Mapping class groups of highly connected $(4k+2)$-manifolds

We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.

The classical mapping class group Γ of a genus surface naturally generalises to all even dimensions 2n as the group of isotopy classes Γ n = π 0 Diff + (W ) of orientation-preserving diffeomorphisms of the -fold connected sum W = ♯ (S n × S n ). Its action on the middle cohomology H ( ) ≔ H n (W ; Z) Z 2 provides a homomorphism Γ n → GL 2 (Z) whose image is the symplectic group Sp 2 (Z) in the surface case 2n = 2, and a certain arithmetic subgroup G ⊂ Sp 2 (Z) or G ⊂ O , (Z) in general, the description of which we shall recall later. The kernel T n ⊂ Γ n of the resulting extension 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 2n ≥ 6 is comparatively manageable: there is an extension (2) 0 −→ Θ 2n+1 −→ T n −→ H ( ) ⊗ Sπ n SO(n) −→ 0 due to Kreck [Kre79], which relates T n to the finite abelian group of homotopy spheres Θ 2n+1 and the image of the stabilisation map S : π n SO(n) → π n SO(n + 1), shown in Table 1. The description of Γ n 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 n [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 n ≥ 3 odd, resulting in a complete description of the mapping class group Γ n and the Torelli group T n in terms of the arithmetic group G and the group of homotopy spheres Θ 2n+1 .
To explain our results, note that (1) and (2) induce further extensions which express Γ n in terms of G and Θ 2n+1 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. T 1. The groups Sπ n SO(n) for n ≥ 3, except that Sπ 6 SO(6) = 0.
Moreover, this extension splits if and only if n ≡ 1 (mod 4) and = 1.
The extension (2) describing the Torelli group T n is the pullback of the extension determined in Theorem B along the map H ( ) ⊗Sπ n SO(n) → Γ n /Θ 2n+1 , so the combination of the previous result with our identification of Γ n /Θ 2n+1 provides an algebraic description of both Γ n and T n in terms of G and Θ 2n+1 . We derive several consequences from this, beginning with deciding when the more commonly considered extensions (1) and (2) split.
Abelian quotients. The second application of our description of the groups Γ n and T n is a computation of their abelianisations.
Although Theorem A exhibits the extension (4) as being nontrivial in some cases, its abelianisation turns out to split nevertheless (see Corollary 2.4), so there exists a splitting H 1 (Γ n /Θ 2n+1 ) H 1 (G ) ⊕ (H ( ) ⊗ Sπ n SO(n)) G , which participates in the following identification of the abelianisation of Γ n and T n .
(i) The extension (3) induces a split short exact sequence where K = Σ P , Σ Q for ≥ 2 and K = Σ Q for = 1. (ii) The extension (4) induces a split short exact sequence of G -modules In particular, the commutator subgroup of T n is generated by Σ Q .
These splittings of H 1 (Γ n ) and H 1 (T n ) are constructed abstractly, but can often be made more concrete by means of a refinement of the mapping torus construction as a map t : Γ n −→ Ω τ >n 2n+1 to the bordism group of closed (2n + 1)-manifolds M equipped with a lift of their stable normal bundle M → BO to the n-connected cover τ >n BO → BO. To state the resulting more explicit description of the abelianisations of Γ n and T n , we write σ ′ n for the minimal positive signature of a closed smooth n-connected (2n + 2)-dimensional manifold. For n 1, 3, 7 odd, the intersection form of such a manifold is even, so σ ′ n is divisible by 8. Corollary E. Let n ≥ 3 odd and ≥ 1.

Remark.
(i) In Theorem G below, we determine the abelianisation of Γ n , 1 and T n , 1 for n ≥ 4 even in which case the morphisms t * ⊕ p * and t * ⊕ ρ * are isomorphisms for all ≥ 1 .
(ii) As shown in [KR20,Prop. 2.15], the minimal signature σ ′ n is nontrivial for n odd, grows at least exponentially with n, and can be expressed in terms of Bernoulli numbers. (iii) For some values of and n, Corollary E leaves open whether t * ⊕ p * and t * ⊕ ρ * split.
The morphisms p * and ρ * always split by Corollary D, and in Section 4.1 we relate the question of whether there exist compatible splittings of t * 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. (iv) Work of Thurston [Thu74] shows that the component Diff 0 (W ) ⊂ Diff + (W ) of the identity is perfect as a discrete group, so the abelianisation of the full diffeomorphism group Diff + (W ) considered as a discrete group agrees with H 1 (Γ n ).
T 2. The abelianisation of G for n odd.
In view of Corollary E, it is of interest to determine the bordism groups Ω τ >n 2n+1 , the abelianisation H 1 (G ), and the coinvariants (H ( ) ⊗ Sπ n SO(n)) G . The computation ≥ 2 or n = 3, 6, 7 or n ≡ 5 (mod 8) Z/2 2 = 1 and n ≡ 0 (mod 8) Z/2 otherwise is straightforward (see Lemma A.2 and Table 1). The abelianisation of G is known and summarised in Table 2 (see Lemma A.1). Finally, the bordism groups Ω τ >n 2n+1 are closely connected to the stable homotopy groups of spheres: the canonical map 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 Corollary 3.6).
Combined with Corollary E, this reduces the computation of the abelianisation of Γ n and T n 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 Γ n and T n for the first few values of n .

