomorphic to $S^{4}$ (Proposition 2.29 of [27]).

This is true because the strata of $K$ are $\sigma^{0}$ (the stratum consisting of all elements of $K$ whose stabilizer is conjugate to $T$ under the adjoint action) and $\sigma^{\pm}=\pm I$ (the stratum consisting of the elements of the center of $K$). We have

[TABLE]

Of course $\Phi^{-1}(\sigma^{\pm})$ is a copy of $K$ and $\Phi^{-1}(\sigma^{\pm})/K$ is a point. These fit together to form the suspension of $K=SU(2)=S^{3}$, which is $S^{4}$. So the imploded cross-section of $K\times K$ is $S^{4}$.

4. Further work on symplectic implosion

In this section we describe some other examples of work on symplectic implosion.

4.1. Hyperkähler implosion: the work of Dancer, Kirwan and Swann

Dancer, Kirwan, Swann and their collaborators defined hyperkähler analogues of symplectic implosion. In [9] these authors define a hyperkähler reduction of the universal example for $SU(n)$ in terms of quiver varieties. For example, the hyperkähler reduction of the universal example is also a quiver variety. For the general definition of a quiver variety, see for example King [31] or Ginzburg [17].

In [10] these authors extend this treatment to the orthogonal and symplectic groups. They discuss some of the ways in which these cases are different from $SU(n)$. In [12] the authors show that the universal hyperkähler imploded cross-section contains a hypertoric variety (in other words a submanifold with a hyperkähler structure which is preserved by a torus action). The last section of this paper outlines an alternative description involving gauge theory and Nahm's equations.

In [14], the hyperkähler quotient of a space with a group-valued moment map is defined. In [13], these authors relate hyperkähler implosion to Nahm's equations. In [11] the authors study the twistor space (see for example [41]) associated to the hyperkähler implosion of a Hamiltonian $K$-space, where $K=SU(n)$.

4.2. Derived geometry and implosion: the work of Safronov

Let $G$ be a reductive Lie group. In [44], Safronov shows that the universal implosion is equivalent to the hyperkähler implosion of a stacky quotient. The idea of shifted symplectic geometry was developed for a stack by Pantev et al. [43]. Safronov interprets symplectic implosion in this context. Symplectic implosion replaces a Hamiltonian $G$-space by a Hamiltonian $H$-space (where $H$ is the maximal torus of $G$) so that the symplectic quotients at all level sets of the moment map are the same. Group-valued implosions are defined.

Safronov gives a characterization of the universal symplectic implosion in terms of stacks (using a compact Lie group $K$ and its complexification $G$, with Borel subgroup $B$ with Lie algebra $b$ and nilpotent subgroup $N$ with Lie algebra $n$, so that $n=[b,b]$). Safronov obtains that the universal symplectic implosion of $G$ is the stack $G\times_{N}b$.

Safronov also provides an adjoint map for implosion, which maps $H$-spaces to $G$-spaces. This could be thought of as a one-sided inverse map.

Safronov shows in Theorem 3.11 [44] that quasi-Hamiltonian reductions of imploded cross-sections with respect to a group $K$ are the same as covers of quasi-Hamiltonian reductions with respect to the maximal torus $H$ along a coadjoint orbit.

4.3. Symplectic cuts: the work of Martens and Thaddeus

The work of Martens and Thaddeus [40] defines a universal nonabelian symplectic cut", the nonabelian symplectic cut" of the cotangent bundle of a compact Lie group $K$. The nonabelian symplectic cut of a Hamiltonian $K$-manifold $M$ is then defined as the symplectic quotient of the product of $M$ and the universal nonabelian symplectic cut. The universal nonabelian symplectic cut has a symplectic description as the symplectic cut of $T^{*}K$ according to a polytope $P$ (see equation (12) of [40] for the definition). There is also an algebraic-geometric characterization of the universal nonabelian symplectic cut (see equation (14) of [40]).

4.4. The work of Howard-Manon-Millson

In [26], these authors consider the space of polygonal linkages in ${\bf R}^{3}$, in other words $m$-sided polygons in ${\bf R}^{3}$ with fixed side lengths. In this paper, the authors restrict to the group $K=SU(2)$ whose Lie algebra is ${\bf R}^{3}$. This space can be identified with the symplectic quotient of the Grassmannian of two-planes in ${\bf C}^{m}$ by the action of $T^{m}$, where $T$ is the maximal torus of $K$.

