Mapping class groups of highly connected $(4k+2)$-manifolds
Manuel Krannich
TL;DR
This paper computes the mapping class groups of certain highly connected manifolds, revealing their structure in relation to homology automorphisms and homotopy spheres, and explores their Torelli subgroup and related groups.
Contribution
It provides explicit descriptions of the mapping class groups of connected sums of sphere products, including their Torelli subgroups and abelianizations, extending understanding of their algebraic structure.
Findings
Computed the mapping class group in terms of automorphisms of middle homology.
Identified the Torelli subgroup of these manifolds.
Determined the abelianizations of the mapping class groups.
Abstract
We compute the mapping class group of the manifolds for in terms of the automorphism group of the middle homology and the group of homotopy -spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.
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Mapping class groups of highly connected -manifolds
Manuel Krannich
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
Abstract.
We compute the mapping class group of the manifolds for in terms of the automorphism group of the middle homology and the group of homotopy -spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.
2010 Mathematics Subject Classification:
57R50, 55N22, 57R60, 11F75, 20J06
The classical mapping class group of a genus surface naturally generalises to all even dimensions as the group of isotopy classes
[TABLE]
of orientation-preserving diffeomorphisms of the -fold connected sum . Its action on the middle cohomology provides a homomorphism whose image is the symplectic group in the surface case , and a certain arithmetic subgroup or in general, the description of which we shall recall later. The kernel of the resulting extension
[TABLE]
is known as the Torelli group—the subgroup of isotopy classes acting trivially on homology. In contrast to the surface case, the Torelli group in high dimensions is comparatively manageable: there is an extension
[TABLE]
due to Kreck [Kre79], which relates to the finite abelian group of homotopy spheres and the image of the stabilisation map , shown in Table 1.
The description of up to these two extension problems has found a variety of applications [KK05, Kry07, BM13, BM14, ERW15, BBP*+*17, BEW20, Kup19, Gre19, KRW20], especially in relation to the study of moduli spaces of manifolds [GRW20]. The remaining extensions (1) and (2) have been studied more closely for particular values of and [Sat69, Fri86, Kry02, Kry03, Cro11, GRW16] but are generally not well-understood (see e.g. [Cro11, p.1189],[GRW16, p.873],[BBP*+*17, p.425]). In the present work, we resolve the remaining ambiguity for odd, resulting in a complete description of the mapping class group and the Torelli group in terms of the arithmetic group and the group of homotopy spheres .
To explain our results, note that (1) and (2) induce further extensions
[TABLE]
[TABLE]
which express in terms of and up to two extension problems that are similar to (1) and (2), but are more convenient to analyse as both of their kernels are abelian. We resolve these two extension problems completely, beginning with an algebraic description of the second one in Section 2, which enables us in particular to decide when it splits.
Theorem A**.**
For odd and , the extension
[TABLE]
splits for . For , it splits if and only if .
Unlike (4), the extension (3) is central and thus classified by a class in , which our main result, B below, identifies in terms of three cohomology classes
- (i)
for odd, 2. (ii)
for and , 3. (iii)
for ,
which can be expressed algebraically, using our description of mentioned above. Referring to Section 3 for the precise definition of these classes, we encourage the reader to think of them geometrically in terms of their pullbacks along the composition
[TABLE]
induced by taking path components and quotients: the pullback of the first class, which is closely related to Meyer’s signature cocycle [Mey73], evaluates a class represented by an oriented smooth fibre bundle over a closed oriented surface with fibre to an eighth of the signature of its total space, the pullback of the second one assigns such a bundle the Pontryagin number up to a fixed constant, and the pullback of the third class evaluates to a certain linear combination of and . In addition to these three classes, our identification of the extension (3) for odd involves two particular homotopy spheres: the first one, , is the Milnor sphere—the boundary of the -plumbing [Bro72, Sect. V], and the second one, , arises as the boundary of the plumbing of two copies of a linear -bundles over classified by a generator of . We write for the change of coefficients induced by .
Theorem B**.**
For odd and , the central extension
[TABLE]
is classified by
- (i)
* for ,* 2. (ii)
* for if ,* 3. (iii)
* for .*
Moreover, this extension splits if and only if and .
The extension (2) describing the Torelli group is the pullback of the extension determined in B along the map , so the combination of the previous result with our identification of provides an algebraic description of both and in terms of and . We derive several consequences from this, beginning with deciding when the more commonly considered extensions (1) and (2) split.
Corollary C**.**
Let odd and .
- (i)
The extension
[TABLE]
does not split for , but admits a splitting for and . 2. (ii)
The extension
[TABLE]
does not split for , but splits -equivariantly for .
Abelian quotients
The second application of our description of the groups and is a computation of their abelianisations.
Although A exhibits the extension (4) as being nontrivial in some cases, its abelianisation turns out to split nevertheless (see 2.4), so there exists a splitting
[TABLE]
which participates in the following identification of the abelianisation of and .
Corollary D**.**
Let and odd.
- (i)
The extension (3) induces a split short exact sequence
[TABLE]
where for and for . 2. (ii)
The extension (4) induces a split short exact sequence of -modules
[TABLE]
In particular, the commutator subgroup of is generated by .
These splittings of and are constructed abstractly, but can often be made more concrete by means of a refinement of the mapping torus construction as a map
[TABLE]
to the bordism group of closed -manifolds equipped with a lift of their stable normal bundle to the -connected cover . To state the resulting more explicit description of the abelianisations of and , we write for the minimal positive signature of a closed smooth -connected -dimensional manifold. For odd, the intersection form of such a manifold is even, so is divisible by .
Corollary E**.**
Let odd and .
- (i)
The morphism
[TABLE]
is an isomorphism for , and for if . For and , it is surjective, has kernel of order generated by , and splits for . 2. (ii)
The morphism
[TABLE]
is an isomorphism for . For , it is surjective, has kernel of order generated by , and splits -equivariantly for .
Remark*.*
xyz
- (i)
In G below, we determine the abelianisation of and for even in which case the morphisms and are isomorphisms for all . 2. (ii)
As shown in [KR20, Prop. 2.15], the minimal signature is nontrivial for odd, grows at least exponentially with , and can be expressed in terms of Bernoulli numbers. 3. (iii)
For some values of and , E leaves open whether and split. The morphisms and always split by D, and in Section 4.1 we relate the question of whether there exist compatible splittings of to a known open problem in the theory of highly connected manifold, showing in particular that such splittings do exist when assuming a conjecture of Galatius–Randal-Williams. 4. (iv)
Work of Thurston [Thu74] shows that the component of the identity is perfect as a discrete group, so the abelianisation of the full diffeomorphism group considered as a discrete group agrees with .
In view of E, it is of interest to determine the bordism groups , the abelianisation , and the coinvariants . The computation
[TABLE]
is straightforward (see A.2 and Table 1). The abelianisation of is known and summarised in Table 2 (see A.1).
Finally, the bordism groups are closely connected to the stable homotopy groups of spheres: the canonical map
[TABLE]
factors through the cokernel of the -homomorphism and work of Schultz and Wall [Sch72, Wal62b] implies that the induced morphism is often an isomorphism (see 3.6).
Theorem** (Schultz, Wall).**
For odd, the natural morphism is surjective with cyclic kernel. It is an isomorphism for and for .
Combined with E, this reduces the computation of the abelianisation of and in many cases to determining the cokernel of the -homomorphism—a well-studied problem in stable homotopy theory. Table 3 shows the resulting calculation of the abelianisations of the groups and for the first few values of .
Remark*.*
After the completion of this work, Burklund–Hahn–Senger [BHS19] and Burklund–Senger [BS20] showed that for odd, the homotopy sphere bounds a parallelisable manifold if and only if . This implies in particular that aside from
- (i)
the canonical map is an isomorphism, which extends the theorem attributed to Schultz and Wall above, 2. (ii)
the conjecture of Galatius–Randal-Williams mentioned in the third part of the previous remark holds, and 3. (iii)
the minimal signature appearing in E is computable from [KR20, Prop. 2.15].
Homotopy equivalences
As an additional application of our results, we briefly discuss the group of homotopy classes of orientation-preserving homotopy equivalences.
The natural map can be seen to factor through the quotient and to induce a commutative diagram of the form
[TABLE]
which exhibits the lower row—an extension describing due to Baues [Bau96]—as the extension pushout of the extension (4) along the left vertical morphism, which is induced by the restriction of the unstable -homomorphism, where is the image of the suspension map . By A, the upper row splits in most cases and thus induces a compatible splitting of Baues’ extension. In the cases in which the upper row does not split, we show that Baues’ extension cannot split either.
Corollary F**.**
Let odd and .
- (i)
For , the two extensions in (6) admit compatible splittings. For , either of the extensions splits if and only if . 2. (ii)
The induced morphism is an isomorphism for , and a split epimorphism with kernel the coinvariants , which vanish for or , and agree with otherwise.