T
3. Some abelianisations of T n and Γ n .
H 1 (T n ) n = 3 n = 5 n = 7 n = 9 = 0 Z/28 Z/992 Z/2 ⊕ Z/8128 Remark. After the completion of this work, Burklund-Hahn-Senger [BHS19] and Burklund-Senger [BS20] showed that for n odd, the homotopy sphere Σ Q ∈ Θ 2n+1 bounds a parallelisable manifold if and only if n 11. This implies in particular that aside from n = 11 (i) the canonical map coker( ) 2n+1 → Ω τ >n 2n+1 is an isomorphism, which extends the theorem attributed to Schultz and Wall above, (ii) the conjecture of Galatius-Randal-Williams mentioned in the third part of the previous remark holds, and (iii) the minimal signature σ ′ n appearing in Corollary E is computable from [KR20, Prop. 2.15].
Homotopy equivalences. As an additional application of our results, we briefly discuss the group π 0 hAut + (W ) of homotopy classes of orientation-preserving homotopy equivalences. The natural map Γ n → π 0 hAut + (W ) can be seen to factor through the quotient Γ n /Θ 2n+1 and to induce a commutative diagram of the form which exhibits the lower row-an extension describing π 0 hAut + (W ) due to Baues [Bau96]-as the extension pushout of the extension (4) along the left vertical morphism, which is induced by the restriction : Sπ n SO(n) → Sπ 2n S n of the unstable -homomorphism, where Sπ 2n S n is the image of the suspension map S : π 2n S n → π 2n+1 S n+1 . By Theorem 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.
The groups Γ n , 1 for n even. Some parts in our analysis of Γ n , 1 go through when n ≥ 4 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 Theorem 3.12 to incorporate the Arf invariant. The abelianisation of the groups Γ n , 1 and T n , 1 , however, can be determined without fully solving the extensions (3) and (4) if n is even. It turns out that in this case, the morphisms considered in Corollary E are isomorphisms for all ≥ 1, which we shall prove as part of Section 4.2.
Other highly connected manifolds. Instead of restricting to W , one could consider any (n − 1)-connected almost parallelisable manifold M of dimension 2n ≥ 6. Baues' and Kreck's work [Bau96,Kre79] applies in this generality, so there are analogues of the sequences (1)-(6) describing π 0 Diff + (M) and π 0 hAut + (M). However, for n 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 W ♯Σ with an exotic sphere Σ ∈ Θ 2n , aside from those of Kervaire invariant 1, which only exist in dimensions 6, 14, 30, 62, and possibly 126 by work of Hill-Hopkins-Ravenel [HHR16]. The mapping class group π 0 Diff + (W ♯Σ) for Σ ∈ Θ 2n and n odd in turn is completely understood in terms of Γ n : Kreck's work [Kre79,Lem. 3,Thm 3] shows that the former is a quotient of the latter by a known element Σ ′ ∈ Θ 2n+1 of order at most 2, which is trivial if and only if η · [Σ] ∈ coker( ) 2n+1 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 Γ n for ≥ 5 and used this to determine the extension (3) for n ≡ 5 (mod 8) up to automorphisms of Θ 2n+1 as long ≥ 5.
Our work recovers and extends their results, also applies to low genera < 5, and does not rely on their work on moduli spaces of manifolds. (ii) Theorems A and F for n = 3, 7 reprove results due to Crowley [Cro11]. (iii) Baues [Bau96, Thm 8.14, Thm 10.3] showed that the lower extension in (6) splits for n 3, 7 odd, which we recover as part of the first part of Corollary F. (iv) The case ( , n) = (1, 3) of Theorem A and Corollary C (ii) can be deduced from work of Krylov [Kry03] and Fried [Fri86], who also showed that the extension of Corollary C (i) does not split in this case. Krylov [Kry02, Thms 2.1, 3.2, 3.3] moreover established the case n ≡ 5 (mod 8) of Corollary C (i) for = 1. For n 3, 7, he also proved the case n ≡ 3 (mod 4) of Theorem A and the case n ≡ 3 (mod 4) of Corollary C (ii) for = 1.
Further applications. Our main result Theorem 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 Sp 2 (Z) (see Section 1.2), or equivalently, the second quadratic symplectic algebraic K-theory group of the integers KSp q 2 (Z). 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 Γ n on the set of stable framings of W to identify the extension (3) and prove Theorem A. Section 3 aims at the proof of our main result Theorem 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 W , 1 -bundles over surfaces with certain boundary conditions in Section 3.3 (which incidentally is the key geometric insight to prove Theorem B), construct the cohomology classes appearing in the statement of Theorem B in Sections 3.4 and 3.5, and finish with the proof of Theorem B in Section 3.6. In Section 4, we analyse the extensions (1) and (2) and compute the abelianisation of Γ n and T n , proving Corollaries C-E and Theorem G. Section 5 briefly discusses the group of homotopy equivalences and proves Corollary F. In the appendix, we compute various low-degree (co)homology groups of the symplectic group Sp 2 (Z) and its arithmetic subgroup G ⊂ Sp 2 (Z).