This space is equipped with a torus action (``bending flows''). The torus action fixes one part of a polygon (the part on one side of a diagonal) and rotates the rest around that diagonal. The difficulty is that there is a set of measure zero where the torus action fails to be defined, the set where some diagonal has length [math].

The space of $m$-gon linkages was originally described in [30] and was given a symplectic structure. In the paper [26], the authors identify the space of polygonal linkages in ${\bf R}^{3}$ with the symplectic quotient of a Grassmannian by a torus action (as described above).

They show in Section 3 of their paper that this topological space may be given an alternative description as an imploded cross-section. One may use the homeomorphism between the Grassmannian of 2-planes in ${\bf C}^{m}$ and the symplectic quotient of the imploded cross-section of cotangent bundle of $K^{m}$ by the left diagonal action of $K$.

4.5. Symplectic contractions: the work of Hilgert-Manon-Martens and Lane

Hilgert, Manon and Martens [22] define a symplectic contraction from one Hamiltonian space to another, which is a continuous surjective map to a new Hamiltonian space whose restriction to an open dense subset is a symplectomorphism. They give an explicit formula for the symplectic contraction map. They are able to show that the symplectic contraction map $\Phi$ maps a dense subset of $M$ symplectomorphically onto a dense subset of the image of $\Phi$. They interpret the Gelfand-Zeitlin system on a coadjoint orbit in this language. Their construction uses symplectic reduction and symplectic implosion.

Jeremy Lane [34] gives an alternative definition of a symplectic contraction $M^{sc}$ of a symplectic manifold $M$. He shows that his definition is equivalent to the definition given by Hilgert, Manon and Martens. He exhibits the symplectic contraction map $\Phi:M\to M^{sc}$ as a surjective Poisson map, a property that is not immediately obvious from the definition given in [22]. Lane also shows that the symplectic contraction is equipped with a Poisson algebra of smooth functions. He identifies the symplectic contraction with the quotient space obtained by subdividing $M$ into suitable coisotropic submanifolds and quotienting each coisotropic submanifold by the null foliation of the restriction of the symplectic form to it.

Lane proves that the symplectic contraction map sends $M$ to $M^{sc}$ in a way that is a homeomorphism which preserves strata. This enables him to define a smooth structure on the symplectic contraction (the quotient space), so he is able to define a smooth structure on a singular space.

The Poisson structure on the symplectic contraction pulls back from the Poisson structure on the original manifold. Lane uses symplectic contractions to study Gelfand-Zeitlin systems: these are symplectic manifolds equipped with a multiplicity-free Hamiltonian $U(n)$ action. Gelfand-Zeitlin systems have many similarities with toric manifolds but are not always equipped with the Hamiltonian action of a torus of half the dimension of the manifold.

4.6. Canonical bases and collective integrable systems: the work of Hoffman and Lane

[25, 24, 36]

Let $K$ be a compact connected Lie group and let $(M,\omega,\mu)$ be a Hamiltonian $K$-manifold. A classical problem in symplectic geometry asks: Beginning with the action of $K$, can one construct a Hamiltonian action of the compact torus $\mathbb{T}=(S^{1})^{m}\times T$ on $M$ (with as small a kernel as possible)? Here $m$ is half the dimension of a regular coadjoint orbit of $K$ and $T$ is the maximal torus of $K$.

This question was first answered by Guillemin and Sternberg [20] in the case when $K$ is a unitary group $U(n)$ or an orthogonal group $O(n)$. Their solution involved the construction of the now-famous Gelfand-Zeitlin integrable system on ${\bf k}^{*}$; they show that there is a Hamiltonian $\mathbb{T}$ action on $M$ whose moment map is the composition of $M\xrightarrow{\mu}{\bf k}^{*}$ with the moment map ${\bf k}^{*}\xrightarrow{\Psi}\operatorname{Lie}(\mathbb{T})^{*}$ for the Gelfand-Zeitlin system. A second approach, involving toric degenerations, was used by Harada-Kaveh [21] in the case that $M$ is a smooth projective variety and $\omega$ is the Fubini-Study form.

The following result of Hoffman and Lane shows how to solve this problem in the general case.