The groups for even
Some parts in our analysis of go through when is even as well, but a few key steps do not and would require new arguments. For instance, a different approach to the extension problem (4) would be necessary, as well as an extension of 3.12 to incorporate the Arf invariant. The abelianisation of the groups and , however, can be determined without fully solving the extensions (3) and (4) if is even. It turns out that in this case, the morphisms considered in E are isomorphisms for all , which we shall prove as part of Section 4.2.
Theorem G**.**
For even and , the morphisms
[TABLE]
are isomorphisms for . The coinvariants \big{(}H(g)\otimes S\pi_{n}\mathrm{SO}(n)\big{)}_{G_{g}} are described in (5).
Other highly connected manifolds
Instead of restricting to , one could consider any -connected almost parallelisable manifold of dimension . Baues’ and Kreck’s work [Bau96, Kre79] applies in this generality, so there are analogues of the sequences (1)–(6) describing and . However, for odd—the case of our main interest—Wall’s classification of highly connected manifolds [Wal62b] implies that any such manifold is diffeomorphic to a connected sum with an exotic sphere , aside from those of Kervaire invariant , which only exist in dimensions , , , , and possibly by work of Hill–Hopkins–Ravenel [HHR16]. The mapping class group for and odd in turn is completely understood in terms of : Kreck’s work [Kre79, Lem. 3, Thm 3] shows that the former is a quotient of the latter by a known element of order at most , which is trivial if and only if vanishes.
Previous results
The extensions (1) and (2) and their variants (3) and (4) have been studied by various authors before, and some special cases of our results were already known:
- (i)
As an application of their programme on moduli spaces of manifolds, Galatius–Randal-Williams [GRW16] determined the abelianisation of for and used this to determine the extension (3) for up to automorphisms of as long . Our work recovers and extends their results, also applies to low genera , and does not rely on their work on moduli spaces of manifolds. 2. (ii)
Theorems A and F for reprove results due to Crowley [Cro11]. 3. (iii)
Baues [Bau96, Thm 8.14, Thm 10.3] showed that the lower extension in (6) splits for odd, which we recover as part of the first part of F. 4. (iv)
The case of A and C (ii) can be deduced from work of Krylov [Kry03] and Fried [Fri86], who also showed that the extension of C (i) does not split in this case. Krylov [Kry02, Thms 2.1, 3.2, 3.3] moreover established the case of C (i) for . For , he also proved the case of A and the case of C (ii) for .
Further applications
Our main result B has been used in [KK20] in conjunction with Galatius–Randal-Williams’ work on moduli spaces of manifolds [GRW20] to compute the second stable homology of the theta-subgroup of (see Section 1.2), or equivalently, the second quadratic symplectic algebraic -theory group of the integers .
Outline
Section 1 serves to recall foundational material on diffeomorphism groups and their classifying spaces, as well as to introduce different variants of the extensions (1) and (2) and to establish some of their basic properties. In Section 2, we study the action of on the set of stable framings of to identify the extension (3) and prove A. Section 3 aims at the proof of our main result B, which requires some preparation. We recall the relation between relative Pontryagin classes and obstruction theory in Section 3.1, discuss aspects of Wall’s classification of highly connected manifolds in Section 3.2, relate this class of manifolds to -bundles over surfaces with certain boundary conditions in Section 3.3 (which incidentally is the key geometric insight to prove B), construct the cohomology classes appearing in the statement of B in Sections 3.4 and 3.5, and finish with the proof of B in Section 3.6. In Section 4, we analyse the extensions (1) and (2) and compute the abelianisation of and , proving Corollaries C–E and G. Section 5 briefly discusses the group of homotopy equivalences and proves F. In the appendix, we compute various low-degree (co)homology groups of the symplectic group and its arithmetic subgroup .
Acknowledgements
I would like to thank Oscar Randal-Williams for several valuable discussions, Aurélien Djament for an explanation of an application of a result of his, and Fabian Hebestreit for many useful comments on an earlier version of this work. I was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 756444).
Contents
1. Variations on two extensions of Kreck
1.1. Different flavours of diffeomorphisms
Throughout this work, we write
[TABLE]
for the -fold connected sum of , including , and the manifold obtained from removing the interior of an embedded disc . Occasionally, we view the manifold alternatively as the iterated boundary connected sum of . We call the genus of or and denote by and the groups of orientation-preserving diffeomorphisms, not necessarily fixing the boundary in the case of . We shall also consider the subgroups
[TABLE]
of diffeomorphisms required to fix a neighbourhood of the boundary or a neighbourhood of a chosen disc in the boundary, respectively. All groups of diffeomorphisms are implicitly equipped with the smooth topology so that
- (i)
and classify smooth oriented -bundles or -bundles, 2. (ii)
classifies -bundles, i.e. smooth -bundles with a trivialisation of their -bundle of boundaries, and 3. (iii)
classifies -bundles, that is, smooth -bundles with a trivialised -subbundle of its -bundle of boundaries.
Taking path components, we obtain various groups of isotopy classes
[TABLE]
Extending diffeomorphisms by the identity provides a map , which induces an isomorphism on path components by work of Kreck as long as .
Lemma 1.1** (Kreck).**
The canonical map is an isomorphism for .
Proof.
Taking the differential at the centre of the disc induces a fibration to the oriented frame bundle of . Its fibre is the subgroup of diffeomorphisms that fix a point and its tangent space, so it is equivalent to the subgroup of diffeomorphisms fixing a small disc around that point, which is in turn equivalent to . We thus have fibration sequences of the form
[TABLE]
whose long exact sequences show that the morphism in question is surjective and also that its kernel is generated by a single isotopy class given by “twisting” a collar using a smooth based loop in that represents a generator of (see [Kre79, p. 647]). It follows from [Kre79, Lem. 3 b), Lem. 4] that this isotopy class is trivial since bounds the parallelisable handlebody . ∎
For the purpose of studying the mapping class group , we can thus equally well work with instead of , which is advantageous since there is a stabilisation map by extending diffeomorphisms over an additional boundary connected summand via the identity, which restricts to a map and thus induces stabilisation maps of the form
[TABLE]
that allow us to compare mapping class groups of different genera. The group has a convenient alternative description: gluing two closed -discs along their boundaries via a diffeomorphism supported in a disc gives a morphism to Kervaire–Milnor’s [KM63] finite abelian group of oriented homotopy -spheres up to -cobordism. By work of Cerf [Cer70], this is an isomorphism for , so we identify these two groups henceforth. Iterating the stabilisation map yields maps
[TABLE]
Lemma 1.2**.**
For , the image of in is central and becomes trivial in . The induced morphism
[TABLE]
is an isomorphism.
Proof.
Every diffeomorphism of supported in a disc is isotopic to one that is supported in an arbitrary small neighbourhood of the boundary and thus commutes with any diffeomorphism in up to isotopy, which shows the first part of the claim. For the others, we consider the sequence of topological groups
[TABLE]
induced by restricting diffeomorphisms in to the moving part of the boundary. This is a fibration sequence by the parametrised isotopy extension theorem. Mapping this sequence for into (1.3) via the iterated stabilisation map, we see that the looped map induced by (1.3) factors as the composition
[TABLE]
of the map defining the Gromoll filtration with the iterated stabilisation map. Since the first map in this factorisation is surjective on path components by Cerf’s work [Cer70], the claim will follow from the long exact sequence on homotopy groups of (1.3) once we show that the map has trivial image. Using that any orientation preserving diffeomorphism fixes any chosen oriented codimension [math] disc up isotopy by the isotopy extension theorem, one easily sees that this image agrees with the the inertia group of , which is known to vanish by work of Kosinski and Wall [Kos67, Wal62a]. ∎
1.2. Wall’s quadratic form
We recall Wall’s quadratic from associated to an -connected -manifold [Wal62b], specialised to the case of our interest—the iterated connected sums in dimension .
The intersection form on the middle cohomology is a nondegenerate -symmetric bilinear form. We use Poincaré duality to identify with and a result of Haefliger [Hae61] to represent classes in by embedded spheres , unique up to isotopy as long as . As is stably parallelisable, the normal bundle of such is stably trivial and hence gives a class
[TABLE]
where is the image of the usual map (see e.g. [Lev85, §1.B]).111Note that is the trivial in the case where Haefliger’s result does not apply. This defines a function , which Wall [Wal62b] showed to satisfy
- (i)
and 2. (ii)
.
Note that for even, (i) and (ii) together imply , so can in this case be recovered from . The triple is the quadratic form associated to . The decomposition into connected summands induces a basis of with respect to which and have the form
[TABLE]
so the automorphism group of the quadratic form can be identified as
[TABLE]
where
[TABLE]
In the theory of theta-functions, the finite index subgroup is known as the theta group; it is the stabiliser of the standard theta-characteristic with respect to the transitive -action on the set of even characteristics (see e.g. [Wei91]). Using this description, it is straightforward to compute its index in to be .