Theorem 4.1**.**

[25]** Let $K$ be a compact connected Lie group. There exists a continuous map $\Psi\colon{\bf k}^{*}\to\operatorname{Lie}(\mathbb{T})^{*}$ so that, for any Hamiltonian $K$-manifold $(M,\omega,\mu),$ there is a commuting diagram

[TABLE]

Here,

• •

$X$* is a singular Hamiltonian $\mathbb{T}$-space and the vertical arrow on the right is its moment map.*

• •

$\phi$* is a continuous, proper, $T$-equivariant, surjective map that is a symplectomorphism from a dense subset of $M$ onto its image.*

Moreover:

• •

The map $\Psi\circ\mu$ generates a Hamiltonian $\mathbb{T}$-action on a dense subset of $M$.

• •

If $M$ is multiplicity-free, then the action of $\mathbb{T}$ on a dense subset of $M$ is completely integrable.

• •

If $M$ is compact and connected, then its image in $\operatorname{Lie}(\mathbb{T})^{*}$ is a convex polytope.

Although we do not go into details of their construction here, the proof of this result uses symplectic implosion in a fundamental way. The key ingredient is the identification of the universal symplectic implosion of $T^{*}K$ with the geometric invariant theory quotient $G\sslash N$, where $G$ is the complexification of $K$ and $N$ is a maximal unipotent subgroup of $G$ [18]. It is known that the affine variety $G\sslash N$ admits toric degenerations. Hoffman and Lane show that it is possible to integrate gradient-Hamiltonian flows of these degenerations and thus obtain completely integrable Hamiltonian torus actions on the universal symplectic implosion of $T^{*}K$. In contrast with previous works [42, 21] on gradient-Hamiltonian flows and toric degenerations, the variety $G\sslash N$ is neither smooth nor projective. In order to prove their result, Hoffman and Lane develop new techniques for controlling gradient-Hamiltonian vector fields in this setting.

The degenerations of $G\sslash N$ used in the construction are degenerations to specific affine toric varieties $X_{C}$, as in [8]. Each $X_{C}$ is associated with the semigroup of integral points in a convex rational polyhedral cone $C\subset\operatorname{Lie}(\mathbb{T})^{*}$ described by Berenstein-Zelevinsky and Littelmann [7, 38]. The lattice points of $C$ are in bijection with the elements of the Kashiwara-Lusztig dual canonical basis [29, 39] of $\mathbb{C}[G\sslash N]$. Under the map $\Psi$ constructed by Hoffman and Lane, the image $\Psi({\bf k}^{*})$ is equal to $C$. The cone $C$ comes with a natural linear projection to the positive Weyl chamber ${\bf t}^{*}_{+}$. This projection fits into a commuting diagram:

[TABLE]

Combining this diagram with the one in the theorem above, one readily observes that when $M$ is compact and connected, the image $\Psi\circ\mu(M)$ is precisely the intersection of $C$ with the pre-image of the Kirwan polytope of $M$ under the projection $\operatorname{Lie}(\mathbb{T})^{*}\to{\bf t}^{*}_{+}$.

5. Topological invariants of the universal imploded cross-section

In this section we describe some invariants of imploded cross-sections. We specialize to the universal imploded cross-section, because general symplectic quotients of imploded cross-sections can be obtained as symplectic quotients of the direct product of an arbitrary Hamiltonian $K$-manifold and the universal imploded cross-section of $K$. Because the examples are easier to compute, we specialize to the universal imploded cross-section of $K=SU(3)$. We compare intersection homology ($IH$) with homology intersection spaces ($HI$).

This section is subdivided as follows. In subsection 5.1 we describe the conifold transition. In subsection 5.2 we survey intersection homology and perversities. In subsection 5.3 we summarize homology intersection spaces. In subsection 5.4 we compute the intersection homology of the universal example for $SU(3)$. In subsection 5.5 we summarize the intersection homology of a cone.

5.1. Conifold transition and blow-up manifold

In this section we define the conifold transition and the blow-up spaces, following Banagl [3].

These spaces provide an approach to studying Poincaré duality on singular spaces. An object called a perversity will be defined below in Definition (5.5). Given a perversity $\overline{p}$, one may associate a CW complex $I^{\overline{p}}X$ to a certain class of singular spaces $X$ . For our purposes, we only need to understand this construction in the case that $X$ is a Thom-Mather pseudomanifold of depth 1 with a trivial link bundle. The definition of Thom-Mather stratified spaces is given with more generality in [2]. The following definition appeared in [4]:

Definition 5.1**.**

A depth one pseudomanifold $X$ with singularity $\Sigma$ is a pair $(X,\Sigma)$, where

1. (1)

$\Sigma$* is understood to be a closed subspace and a smooth manifold of codimension at least 2.* 2. (2)

$X\setminus\Sigma$* is a smooth manifold which is dense in $X$.* 3. (3)

$\Sigma$* possesses control data consisting of a tube $T\subset X$ around $\Sigma$ which is an open set in $X$ together with two maps:*

[TABLE]

such that $\pi$ is a continuous retraction and $\rho$ is a continuous distance function such that $\rho^{-1}(0)=\Sigma$. Moreover, it is required that $(\pi,\rho):T\setminus\Sigma\longrightarrow\Sigma\times(0,\infty)$ is a smooth submersion.

Definition 5.2**.**

Let $L$ be a simply connected, smooth manifold. Let $c^{\circ}(L)$ be the open cone on $L$.

Remark 5.3**.**

Notice that, when $L$ is a smooth manifold, the cone $c^{\circ}(L)$ on $L$ is a depth 1 Thom-Mather pseudomanifold, with $v$ (the vertex of the cone) as its singularity. The link bundle of a depth 1 pseudomanifold is defined in Proposition $8.2$ in [4]. In the case that $X=c^{\circ}(L)$, the link bundle is as follows:

[TABLE]

Carefully following [5], we write down the construction of the conifold transition and the blow-up manifold associated to $(X,\Sigma)$.

Take a tubular neighborhood $N$ around the singularity $\Sigma$ and fix a diffeomorphism

[TABLE]

Define the blow-up manifold to be:

[TABLE]

Define the conifold transition to be:

[TABLE]

for all $z\in L$, and for all $y,y^{\prime}\in\Sigma$.

Remark 5.4**.**

Following this construction, one can see that when the singularity $\Sigma$ is a point, $CT(X)=\overline{M}$ is a manifold with boundary $L$. In particular, when $X$ is $c^{\circ}(L)$ for some smooth manifold $L$, we have

[TABLE]

5.2. Intersection homology and perversities

In this section we outline intersection homology, which reduces to ordinary homology for smooth manifolds. See the books [15] (Chapter 2) and [33] for general background information on intersection homology.

Taking the quotient of a smooth manifold by a group action produces singularities unless the group acts freely. The resulting topological spaces fail to satisfy standard topological properties such as Poincaré duality. Intersection homology and intersection cohomology were introduced by Goresky and MacPherson [16] in an effort to preserve some of the topological properties that are familiar in the setting of smooth manifolds. Intersection homology satisfies the ``Kähler package", which includes Poincaré duality, the hard Lefschetz theorem and the Lefschetz hyperplane theorem. This is the most we can hope to retain when we leave the the setting of smooth manifolds.

Since imploded cross-sections are singular, it is reasonable to investigate their topological properties using tools like intersection homology which are adapted to singular spaces, rather than to try to study the ordinary homology of these spaces.

Definition 5.5**.**

A perversity is a map

[TABLE]

satisfying

[TABLE]

and

[TABLE]

for all $k\geq 2.$

Throughout this article $\overline{p},\overline{q}$ are considered to be extended perversities, which are just sequences of integers (see [5], Section 3).

Definition 5.6**.**

The intersection homology of $X$ is the homology of the chain complex of $p$-allowable chains of $X$, where $p$ is a perversity. Here a $j$-simplex $\sigma$ is $p$-allowable if

[TABLE]

for all $k\geq 2$.

5.3. Homology intersection spaces

The definition of intersection spaces was given in [3]. In this subsection we will describe how to construct $I^{\overline{p}}X$, the perversity $\overline{p}$ intersection space associated to $X$ when $(X,\Sigma)$ is a depth 1 pseudomanifold where the link $L$ of the singular stratum $\Sigma$ is simply connected and the link bundle is the product bundle $L\times\Sigma\to\Sigma$ (as defined in the first section of [5]). The definition of the intersection space is given below in equation (5.8).