1.3. Kreck’s extensions
To recall Kreck’s extensions [Kre79, Prop. 3] describing for , note that an orientation-preserving diffeomorphism of induces an automorphism of the quadratic form . This provides a morphism , which Kreck proved to be surjective using work of Wall [Wal63].222Kreck phrases his results in terms of pseudo-isotopy instead of isotopy. By Cerf’s “pseudo-isotopy implies isotopy” [Cer70], this does not make a difference as long as . This explains the first extension
[TABLE]
The second extension describes the Torelli subgroup and has the form
[TABLE]
where denotes the image of the morphism induced by the usual inclusion . The isomorphism type of this image can be extracted from work of Kervaire [Ker60] to be as shown in Table 1. As a diffeomorphism supported in a disc acts trivially on cohomology, the morphism in (1.2) has image in , which explains first map in the extension (1.5). To define the second one, we canonically identify with using that is free and note that for a given isotopy class and a class represented by an embedded sphere , the embedding is isotopic to , so we can assume that fixes pointwise by the isotopy extension theorem. The derivative of thus induces an automorphism of the once stabilised normal bundle , which after choosing a trivialisation defines the image of under the morphism , noting that is independent of all choices and actually lies in the subgroup (see [Kre79, Lem. 1]).
Instead of the extensions (1.4) and (1.5), we shall mostly be concerned with two closely related variants which we describe now. By Kreck’s result, the morphism is injective, so gives rise to an extension , which combined with the canonical identification of 1.2 has the form
[TABLE]
and agrees with the extension induced by taking path components of the chain of inclusions . The action of on preserves the quadratic form as is surjective by 1.2, so this action yields an extension
[TABLE]
which, via the isomorphism induced by (1.5), corresponds to the quotient of the extension (1.4) by , using once more.
Lemma 1.3**.**
The action of on induced by the extension (1.7) is through the standard action of on .
Proof.
In view of the commutative diagram
{0}$${\mathrm{T}_{g,1}^{n}}$${\Gamma_{g,1}^{n}}$${G_{g}}$${0}$${0}$${H(g)\otimes S\pi_{n}\mathrm{SO}(n)}$${\Gamma^{n}_{g,1/2}}$${G_{g}}$${0,}$$\scriptstyle{\rho}$$\scriptstyle{p}
it suffices to establish the identity for all and . Unwrapping the definition of , the image of on a homology class in is given by the automorphism
[TABLE]
where is an embedded sphere which represents the homology class and is pointwise fixed by and is any choice of framing of . Choosing the framing
[TABLE]
to compute the image of under , the claimed identity is a consequence of the chain rule for the differential. ∎
1.4. Stabilisation
Iterating the stabilisation map (1.1) induces a morphism
{0}$${\Theta_{2n+1}}$${\Gamma^{n}_{h,1}}$${\Gamma^{n}_{h,1/2}}$${0}$${0}$${\Theta_{2n+1}}$${\Gamma^{n}_{g,1}}$${\Gamma^{n}_{g,1/2}}$${0,}
of group extensions for , which exhibits the upper row as the pullback of the lower row, so the extension (1.6) for a fixed genus determines those for all . The situation for the extension (1.7) is similar: writing , we obtain a decomposition , which yields a stabilisation map
[TABLE]
and two morphisms of -modules, an inclusion and a projection . These morphisms express the extension (1.7) for genus as being obtained from that for genus by pulling back along followed by forming the extension pushout along . They also induce a morphism of the form
{0}$${H(h)\otimes S\pi_{n}\mathrm{SO}(n)}$${\Gamma^{n}_{h,1/2}}$${G_{h}}$${0}$${0}$${H(g)\otimes S\pi_{n}\mathrm{SO}(n)}$${\Gamma^{n}_{g,1/2}}$${G_{g}}$${0.}
2. The action on the set of stable framings and A
This section serves to resolve the extension problem
[TABLE]
described in the previous section. Our approach is in parts inspired by work of Crowley [Cro11], who identified this extension in the case .
The group acts on the set of equivalence classes of stable framings
[TABLE]
that extend the standard stable framing on , by pulling back stable framings along the derivative. As the equivalence classes of such framings naturally form a torsor for the group of pointed homotopy classes , the action of on a fixed choice of stable framing as above yields a function
[TABLE]
where the first isomorphism is induced by and the Hurewicz isomorphism, and the second one by the universal coefficient theorem. This function is a -cocycle (or crossed homomorphism) for the canonical action of on (cf. [Cro11, Prop. 3.1]) and as this action factors through the map , we obtain a morphism of the form
[TABLE]
which is independent of up to conjugation in the target by a straightforward check. This induces a morphism from (2.1) to the trivial extension of by the -module ,
[TABLE]
The left vertical map is induced by the natural map originating from the inclusion and is an isomorphism for odd as a consequence of the following lemma whose proof is standard (see e.g. [Lev85, §1B)]).
Lemma 2.1**.**
For odd, the morphism induced by the inclusion is an isomorphism for odd. For , it is injective with cokernel .
As a result, the diagram (2.2) induces a splitting of (2.1) for odd, since all vertical maps are isomorphisms. This proves the cases of the following reformulation of A (see Section 1.3). We postpone the proof of the cases to Section 3.6.
Theorem 2.2**.**
For odd, the extension
[TABLE]
splits for . For , it splits if and only if .
Even though the extension does not split for , the morphism (2.2) is still injective by 2.1 and thus expresses the extension in question as a subextension of the trivial extension of by the -module . Crowley [Cro11, Cor. 3.5] gave an algebraic description of this subextension and concluded that it splits if and only if . We proceed differently and prove this fact in Section 3.3 directly, which can in turn be used to determine the extension in the following way: by the discussion in Section 1.4, it is sufficient to determine its extension class in for . For , we have with its usual action on . Using work of Djament [Dja12, Thm 1], one can compute for , so there is only one nontrivial extension of by , which must be the one in consideration because of the second part of 2.2. Note that this line of argument gives a geometric proof for the following useful fact on the twisted cohomology of as a byproduct, which can also be derived algebraically (see for instance [Cro11, Sect. 2]).
Corollary 2.3**.**
The pullback of the unique nontrivial class in for to for is trivial if and only if .
We close this section by relating the abelianisation of to that of . The latter is content of A.1.
Corollary 2.4**.**
For odd, the morphism
[TABLE]
is split surjective and has the coinvariants as its kernel, which vanish for . For it vanishes if and only if or , and has order otherwise.
Proof.
The claim regarding the coinvariants follows from A.2 and Table 1. Since they vanish for , the remaining statement follows from the exact sequence
[TABLE]
induced by (2.1), combined with the fact that this extension splits for by 2.2. ∎
3. Signatures, obstructions, and B
By 1.2, the extension
[TABLE]
discussed in Section 1.3 is central and is as such classified by a class in with acting trivially on . In this section, we identify this extension class in terms of the algebraic description of provided in the previous section, leading to a proof of our main result B. Our approach is partially based on ideas of Galatius–Randal-Williams [GRW16, Sect. 7], who determined the extension for and up to automorphisms of .
We begin with an elementary recollection on the relation between Pontryagin classes and obstructions to extending trivialisations of vector bundles, mainly to fix notation.
3.1. Obstructions and Pontryagin classes
Let and be a map to the st connected cover of with a lift over a subspace along the canonical map . Such data has a relative Pontryagin class given as the pullback along the map of the unique lift to of the pullback of the usual Pontryagin class . The class is related to the primary obstruction to solving the lifting problem
{A}$${\tau_{>4k}\mathrm{BO}}$${X}$${\tau_{>4k-1}\mathrm{BO}}$$\scriptstyle{\bar{\xi}}$$\scriptstyle{\xi}
by the equality
[TABLE]
up to the choice of a generator (cf. [MK60, Lem. 2]). We suppress the lift from the notation whenever there is no source of confusing. For us, will usually be a compact oriented -manifold and its boundary, in which case we can evaluate against the relative fundamental class to obtain a number . The following two sources of manifolds are relevant for us.
Example 3.1*.*
Fix an integer .
- (i)
For a compact oriented -connected -manifold whose boundary is a homotopy sphere, there is a (up to homotopy) unique lift of the stable oriented normal bundle. On the boundary , this lifts uniquely further to , so we obtain a canonical class and a characteristic number . 2. (ii)
Consider a -bundle , i.e. a smooth -bundle with a trivialised -subbundle of its -bundle of boundaries. The standard framing of induces a trivialisation of stable vertical tangent bundle over the subbundle , which extends uniquely to a -structure on by obstruction theory. Using that is -parallelisable, another application of obstruction theory shows that the induced -structure on extends uniquely to a -structure on , so the above discussion provides a class , and, assuming is an oriented closed surface, a number .
3.2. Highly connected almost closed manifolds
As a consequence of 3.12, we shall see that -bundles over surfaces are closely connected to -connected almost closed -manifolds. These manifolds were classified by Wall [Wal62b], which we now recall for in a form tailored to later applications, partly following [KR20, Sect. 2].