Let $l:=\dim L$ and set $k:=l-\overline{p}(l+1)$. 111Note that by the dimension of a manifold we always mean the real dimension.

Definition 5.7**.**

Let $L_{<k}$ be the union of all strata of $L$ of degree less than $k$.

Assume $f:L_{<k}\longrightarrow L$ is a stage $k$ Moore approximation of $L$ (see [5], Definition 3.1 for the definition). In other words, the homology groups $H_{i}(L_{<k})$ are [math] for $i\geq k$ and

[TABLE]

is an isomorphism for $i<k$. Define the map $g:L_{<k}\times\Sigma\longrightarrow M$ to be the composition:

[TABLE]

Definition 5.8**.**

In the notation introduced above, the perversity $\overline{p}$ intersection space $I^{\overline{p}}X$ is defined to be:

[TABLE]

The notation $I^{\overline{p}}X$ stands for the perversity $\overline{p}$ intersection space associated to $X$ (as introduced in [3]). See the definition given by equation (5.8) above.

Definition 5.9**.**

(Homology intersection space) [5] The homology $\widetilde{H}I_{i}^{\overline{p}}(X)$ is defined by

[TABLE]

where by $\widetilde{H}_{*}(X)$ we mean the reduced (singular) homology of $X$.

Remark 5.10**.**

When $X$ is a stratified pseudomanifold of dimension $n$ with an isolated singularity, the following formulas are available for the homology of the intersection space $\widetilde{H}I^{p}(X)$ ([6], p. 221):

[TABLE]

where $k:=n-1-p(n)$ and $\overline{M}$ is the blow-up manifold associated to the space $X$ (as defined in subsection 5.1). For the dimension $k$ homology, the following diagram with exact rows and columns exists:

[TABLE]

$\square$**

Remark 5.10 provides a proof of Theorem 5.13. When $X=c^{o}(L)$, the blow-up manifold $\overline{M}$ associated to $X$ is equal to $L\times[0,1)$ (as explained in Remark 5.4). In particular, notice that

[TABLE]

and

[TABLE]

5.4. Universal imploded cross-section for $SU(3)$

The universal imploded cross-section $(T^{*}SU(3))_{\rm{impl}}$ of $SU(3)$ was described in Section 3.2.2 above. The middle perversity intersection homology of the universal imploded cross-section $(T^{*}SU(3))_{\rm{impl}}$ is calculated in [23] and it is given by:

[TABLE]

This was done in [23] by first computing the homology of $Y$ by a Mayer-Vietoris argument, and then applying at result to compute the interseection homology.

Comparing this with the result of Corollary 5.15 below, we observe that the homology theories ${HI}^{m}$ and $IH^{m}$ do not agree on $(T^{*}SU(3))_{\rm{impl}}$.

5.5. Intersection homology of a cone

In this section we describe the intersection homology for a cone.

Definition 5.11**.**

By $c^{\circ}(X)$, the open cone over a topological space $X$, we mean the quotient space

[TABLE]

On the other hand, by $c(X)$ we mean the closed cone over $X$.

Given any smooth manifold $L$ and perversity function $p$, the perversity $p$ intersection homology groups of $c^{\circ}(L)$ are given by  ([33], p. 58):

[TABLE]

By comparing this with the result of Theorem 5.13 below, we see that the homology theories $\widetilde{H}I^{p}$ and $I\widetilde{H}^{p}$ usually do not agree on open cones over simply connected, smooth oriented manifolds.

First we are going to prove two general results.

Definition 5.12**.**

Let $X$ be as defined above. In terms of the notation $L$ introduced above, the intersection space of $X$ is

[TABLE]

where $f:L_{<k}\longrightarrow L$ is a stage $k=l-\overline{p}(l+1)$ Moore approximation of $L$. Here the spaces $L$ and $c_{\circ}(L)$ were defined in Definition 5.2 above, and the space $L_{<k}$ was defined in Definition 5.7.

Theorem 5.13**.**

Let $L$ be a simply connected smooth manifold of dimension $l$, and let

[TABLE]

denote the open cone on $L$, where $L$ is as in Definition 5.12. Assume that $\overline{p}$ is an (extended) perversity. Then the homology intersection space associated to $X$ and $\overline{p}$ is

[TABLE]

In the following Lemma we compute the homology of the middle perversity intersection space associated to the universal imploded cross-section of $SU(3)$, denoted by $(T^{*}SU(3))_{\rm{impl}}$.