A compact manifold is almost closed if its boundary is a homotopy sphere. We write for the abelian group of almost closed oriented -connected -manifolds up to oriented -connected bordism restricting to an -cobordism on the boundary. Recall that denotes the bordism group of closed -manifolds equipped with a -structure on their stable normal bundle , i.e. a lift to the -connected cover. By classical surgery, the group is canonically isomorphic to the bordism group of closed oriented -connected -manifolds up to -connected bordism as long as , so we will use both descriptions interchangeably. There is an exact sequence
[TABLE]
due to Wall [Wal67, p. 293] in which the two outer morphisms are the obvious ones, noting that homotopy -spheres -connected for . The morphisms and are given by cutting out an embedded disc and by assigning to an almost closed manifold its boundary, respectively. By surgery theory, the subgroup
[TABLE]
of homotopy -spheres bounding -connected manifolds contains the cyclic subgroup of homotopy -spheres bounding parallelisable manifolds, so the right end of (3.1) receives canonical a map from Kervaire–Milnor’s exact sequence [KM63],
[TABLE]
which in particular induces a morphism , concretely given by representing a class in by a stably framed manifold and restricting its stable framing to a -structure.
3.2.1. Wall’s classification
For our purposes, Wall’s computation [Wal62b, Wal67] of is for odd is most conveniently stated in terms of two particular almost closed -connected -manifolds, namely
- (i)
Milnor’s -plumbing , arising from plumbing together copies of the disc bundle of the tangent bundle of the standard -sphere such that the intersection form of agrees with the -form (see e.g.[Bro72, Ch. V.2]), and 2. (ii)
the manifold , obtained from plumbing together two copies of a linear -bundle over the -sphere representing a generator of .
The following can be derived from Wall’s work, as explained for instance in [KR20, Thm 2.1].
Theorem 3.2** (Wall).**
For odd, the bordism group satisfies
[TABLE]
The first summand is generated by the class of in all cases but in which it is generated by and . The second summand for is generated by .
From a consultation of Table 1, one sees that the group vanishes for , so is trivial in this case, which shows that the subgroup is for all odd generated by the boundaries
[TABLE]
In the cases in which defines a -summand, its boundary is trivial by a result of Schultz [Sch72, Cor. 3.2, Thm 3.4 iii)]. For on the other hand, it is nontrivial by a calculation of Kosinski [Kos67, p. 238-239].
Theorem 3.3** (Kosinski, Schultz).**
The homotopy sphere is trivial for and nontrivial for
In the exceptional dimensions , the homotopy sphere agrees with the inverse of the Milnor sphere , as explained in [KR20, Cor. 2.8 ii)].
Lemma 3.4**.**
* for .*
For odd, the Milnor sphere is well-known to be nontrival and to generate the cyclic subgroup whose order can be expressed in terms of numerators of divided Bernoulli numbers (see e.g. [Lev85, Lem. 3.5 (2), Cor. 3.20]), so 3.3 has the following corollary.
Corollary 3.5**.**
For odd, the subgroup is nontrivial. It is generated by for , by for , and by and for .
Combining the previous results with the diagram (3.2), we obtain the following result, which we already mentioned in the introduction.
Corollary 3.6**.**
The natural morphism is an isomorphism for and for . For , it is an epimorphism whose kernel is generated by the class .
is conjecturally trivial [GRW16, Conj. A] for all odd. Until recently (see 3.8), this was only known for and by the results above.
Conjecture 3.7** (Galatius–Randal-Williams).**
* for all odd.*
Remark 3.8*.*
As mentioned in the introduction, after the completion of this work, Burklund–Hahn–Senger [BHS19] and Burklund–Senger [BS20] showed that vanishes in for odd if and only if , confirming 3.7 for and disproving it for . This has as a consequence that, for odd, the subgroup is generated by even for and that the morphism discussed in 3.6 is an isomorphism.
3.2.2. Invariants
It follows from 3.2 and 3.3 that the boundary of an -connected almost closed -manifold is determined by at most two integral bordism invariants of . Concretely, we consider the signature and for the characteristic number , explained in 3.1. As discussed for example in [KR20, Sect. 2.1], these functionals evaluate to
[TABLE]
and on the closed manifolds and to
[TABLE]
which results in the following formula for boundary spheres of highly connected manifolds when combined with the discussion above.
Proposition 3.9**.**
For odd, the boundary of an almost closed oriented -connected -manifold satisfies
[TABLE]
3.2.3. The minimal signature
As in the introduction, we denote by the minimal positive signature of a smooth closed -connected -manifold. This satisfies for as witnessed by , , and , and in all other cases, it can be expressed in terms of the subgroup as follows.
Lemma 3.10**.**
For odd, the quotient is a cyclic group generated by the class of . It is trivial if and of order otherwise.
Proof.
For , the claim is a direct consequence of 3.2 and 3.4 and in the case , it follows from taking vertical cokernels in the commutative diagram
{0}$${\langle\mathrm{ord}(\Sigma_{Q})\cdot Q\rangle}$${\Omega_{2n+2}^{\tau_{>n}}/\Theta_{2n+1}}$${\sigma_{n}^{\prime}\cdot\mathbf{Z}}$${0}$${0}$${\langle Q\rangle}$${A_{2n+2}^{\tau_{>n}}}$${8\cdot\mathbf{Z}}$${0}$$\scriptstyle{\mathrm{sgn}}$$\scriptstyle{\mathrm{sgn}}
with exact rows, obtained from a combination of 3.2 with (3.1) and (3.3). ∎
Remark 3.11*.*
In [KR20, Prop. 2.15], the minimal signature was expressed in terms of Bernoulli numbers and the order of , from which one can conclude that for , the signature of such manifolds is divisible by if is odd and by otherwise, where denotes the -adic valuation (see [KR20, Cor. 2.18]).
3.3. Bundles over surfaces and almost closed manifolds
In order to identify the cohomology class in that classifies the central extension
[TABLE]
we first determine how it evaluates against homology classes, i.e. identify its image
[TABLE]
under the map participating in the universal coefficient theorem
[TABLE]
followed by resolving the remaining ambiguity originating from the Ext-term. Indicated by our choice of notation, the morphism can be viewed alternatively as the first possibly nontrivial differential in the -page of the Serre spectral sequence of the extension (3.5) (cf. [HS53, Thm 4]). Before identifying this differential, we remind the reader of two standard facts we shall make frequent use of.
- (i)
The natural map is -connected, so pushing forward fundamental classes induces an isomorphism for and any space , and 2. (ii)
the -truncation of a connected space (in particular the natural map for a topological group ) induces a surjection , whose kernel agrees with the image of the Hurewicz homomorphism .
The key geometric ingredient to identify the differential is the following result.
Theorem 3.12**.**
Let be odd and a -bundle over an oriented closed surface . There exists a class such that
- (i)
its boundary is the image of the class under
[TABLE] 2. (ii)
it satisfies , and if
Proof.
By the isotopy extension theorem, the restriction map to the moving part of the boundary is a fibration. As its image is contained in the component of the identity (see the proof of 1.2), this fibration induces the upper row in a map of fibre sequences
[TABLE]
whose bottom row is induced by the extension (3.5). The two right vertical maps are induced by taking components and the left vertical map is the induced map on homotopy fibres. The latter agrees with the delooping of the Gromoll map followed by taking components, which one checks by looping the fibre sequences and using that
[TABLE]
is given by “twisting” a collar , meaning that it sends a smooth loop to the diffeomorphism that is the identity outside the collar and is given by on the collar. Now consider the square
[TABLE]
obtained from delooping (3.6) once to the right and using . By transgression, the bottom horizontal arrow agrees with the differential in the statement. Combining this with the Hurewicz theorem, the square (3.7) provides a factorisation
[TABLE]
of the map in the first part of the statement, which thus has the following geometric description: a smooth -bundle represents a class and its image under the first map in the composition is the class of its -bundle of boundaries, which in turn maps under the inverse of the Hurewicz homomorphism to a -bundle over the -sphere that is bordant, as a bundle, to . That is, there exists a -bundle over an oriented bordism between and that restricts to over and to over . We claim that the image of the -bundle under the final map in the composition is the homotopy sphere obtained by doing surgery on the total space along the trivialised subbundle . This is most easily seen by thinking of a class in as a smooth bundle together with a trivialisation and a trivialised -subbundle such that and agree on . From this point of view, the morphism induced by the Gromoll map is given by sending such a bundle to the -bundle , and the isomorphism is given by assigning to a disc bundle the manifold . Deccomposing the sphere into half-discs , we see from this description that the composition of the iterated Gromoll map with the isomorphism maps a class represented by an -bundle with a trivialisation and a trivialised subbundle that agree on to the homotopy sphere
[TABLE]
obtained by doing surgery along the trivialised -subbundle, where is glued to Q\backslash\mathrm{int}\big{(}\varphi(D_{+}^{k}\times S^{d})\cup_{\partial(D^{k}\times S^{d})}\psi(S^{k}\times D_{+}^{d})\big{)} along the embedding
[TABLE]
This in particular implies the claim we made above in the case and .
As a consequence of this description of the morphism in consideration, the image of the class comes equipped with a nullbordism, namely where is the trace of the performed surgery. Omitting the trivialised -subbundles, the situation can be summarised schematically as follows
{W_{g,1}}$${\partial W_{g,1}}$${\partial W_{g,1}}$${\partial W_{g,1}} {E}$${E_{+}}$${\bar{E}}$${E_{-}}$${W}$${\Sigma_{\pi}}$${S}$${S}$${K}$${S^{2}.}$$\scriptstyle{\pi}$$\supset$$\scriptstyle{\pi_{+}}$$\subset$$\supset$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi_{-}}$$\subset$$\supset$$=$$\subset$$\supset
A choice of a stable framing of induces stable framings on and and thus a stable isomorphism using which the canonical -structure on and the -one on (see 3.1) induce a -structure on and a -structure on . With these choices, we have . By construction, the restriction of this -structure to agrees with the -structure on obtained from the stable framing of , so we obtain a -structure on , which by obstruction theory extends to one on : the relative Serre spectral sequence shows that vanishes for and thus that for , using and . The restriction of this -structure on to a and the canonical -structure on (see 3.1) assemble to a -structure on . By construction, the canonical restriction map (using excision)
[TABLE]
sends to , so we conclude . To finish the proof, note that the -structure on allows us to do surgery away from the boundary on to obtain an -connected manifold , which gives a class in as aimed for: holds by construction, by the bordism invariance of Pontryagin numbers (see 3.1), and by the additivity and bordism invariance of the signature.∎
Combining the previous result with 3.9, we conclude that the composition
[TABLE]
sends a homology class represented by a bundle to a certain linear combination of and whose coefficients involve the invariants and . In the following two subsections, we shall see that these functionals
[TABLE]
factor through the composition
[TABLE]
and have a more algebraic description in terms of . This uses the morphism
[TABLE]
induced by acting on a stable framing of (agreeing with the usual framing on ) as explained in Section 2.
3.4. Signatures of bundles of symplectic lattices
The standard action of the symplectic group on gives rise to a local system over and the usual symplectic form on gives a morphism of local systems to the constant system. To an oriented closed surface with a map , we can associate a bilinear form
[TABLE]
defined as the composition
[TABLE]
As both the cup-product and the symplectic pairing are antisymmetric, the form is symmetric. The usual argument for the bordism invariance of the signature shows that its signature depends only on the bordism class and thus induces a morphism
[TABLE]
which is compatible with the usual inclusion .
Remark 3.13*.*
As is perfect for (see A.1), the morphism (3.8) determines a unique cohomology class . There is a well-known (purely algebraic) cocycle representative of this class due to Meyer [Mey73], known as the Meyer cocycle.
The morphism (3.8) measures signatures of total spaces of smooth bundles over surfaces (even of fibrations of Poincaré complexes). More precisely, for a compact oriented -manifold , the action of its group of diffeomorphisms on the middle cohomology induces a morphism for and the resulting composition
[TABLE]
can be shown to map a homology class represented by a smooth bundle over a surface to the signature of its total space. This fact can either be proved along the lines of [CHS57] or extracted from [Mey72] and it has in particular the following consequence.
Lemma 3.14**.**
For odd, the composition
[TABLE]
sends the class of an -bundle to the signature of its total space.
We proceed by computing the image of the signature morphism (3.8) and of its pullback to the theta-subgroup as defined in Section 1.2.
Lemma 3.15**.**
The signature morphism satisfies
[TABLE]
Proof.
The signatures realised by classes in are well-known (see e.g. [BCRR20, Lem 6.5, Thm 6.6 (vi)]). To prove that the signature of classes in is divisible by , recall from Sections 1.2 and 1.3 that for odd the morphism lands in the subgroup as long as , so we have a composition
[TABLE]
which maps the class of a bundle by 3.14 to . The latter agrees by 3.12 with the signature of an almost closed -connected -manifold, so it is divisible by as the intersection form of such manifolds is unimodular and even (see e.g. [Wal62b]). This proves the claimed divisibility, since the first two morphisms in the composition are surjective, the first one because of the second reminder at the beginning of Section 3.3 and the second one by 2.4. As the signature morphism vanishes on by the first part, it certainly vanishes on . Consequently, by the compatibility of the signature with the inclusion , the remaining claim follows from constructing a class in of signature for . Using (see e.g. [BCRR20, Lem. A.1(iii)]) and the first part of the claim, the existence of such a class is equivalent to the image of in the torsion free quotient containing . That it contains is ensured by transfer, since the index of is (see Section 1.2). As and are -torsion by A.1, it therefore suffices to show that is nontrivial, for which we consider the level congruence subgroup , i.e. the kernel of the reduction map , which is surjective (see e.g. [NS64, Thm 1] for an elementary proof). From the explicit description of presented in Section 1.2, one sees that it contains the congruence subgroup . As a result, it is enough to prove that is nontrivial, which follows from an application of the Serre spectral sequence of the extension
[TABLE]
using that vanishes by a result of Sato [Sat10, Cor. 10.2] and that the groups are trivial in low degrees, because of the exceptional isomorphism between and the symmetric group in letters (see e.g. [O’M78, p. 37]). ∎
Remark 3.16*.*
xyz
- (i)
There are at least two other proofs for the divisibility of the signature of classes in by . One can be extracted from the proof of [GRW16, Lem. 7.5 i)] and another one is given in [Ben17, Thm 12.1]. The proof in [GRW16] shows actually something stronger, namely that the form associated to a class is always even. We shall give a different proof of this fact as part of the second part of 3.19 below. 2. (ii)
For , the existence of a class in of signature was shown as part of the proof of [GRW16, Thm 7.7], using that the image of
[TABLE]
for is known to be by a result of Putman [Put12, Thm F]. However, this argument breaks for small values of , in which case the image of was not known before, at least to the knowledge of the author.
By 3.15, the signatures of classes in are divisible by , so we obtain a morphism
[TABLE]
To lift this morphism to a cohomology class in , we consider the morphism
[TABLE]
induced by the matrix . By A.1, this is an isomorphism on abelianisations for and thus induces a splitting of the universal coefficient theorem
[TABLE]
This splitting is compatible with the inclusion , so we can define a lift of the divided signature to a class as follows.
Definition 3.17**.**
xyz
- (i)
Define the class
[TABLE]
for via the splitting (3.10) by declaring its image in the first summand to be trivial and to be in the second. For small , the class is defined as the pullback of for . 2. (ii)
Define the class
[TABLE]
as the pullback of the same-named class along the map induced by the action on the middle cohomology.
3.5. Framing obstructions
To describe the invariant
[TABLE]
explained in Section 3.1 more algebraically, note that a map from an oriented closed connected based surface to induces a -cocycle
[TABLE]
and hence a class . The composition defines a bilinear form on as explained in Section 3.4, and hence a number . Varying , this gives a morphism
[TABLE]
which is compatible with natural inclusion and takes for part in a composition
[TABLE]
where the first morphism is induced by acting on a stable framing of as in Section 2. A priori, this requires three choices: a stable framing, a generator , and a symplectic basis . However, the composition turns out to not be affected by these choices and the following proposition shows that it is related to the invariant of -bundles explained in 3.1.
Proposition 3.18**.**
For , the composition
[TABLE]
sends the class of a -bundle to .
Proof.
The relative Serre spectral sequence of
[TABLE]
induces canonical isomorphisms
[TABLE]
where denotes the composition
[TABLE]
By the compatibility of the Serre spectral sequence with the cup-product and after identifying with , it suffices to show that the second isomorphism sends up to signs to the class represented by the cocycle
[TABLE]
involving the choice of stable framing as in Section 2. As a first step, we describe this isomorphism more explicitly:
Note that as is trivial for . Unwinding the construction of the Serre spectral sequence using a skeletal filtration of , one sees that after fixing an identification , the image of a class under the isomorphism in question is represented by the cocycle which maps a loop to the class obtained from a choice of lift by pulling it back along
[TABLE]
where the second morphism is induced by pulling back the bundle along and the first morphism is the unique (up to homotopy) trivialisation relative to of the pullback bundle over ; here we used the canonical isomorphism .
Recall from 3.1 that the class is the primary obstruction to extending the canonical -structure on to a -structure. The choice of framing induces such an extension on the fibre and thus induces a lift of to a relative class in . This uses that the structure on induced by the framing agrees with the restriction of the -structure on to by obstruction theory. Using the above description, we see that the image of under the isomorphism in question is thus represented by the cocyle that sends a loop to the primary obstruction to solving the lifting problem
[TABLE]
in , since this obstruction agrees with the corresponding obstruction when replacing by . Here the left square in the diagram is given by the trivialisation explained earlier. There is a useful alternative description of this obstruction class: relative to the subspace there is a unique (up to homotopy) lift in (3.12), so the obstruction to finding a lift relative to the subspace can be seen as an element in the group of path-components of the fibre of the principal fibration , which is exactly .