Lemma 5.14**.**

As shown in [23], the imploded cross-section $(T^{*}SU(3))_{\rm{impl}}$ is homeomorphic to the open cone $c^{\circ}(Y)$ where

[TABLE]

is a compact Riemannian manifold of $\dim_{\mathbb{R}}(Y)=9$. (Here $z\cdot w$ denotes the usual inner product on ${\bf C}^{3}.$) Moreover, the reduced homology groups of $Y$ are given by:

[TABLE]

Using the Lemma 5.14, the corollary below is thus a direct consequence of Theorem 5.13.222Throughout this paper, the letter $m$ denotes the lower middle perversity – see for example [33].

Corollary 5.15**.**

The homology intersection space of the imploded cross-section of $SU(3)$ is

[TABLE]

Remark 5.16**.**

Observe that homology of intersection spaces is a homotopy invariant, whereas intersection homology is not.

5.6. Proof of Theorem 5.13

In this section we give the proof of Theorem 5.13, which gives the intersection homology of cones of smooth oriented manifolds $L$.

Proof.

Following Remark 5.4, we have

[TABLE]

where $k=l-\overline{p}(l+1)$ and the equivalence relation is given by:

[TABLE]

Here $f:L_{<k}\longrightarrow L$ is a stage $k$ Moore approximation of $L$ (see for example [4]). Define two open sets $A$ and $B$ as follows:

[TABLE]

[TABLE]

We observe that $B$ is contractible, as it is the preimage of $[1,1/2+\epsilon)$ in $c(L_{<k})$ under the cone map sending $(x,y)\in[0,1]\times L_{<k}$ to $[0,1]$. Hence it is homeomorphic to the cone on $L_{<k}$, so it is contractible. The set $A$ is homotopy equivalent to $L$, because the identification map $f$ identifies each point in $L_{<k}$ to a point in $L$. Moreover, we observe that $A\cap B$ deformation retracts to $L_{<k}$, as $A\cap B$ is homeomorphic to $L_{<k}\times(1/2-\epsilon,1/2+\epsilon)$.

Writing the Mayer-Vietoris sequence, we have the following:

Case I: $j-1\geq k$

In this case, the Mayer-Vietoris sequence gives $0\to\widetilde{H}_{j}(L)\to\widetilde{H}_{j}(I^{\bar{p}}(X))\to 0$ (because $\widetilde{H}_{j}(L_{<k})=\widetilde{H}_{j-1}(L_{<k})=0$). Hence $I^{\bar{p}}(X)\cong\widetilde{H}_{j}(L_{<k})=0$.

Case II: $j=k$

The Mayer-Vietoris sequence gives

[TABLE]

The maps $f_{*}:\widetilde{H}_{j}(L_{<k})\to\widetilde{H}_{j}(L)$ are isomorphisms for $j\leq k-1$. This implies $\alpha_{j}$ $=\beta_{j}=0$ for $j\leq k-1$. Also $\beta_{k}=0$, so $\widetilde{H}_{k}(I^{\bar{p}}(X))\cong\widetilde{H}_{k}(L)$.

Case III: $j<k$

Since $f_{*}:\widetilde{H}_{j}(L_{<k})\rightarrow\widetilde{H}_{j}(L\ )$ is an isomorphism for $j<k$, $\alpha_{j}=$ $\beta_{j}=0$ in this range. This implies the Mayer-Vietoris sequence gives

[TABLE]

which implies $\widetilde{H}_{j}(I^{\bar{p}}(X))=0$ for $j\leq k-1$.  This completes the proof of the theorem. ∎

Remark 5.17**.**

When $X=c^{\circ}(L)$, $I^{\overline{p}}X$ deformation retracts to ${\rm Cone}(f)$ where by ${\rm Cone}(f)$ we mean the mapping cone of $f:L_{<k}\longrightarrow L$. Therefore Theorem 5.13 gives the homology groups of ${\rm Cone}(f)$.

Appendix A Intersection space associated to a suspension

The authors are not aware of examples of imploded cross-sections which are suspensions. The material below is included nonetheless because the homology of intersection spaces of suspensions can be treated using the same techniques as are used to characterize intersection homology and cohomology of intersection spaces of the universal example for $SU(3)$.