To see that the cocyle we just described agrees with (3.11), note that the function induced by acting on the stable framing arises as the connecting map of the fibration
[TABLE]
where is the space that classifies -bundles with a -structure on the vertical tangent bundle extending the given -structure on the restriction to the trivial -subbundle induced by the standard framing of ; here we identified the space of -structures of relative to with the mapping space by using the choice of stable framing , which also induces the basepoint . This shows that the value of (3.11) on a loop is given by the component in obtained by evaluating a choice of path-lift
{\{0\}}$${*}$${\mathrm{BDiff}^{\tau_{>n+1}}_{\partial/2}(W_{g,1})}$${{\left[0,1\right]}}$${S}$${\mathrm{BDiff}_{\partial/2}(W_{g,1}).}$$\scriptstyle{\subset}$$\scriptstyle{F}$$\scriptstyle{\omega}
at the end point. Such a path-lift precisely classifies a lift as in (3.12) relative to the subspace , so the claim follows from the second description of the obstruction to solving the lifting problem (3.12) mentioned above. ∎
Lemma 3.19**.**
Let .
- (i)
The image of the composition induced by the inclusion
[TABLE]
contains and agrees with for . 2. (ii)
We have
[TABLE] 3. (iii)
For , we have
[TABLE]
Proof.
By the compatibility of with the stabilisation maps, it suffices to show the first part for , which follows from checking that the image of a generator in is mapped to under by chasing through the definition. As the signature morphism pulls back from , the second part follows from the first part and 3.15 by showing that the image of is for always divisible by and for divisible by if and only if . In the case of , this can be shown “geometrically” as in the proof of 3.15: choose and consider the composition
[TABLE]
The first morphism is surjective by the second reminder at the beginning of Section 3.3 and the second morphism is an isomorphism as a result of Section 2, so it suffices to show that the composition is divisible by , which in turn follows from a combination of 3.18, 3.12 and the fact that in these dimensions, an -connected almost closed -manifold satisfies by a combination of 3.2 and (3.3). For , we argue as follows: in the case , we show that the morphism is surjective, which will exhibit the claimed divisibility as a consequence of (i). It follows from A.3 that the group vanishes and as , we also have 333This can be seen for instance from the Meyer–Vietoris sequence of the well-known decomposition , where is generated by , by , and by ., so an application of the Serre spectral sequence to shows the claimed surjectivity. This leaves us with proving that is not divisible by for for which we use that there is class of signature by 3.15, so the form cannot be even and hence there is a -cocycle for which is odd, which means that the image of the class induced by the morphism is odd. For the last part, note that the argument we gave for the divisibility in the second part for shows for that the image of the composition in (iii) is contained in , since is divisible by if . Hence, to finish the proof, it suffices to establish the existence of a class in for on which the composition evaluates to . To this end, we consider the square
{\mathrm{H}_{2}(\mathbf{Z}^{2g}\otimes S\pi_{n}\mathrm{SO}(n);\mathbf{Z})}$${\mathrm{H}_{2}(\Gamma^{n}_{g,1/2};\mathbf{Z})}$${\mathrm{H}_{2}(\mathbf{Z}^{2g}\otimes\pi_{n}\mathrm{SO};\mathbf{Z})}$${\mathrm{H}_{2}((\mathbf{Z}^{2g}\otimes\pi_{n}\mathrm{SO})\rtimes\mathrm{Sp}_{2g}(\mathbf{Z});\mathbf{Z}),}$$\scriptstyle{(s_{F},p)}
induced by the embedding of the extension describing into the trivial extension of by (see Section 2). By the first part, there is a class with and trivial signature, since the signature morphism pulls back from . As a result of 2.1, the cokernel of the left vertical map in the square is -torsion if , so lifts to and provides a class as desired. ∎
Similar to the construction of , we would like to lift the morphism
[TABLE]
resulting from the second part of 3.19 to a class . To this end, observe that induces an isomorphism on abelianisations for since the coinvariants vanish in this range by A.2. The morphism
[TABLE]
considered in Section 3.4 thus induces a splitting
[TABLE]
for , analogous to the splitting (3.10). As before, this splitting is compatible with the inclusions , so the following definition is valid.
Definition 3.20**.**
xyz
- (i)
Define the class
[TABLE]
for via the splitting (3.13) by declaring its image in the first summand to be trivial and to be in the second. For small , the class is defined as the pullback of the class for . 2. (ii)
Define the class
[TABLE]
as the pullback of the same-named class along the map
[TABLE] 3. (iii)
Define the class
[TABLE]
for as image of ensured by 3.19, and for small as the pullback of for .
The isomorphism for in the previous definition is assured by 2.4 and A.1, as vanishes for .
We finish this subsection with an auxiliary lemma convenient for later purposes.
Lemma 3.21**.**
For , the class and the pullback of the class to are trivial.
Proof.
Both classes evaluate trivially on . For the first class, this is a consequence of 3.15, for the second this holds by construction. Moreover, both classes pull back trivially to along the morphism by definition. Although this morphism does not induce an isomorphism on abelianisations for , it still induces one on the torsion subgroups of the abelianisations by A.1 and this is sufficient to deduce the claim from the universal coefficient theorem. ∎
3.6. The proof of B
We are ready to prove our main result B, which we state equivalently in terms of the central extension
[TABLE]
explained in Section 1.3. Our description of its extension class in involves the cohomology classes in of Definitions 3.17 and 3.20, and the two homotopy spheres and in the subgroup examined in Section 3.2.1. We write for the change of coefficients induced by .
Theorem 3.22**.**
For odd, the extension
[TABLE]
is classified by the class
[TABLE]
and its induced differential has image
[TABLE]
Moreover, the extension splits if and only if and .
Proof.
As all cohomology classes involved are compatible with the stabilisation map , it is sufficient to show the first part of the claim for (see Section 1.4). We assume first. Identifying with via the isomorphism of Section 2, the morphism of the previous section induces a morphism between the sequences of the universal coefficient theorem
{0}$${\mathrm{Ext}(\mathrm{H}_{1}(\Gamma^{n}_{g,1/2};\mathbf{Z}),\Theta_{2n+1})}$${\mathrm{H}^{2}(\Gamma^{n}_{g,1/2};\Theta_{2n+1})}$${\mathrm{Hom}(\mathrm{H}_{2}(\Gamma^{n}_{g,1/2};\mathbf{Z}),\Theta_{2n+1})}$${0}$${0}$${\mathrm{Ext}(\mathbf{Z}/4,\Theta_{2n+1})}$${\mathrm{H}^{2}(\mathbf{Z}/4;\Theta_{2n+1})}$${\mathrm{Hom}(\mathrm{H}_{2}(\mathbf{Z}/4;\mathbf{Z}),\Theta_{2n+1})}$${0.}
By the exactness of the rows and the vanishing of , it is sufficient to show that the extension class in consideration agrees with the classes in the statement when mapped to and . Regarding the images in the Hom-term, it is enough to identify them after precomposition with the epimorphism
[TABLE]
so from the construction of and together with Lemmas 3.15 and 3.19, we see that it suffices to show that induced by the extension class maps the class of a bundle to if and to otherwise, which is a consequence of 3.12 combined with 3.9. By construction, the classes and vanish in , so the claim for follows from showing that the extension class is trivial in , i.e. that the pullback of the extension to splits, which is in turn equivalent to the existence of a lift
{\Gamma^{n}_{g,1}}$${\mathbf{Z}/4}$${G_{g}\subset{(H(g)\otimes\pi_{n}\mathrm{SO})\rtimes G_{g}}.}$$\scriptstyle{(s_{F},p)}
Using the standard embedding , we consider the diffeomorphism
[TABLE]
which is of order , maps to , and has constant differential, so it vanishes in . As the natural map is an isomorphism by 1.1, this diffeomorphism induces a lift as required for , which in turn provides a lift for all via the stabilisation map . For , the abelianisation of vanishes for due to 2.4, so it suffices to identify the extension class with the classes in the statement in which follows as in the case .
3.19 (iii) implies that the image of for is generated by for all . For , the map is an isomorphism (see Section 2), so 3.19 tells us that the image of for is generated by and if and by if . For , it follows from 3.15 that the image of is generated by if and that it is trivial for . In sum, this implies the second part of the claim by 3.5, and also that the differential does not vanish for , so the extension is nontrivial in these cases. For , the homotopy sphere is nontrivial by 3.3, so does not vanish for either. Finally, in the case , the extension is classified by , which is trivial for by 3.21, so the extension splits and the proof is finished. ∎
It is time to make good for the missing part of the proof of 2.2.
Proof of 2.2 for .
We have , so the case follows from the fact that vanishes by A.3. To prove the case , note that a hypothetical splitting of the upper row of the commutative diagram
{0}$${\mathbf{Z}^{2g}\otimes S\pi_{n}\mathrm{SO}(n)}$${\Gamma_{g,1/2}^{n}}$${\mathrm{Sp}_{2g}(\mathbf{Z})}$${0}$${0}$${\mathbf{Z}^{2g}\otimes\pi_{n}\mathrm{SO}}$${(\mathbf{Z}^{2g}\otimes\pi_{n}\mathrm{SO})\rtimes\mathrm{Sp}_{2g}(\mathbf{Z})}$${\mathrm{Sp}_{2g}(\mathbf{Z})}$${0}$$\scriptstyle{(s_{F},p)}
induces a splitting of the lower row, which agrees with the canonical splitting of the lower row up to conjugation with , because such splittings up to conjugation are a torsor for which vanishes by A.3. 3.15 on the other hand ensures that there is a class with signature , so satisfies and , which contradicts 3.19 (iii).∎
4. Kreck’s extensions and their abelian quotients
The inclusion of the Torelli group extends to a pullback diagram
[TABLE]
of extensions whose bottom row we identified in 3.22. We now apply this to obtain information about the top row, and moreover to compute the abelianisations of and in terms of the homotopy sphere , the subgroup of Section 3.2, and the abelianisation of as computed in 2.4.
Theorem 4.1**.**
Let be odd and .
- (i)
The kernel of the morphism is generated by for and agrees with the subgroup for . The induced extension
[TABLE]
splits. 2. (ii)
The extension
[TABLE]
is nontrivial for and splits -equivariantly for . The image of its differential is generated by . 3. (iii)
The commutator subgroup of is generated by and the resulting extension
[TABLE]
splits -equivariantly.
Proof.
By the naturality of the Serre spectral sequence, the morphism of extension (4.1) induces a ladder of exact sequences
{\mathrm{H}_{2}(H(g)\otimes S\pi_{n}\mathrm{SO}(n);\mathbf{Z})}$${\Theta_{2n+1}}$${\mathrm{H}_{1}(\mathrm{T}_{g,1}^{n})}$${H(g)\otimes S\pi_{n}\mathrm{SO}(n)}$${0}$${\mathrm{H}_{2}(\Gamma_{g,1/2}^{n};\mathbf{Z})}$${\Theta_{2n+1}}$${\mathrm{H}_{1}(\Gamma_{g,1}^{n})}$${\mathrm{H}_{1}(\Gamma_{g,1/2}^{n})}$${0}$$\scriptstyle{d_{2}^{\mathrm{T}}}$$\scriptstyle{d_{2}^{\Gamma}}
from which we see that the kernel in question agrees with the image of the differential , which we described in 3.22. By the universal coefficient theorem, the pushforward of the extension class of the bottom row of (4.1) along the quotient map is classified by a class in , which also describes the exact sequence in (i). By a combination of 3.22 and 3.5, this class is trivial for , and for as long as . For and , this class agrees with the image of in and therefore vanishes as is trivial by 3.21.
The extension class of the bottom row of (4.1), determined in 3.22, pulls back to the extension class of the top one. Since is pulled back from by construction, it is trivial in , so the extension in (ii) is trivial for , classified by for if , and by if . From this, the claimed image of follows from 3.19 and its proof. For , the homotopy sphere is nontrivial, so the extension does not split. This shows the second part of the statement, except for the claim regarding the equivariance, which will follow from the third part, since is trivial for by 3.3.
The diagram above shows that the commutator subgroup of agrees with the image of , which we already showed to be generated by . To construct a -equivariant splitting as claimed, note that the quotient of the lower row of (4.1) by pulls back from by 3.22 because has this property. Consequently, there is a central extension fitting into a commutative diagram
{0}$${\Theta_{2n+1}/\Sigma_{Q}}$${\mathrm{T}_{g,1}/\Sigma_{Q}}$${H(g)\otimes S\pi_{n}\mathrm{SO}(n)}$${0}$${0}$${\Theta_{2n+1}/\Sigma_{Q}}$${\Gamma^{n}_{g,1}/\Sigma_{Q}}$${\Gamma^{n}_{g,1/2}}$${0}$${0}$${\Theta_{2n+1}/\Sigma_{Q}}$${E}$${G_{g}}$${0}$$\scriptstyle{p}
whose middle vertical composition induces a splitting as claimed, using that the right column in the diagram is exact. ∎
The previous theorem, together with 1.1, 2.4, and 3.2 has C (ii) and D as a consequence. It also implies the following.
Corollary 4.2**.**
For odd and , is abelian if and only if .
4.1. A geometric splitting
The splittings of the abelianisations of and provided by 4.1 are of a rather abstract nature. Aiming towards splitting these sequences more geometrically, we consider the following construction.
A diffeomorphism fixes a neighbourhood of the boundary pointwise, so its mapping torus comes equipped with a canonical germ of a collar of its boundary using which we obtain a closed oriented -connected -manifold by gluing in . By obstruction theory and the fact that is -parallelisable, the stable normal bundle has a unique lift to compatible with the lift on induced by its standard stable framing. This gives rise to a morphism
[TABLE]
which is compatible with the stabilisation map since
[TABLE]
where carries the -structure induced by the standard stable framing, which bounds. Using this, it is straight-forward to see that the composition
[TABLE]
of the iterated stabilisation map with agrees with the canonical epimorphism appearing in Wall’s exact sequence (3.1), so its kernel agrees with the subgroup . Together with the first part of 4.1, we conclude that the dashed arrow in the commutative diagram
{0}$${\Theta_{2n+1}/K_{g}}$${\mathrm{H}_{1}(\Gamma_{g,1}^{n})}$${\mathrm{H}_{1}(\Gamma^{n}_{g,1/2})}$${0}$${\Omega_{2n+1}^{\tau_{>n}},}$$\scriptstyle{p_{*}}$$\scriptstyle{t_{*}}
is an isomorphism if and only if which is the case for , and for as long as since generates for by 3.5. Consequently, in these cases, the morphism
[TABLE]
is an isomorphism, whereas its kernel for coincides with the quotient , which is nontrivial as long as , generated by , and whose order can be interpreted in terms of signatures (see 3.10). Since splits by 4.1 and the natural map is an isomorphism by 3.6, the morphism (4.2) splits for if and only if the natural map
[TABLE]
does. Brumfiel [Bru68, Thm 1.3] has shown that this morphism always splits before taking quotients, so the map (4.3) (and hence also (4.2)) in particular splits whenever is trivial, which is conjecturally always the case (see 3.7) and known in many cases as a result of 3.3 (see also 3.8).
The situation for is similar. By 4.1, the morphism in the diagram
{0}$${\Theta_{2n+1}/\Sigma_{Q}}$${\mathrm{H}_{1}(\mathrm{T}_{g,1}^{n})}$${H(g)\otimes S\pi_{n}\mathrm{SO}(n)}$${0}$${\Omega_{2n+1}^{\tau_{>n}}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{t_{*}}
splits -equivariantly and since the morphism is defined on , its restriction to is -equivariant when equipping with the trivial action. By an analogous discussion to the one above, the kernel of the resulting morphism of -modules
[TABLE]
is trivial for and given by the quotient for . Moreover, this morphism splits if and only if it splits -equivariantly,444If is a non-equivariant splitting of , then is an equivariant splitting of , where an equivariant splitting of ensured by 4.1. This uses that is already equivariant since , so the image of is contained in the image of which is fixed by the action. which is precisely the case if the natural map (4.3) admits a splitting. We summarise this discussion in the following corollary, which has E as a consequence when combined with 3.10.
Corollary 4.3**.**
Let and odd.
- (i)
The morphism
[TABLE]
is an isomorphism for . For , it is an epimorphism and has kernel . 2. (ii)
The morphism
[TABLE]
is an epimorphism and has kernel . 3. (iii)
The morphism splits for if and only if
[TABLE]
splits, which is the case for . The same holds for for all .
4.2. Abelianising and for even
For even, the abelianisations of and can be computed without fully determining the extensions
[TABLE]
Indeed, arguing similarly as in the proof of 2.4, the second extension provides an isomorphism , using that the coinvariants vanish also for as long as by A.2 and that the extension splits for , which is straightforward to check by noting that it is easy to lift elements of
[TABLE]
to (so in particular to ) using the flip of the factors and a diffeomorphism of of degree 555There are several mistakes in the literature related to the fact that is isomorphic to and not : in [Kre79, p. 645], it should be , in [Sat10, Thm 1] it should be for even, and finally in the proof of [Kry02, Thm 2.6] it should be for even.. In contrast to the case , the resulting analogues of the morphisms (4.2) and (4.4) for even are isomorphisms for all ; this is G.
Proof of G.