In this Appendix we will prove a theorem related to the suspension over a smooth manifold.

Theorem A.1**.**

Let $L$ be a smooth, simply connected, oriented manifold of dimension $l$ and let $\overline{p}$ denote an extended perversity. Then:

[TABLE]

where by $\rm{susp}(L)$ we mean the suspension over $L$.

Remark A.2**.**

Note that the suspension of a smooth manifold $M$ is not normally a smooth manifold itself (the suspension of $M$ is only a smooth manifold only if $M$ is a sphere), so we would not expect the ordinary cohomology of the suspension of $M$ to satisfy Poincaré duality.

Let $L$ be a smooth, simply connected manifold of dimension $L$. By $X:={\rm susp}(L)$ we mean the quotient space obtained from $L\times[-1,1]$ by collapsing $L\times\{1\}$ to one point (denote this point by $v$), and $L\times\{-1\}$ to another point (denoted by $u$). We observe that $(X,\{u,v\})$ is a depth 1 Thom-Mather pseudomanifold with trivial link bundle

[TABLE]

Following the construction given in subsection 5.6, we have $\overline{M}=CT(X)\cong[-1,1]\times L$.

Fix a perversity $\overline{p}$ and set $k:=l-\overline{p}(l+1)$, then

[TABLE]

where the map $g$ is defined to be the composition

[TABLE]

where $f:L_{<k}\rightarrow L$ is a stage $k$ Moore approximation of $L$.

A.1. Proof of Theorem A.1

Throughout this section, $X={\rm susp}(L)$ where $L$ is a smooth manifold satisfying the conditions given in Theorem A.1. Moreover, we set $k=l-\overline{p}(l+1)$.

First we are going to prove the following lemma:

Lemma A.3**.**

For $i>k$ we have

[TABLE]

Proof.

The proof of this lemma is very similar to the proof of Theorem 5.13. Define two open sets $A$ and $B$ as follows. The space $A$ is defined by

[TABLE]

where the equivalence relation is given by:

[TABLE]

On the other hand, the space $B$ is defined by

[TABLE]

By similar reasoning as in the proof of Theorem 5.13, we see that $A$ deformation retracts to $L$, $B$ is contractible and $A\cap B$ is homotopy equivalent to $L_{<k}\sqcup L_{<k}.$

For $j>k$ the Mayer-Vietoris sequence gives

[TABLE]

as $\widetilde{H}(L_{<k})\oplus\widetilde{H}(L_{<k})=0$ for $j\geq k$. ∎

We now require the following lemma:

Lemma A.4**.**

For $i<k$

[TABLE]

Proof.

This time we cover $I^{\overline{p}}X$ with two different open sets $C,D$ as follows:

[TABLE]

Now we have that $C$ and $D$ deformation retract to ${\rm Cone}(f)$. Moreover, $C\cap D$ is homotopy equivalent to $L$. For $i<k$ the Mayer-Vietoris sequence with respect to the cover $\{C,D\}$ gives

[TABLE]

as $\widetilde{H}_{j}({\rm Cone}(f))\oplus\widetilde{H}_{j}({\rm Cone}(f))=0$ for $j<k$. ∎

Conclusion of proof of Theorem A.1: In order to complete the proof of Theorem A.1, we need to calculate $H_{k}(I^{\overline{p}}X)$. Once again we consider the Mayer-Vietoris sequence with respect to the cover $I^{\overline{p}}X=A\cup B$ given in the proof of Lemma A.3.

[TABLE]

Using this diagram, we observe that $\alpha_{k}$ is injective, as the top line of the diagram gives

[TABLE]

The map $\gamma_{i}$ is given by

[TABLE]

For $i<k$, $f_{*}:\widetilde{H}_{i}(L_{<k})\longrightarrow\widetilde{H}_{i}(L)$ is an isomorphism. This implies that $\gamma_{i}$ is a surjective map with

[TABLE]

In particular we get $\ker(\gamma_{k-1})\cong\widetilde{H}_{k-1}(L).$ Now we can calculate $\widetilde{H}_{k}(I^{\overline{p}}X)$.

[TABLE]

$\square$

This completes the proof of Theorem A.1.