Wall’s exact sequence (3.1) is also valid for even, so implies similarly to the case odd that both morphisms in question are surjective and that their kernels agree with the quotients of by the images of the differentials
[TABLE]
induced by the extensions (1.5) and (1.6), so we have to show that these images agree with . By comparing the extensions for different via the stabilisation map (see Section 1.4) and noting that the second differential factors through the first, we see that it suffices to show this for the second differential in the case . Plumbing disc bundles over defines a pairing of the form
[TABLE]
which can be seen to be surjective for even, unlike in the case odd [Wal67, p. 295]. The composition of the pairing with the boundary map with image is usually called the Milnor pairing. Using the surjectivity, it is enough to show that the image of the second differential contains the image of the Milnor pairing. To do so, we rewrite the extension inducing the second differential via the canonical isomorphism resulting from 1.1 as
[TABLE]
and constructsin an explicit cocycle classifying this extension as follows: define morphisms
[TABLE]
by assigning to a class represented by the diffeomorphism and , where is the upper half-disc. Choosing the centre as the base point, the diffeomorphisms and fix both spheres , one because is a based map and the other because it factors through the stabilisation map. From the description of in Section 1.3, we see that and , so the function mapping to is a set-theoretical section to , which implies that
[TABLE]
defines a -cocycle that represents the extension class of (4.5) in (see e.g. [Bro82, Ch. IV.3]); here we identified with its image in . The differential is the image of this class under the map participating in the universal coefficient theorem (see Section 3.3), so with respect to the canonical isomorphism (see e.g. [Bro82, Thm V.6.4 (iii)], the differential takes the form (see e.g. [Bro82, Ex. IV.4.8 (c), Ex. V.6.5])
[TABLE]
The precomposition of this differential with the map mapping to thus agrees with the pairing
[TABLE]
that sends to the commutator . By the discussion in [Law72, p. 834], this coincides with the Milnor pairing, so its image is by the discussion above, which concludes the claim. ∎
4.3. Splitting the homology action
We conclude our study of Kreck’s extensions and its abelianisations with the following result, which proves the remaining first part of C. The reader shall be reminded once more of 1.1, saying that the natural map is an isomorphism for .
Theorem 4.4**.**
For odd, the extension
[TABLE]
does not split for , but admits a splitting for and .
Proof.
The case and is clear, since A shows that under this assumption even the quotient of the extension by the subgroup does not split. To deal with the other cases, we consider the morphism of extensions
{0}$${\Theta_{2n+1}}$${\Gamma_{g,1}^{n}}$${\Gamma^{n}_{g,1/2}}$${0}$${0}$${\mathrm{T}_{g,1}^{n}}$${\Gamma_{g,1}^{n}}$${G_{g}}$${0.}
By the naturality of the Serre spectral sequence, we have a commutative square of the form
{\mathrm{H}_{2}(\Gamma^{n}_{g,1/2};\mathbf{Z})}$${\Theta_{2n+1}}$${\mathrm{H}_{2}(G_{g};\mathbf{Z})}$${\mathrm{H}_{1}(\mathrm{T}_{g,1}^{n};\mathbf{Z})_{G_{g}}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{2}}
whose right vertical map has kernel generated by by the computation of as a -module in 4.1. Therefore, to finish the proof of the first claim, it suffices to show that the differential is nontrivial for and , which follows from 3.22 together with the fact that is nontrivial in these cases by 3.10 and 3.11. Turning towards the second claim, we assume and recall that the isomorphism induces a splitting of the right vertical map in the above diagram (see Section 2), so the claim follows from showing that the pullback of along splits for , which is a consequence of B and 3.21. ∎
Remark 4.5*.*
4.4 leaves open whether admits a splitting for in dimensions . Krylov [Kry03, Thm 2.1] and Fried [Fri86, Sect. 2] showed that this can not be the case for and we expect the same to hold for .
5. Homotopy equivalences
Our final result F is concerned with the morphism of extensions
[TABLE]
underlying work of Baues [Bau96, Thm 10.3], relating the mapping class group to the group of homotopy classes of orientation preserving homotopy equivalences. The left vertical morphism is induced by the restriction of the unstable homomorphism to the image of the stabilisation in the source and to the image of the suspension map in the target, justified by the fact that the square
{\pi_{n}\mathrm{SO}(n)}$${\pi_{n}\mathrm{SO}(n+1)}$${\pi_{2n}S^{n}}$${\pi_{2n+1}S^{n+1}}$$\scriptstyle{S}$$\scriptstyle{J}$$\scriptstyle{J}$$\scriptstyle{S}
commutes up to sign (see [Tod62, Cor. 11.2]).
Proof of F.
By A, the upper row of (5.1) splits for odd, so the first part of (i) is immediate. The second part follows from A as well if we show that the existence of a splitting of the lower row for is equivalent to one of the upper row. To this end, note that for , we have isomorphisms and for and with respect to which the -homomorphism is given by reduction by (see e.g. [Tod62, Ch. XIV]). By A.3, the group is annihilated by , so in particular by . Using this, the first claim follows from the long exact sequence on cohomology induced by the exact sequence
[TABLE]
of -modules, since the extension class of the lower row is obtained from that of the upper one by the change of coefficients .
To prove the second part, we consider the exact sequence
[TABLE]
induced by the Serre spectral sequence of the lower row of (5.1). The left morphism in this sequence is split injective as long as the extension splits, which, together with the first part and a consultation of A.2, exhibits to be as asserted. ∎
Appendix A Low-degree cohomology of symplectic groups
This appendix contains various results on the low-degree (co)homology of the integral symplectic group and its theta subgroup (see Section 1.2).
Lemma A.1**.**
xyz**
- (i)
The abelianisations of and satisfy
[TABLE] 2. (ii)
The element generates for . 3. (iii)
The element generates the -summand in for all . 4. (iv)
The -summand in is generated by and the -summand in is generated by , where . 5. (v)
The stabilisation map
[TABLE]
is surjective for all and the stabilisation map
[TABLE]
is surjective for , but has cokernel for .
Proof.
The fact that the abelianisation of is generated by and of order is well-known, and so is the isomorphism type of for (see e.g. [BCRR20, Lem. A.1 (ii)]). The remaining claims regarding follows from showing that is nontrivial, which can for instance be extracted from the proof of [GH98, Thm 2.1]: in their notation and the map identifies with the stabilisation . The isomorphism type of for is determined in [End82, Thm 2] and for in [Wei91, Thm 1]. The first claim of (ii) follows from the main formula of [JM90] for and from [Wei91, Cor. 2] for , which also gives the claimed generator of the -summand in . The proof of [End82, Thm 2] provides the asserted generator of the -summand in . The final claim follows from the first four items once we show that the image of in vanishes, which is another consequence of the formulas in [Wei91, Cor. 2]. ∎
Lemma A.2**.**
The (co)invariants of the standard actions of and on for an abelian group satisfy
[TABLE]
The same applies to the action of , except that the (co)invariants also vanish for .
Proof.
We prove the claim for first. The self-duality of induced by the symplectic form implies
[TABLE]
so the computation of the invariants is a consequence of that of the coinvariants. To settle the case for both and , it thus suffices to prove that the coinvariants with respect to the smaller group vanish. We consider the matrices
[TABLE]
which can be seen to be contained in the subgroup using the description in Section 1.2; here is the unit matrix and the permutation matrix associated to . It suffices to show that and are trivial in the coinvariants for and , where is the standard symplectic basis of . Writing for the residue class of an element in the coinvaraints, acting with the permutation matrices shows and for all , so it suffices to prove . Using the second matrix, we see that and , and finally the third matrix shows , so the coinvariants are trivial. In the case , the first part of the proof is still valid and shows that and are quotients of generated by for . For , we may use to conclude , so the coinvariants vanish. For , one uses that is generated by and (see e.g. [Wei91, p. 385]) to see that the surjection induced by adding coordinates is invariant and surjective, and thus induces an isomorphism
[TABLE]
as claimed. The proof for is almost identical to that for , except that one has to replace the second displayed matrix by , also act with , and use that is generated by and .∎
Lemma A.3**.**
The cohomology groups and with coefficients in the standard module are annihilated by and in low degrees given by
[TABLE]
Proof.
The negative of the identity matrix is central and acts by on , so the first claim follows from the “centre kills trick" which is worth recalling: multiplication by an element of a discrete group on a -module is equivariant with respect to the conjugation by and induces the identity on by the usual argument. If is central, conjugation by is trivial, so is annihilated by the action of on .
Since is -torsion, the self-duality of combined with the universal coefficient theorem for nontrivial coefficients (see e.g. [Span66, p. 283]) implies
[TABLE]
so the computation of for is a consequence of A.2.
From the Lyndon–Hochschild–Serre spectral sequence for the extension
[TABLE]
with coefficients in , we see that
[TABLE]
and the right hand side can easily be computed to be as claimed by applying the Mayer–Vietoris sequence to the presentation
[TABLE]
for
[TABLE]
which is well-known for and appears e.g. in [Wei91, p. 385] for . ∎
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