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 for the -fold connected sum of S n × S n , including W 0 = S 2n , and the manifold obtained from W removing the interior of an embedded disc D 2n ⊂ W . Occasionally, we view the manifold W , 1 alternatively as the iterated boundary connected sum W , 1 = ♮ W 1, 1 of W 1, 1 = S n × S n \int(D 2n ). We call the genus of W or W , 1 and denote by Diff + (W ) and Diff + (W , 1 ) the groups of orientation-preserving diffeomorphisms, not necessarily fixing the boundary in the case of W , 1 . We shall also consider the subgroups of diffeomorphisms required to fix a neighbourhood of the boundary ∂W , 1 S 2n−1 or a neighbourhood of a chosen disc D 2n−1 ⊂ ∂W , 1 in the boundary, respectively. All groups of diffeomorphisms are implicitly equipped with the smooth topology so that (i) BDiff + (W , 1 ) and BDiff + (W ) classify smooth oriented W , 1 -bundles or W -bundles, (ii) BDiff ∂ (W , 1 ) classifies (W , 1 , S 2n−1 )-bundles, i.e. smooth W , 1 -bundles with a trivialisation of their S 2n−1 -bundle of boundaries, and (iii) BDiff ∂/2 (W , 1 ) classifies (W , 1 , D 2n−1 )-bundles, that is, smooth W , 1 -bundles with a trivialised D 2n−1 -subbundle of its S 2n−1 -bundle of boundaries.
Extending diffeomorphisms by the identity provides a map Diff ∂ (W , 1 ) → Diff + (W ), which induces an isomorphism on path components by work of Kreck as long as n ≥ 3.
Proof. Taking the differential at the centre of the disc induces a fibration Diff + (W ) → Fr + (W ) to the oriented frame bundle ofW . 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 Diff ∂ (W , 1 ). We thus have fibration sequences of the form Diff ∂ (W , 1 ) −→ Diff + (W , 1 ) −→ Fr + (W ) and SO(2n) −→ Fr + (W ) −→ W 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 [0, 1] × S 2n−1 ⊂ W , 1 using a smooth based loop in SO(2n) that represents a generator of π 1 SO(2n) Z/2 (see [Kre79,p. 647]). It follows from [Kre79, Lem. 3 b), Lem. 4] that this isotopy class is trivial since W bounds the parallelisable handlebody ♮ (D n+1 × S n ).
For the purpose of studying the mapping class group Γ n , we can thus equally well work with Diff ∂ (W , 1 ) instead of Diff + (W ), which is advantageous since there is a stabilisation map Diff ∂/2 (W , 1 ) → Diff ∂/2 (W +1, 1 ) by extending diffeomorphisms over an additional boundary connected summand via the identity, which restricts to a map Diff ∂ (W , 1 ) → Diff ∂ (W +1, 1 ) and thus induces stabilisation maps of the form (1.1) Γ n , 1/2 −→ Γ n , 1/2 and Γ n , 1 −→ Γ n +1, 1 , that allow us to compare mapping class groups of different genera. The group Γ 0, 1 has a convenient alternative description: gluing two closed d-discs along their boundaries via a diffeomorphism supported in a disc D d ⊂ ∂D d +1 gives a morphism π 0 Diff ∂ (D d ) −→ Θ d +1 to Kervaire-Milnor's [KM63] finite abelian group Θ d of oriented homotopy d-spheres up to hcobordism. By work of Cerf [Cer70], this is an isomorphism for d ≥ 5, so we identify these two groups henceforth. Iterating the stabilisation map yields maps Lemma 1.2. For n ≥ 3, the image of Θ 2n+1 in Γ n , 1 is central and becomes trivial in Γ n , 1/2 . The induced morphism is an isomorphism.
Proof. Every diffeomorphism of W , 1 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 Diff ∂ (W , 1 ) up to isotopy, which shows the first part of the claim. For the others, we consider the sequence of topological groups induced by restricting diffeomorphisms in Diff ∂/2 (W , 1 ) to the moving part of the boundary. This is a fibration sequence by the parametrised isotopy extension theorem. Mapping this sequence for = 0 into (1.3) via the iterated stabilisation map, we see that the looped map

) factors as the composition
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 Γ n , 1/2 → π 0 Diff ∂ (D 2n−1 ) = Θ 2n has trivial image. Using that any orientation preserving diffeomorphism fixes any chosen oriented codimension 0 disc up isotopy by the isotopy extension theorem, one easily sees that this image agrees with the the inertia group of W , 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 (n−1)-connected 2n-manifold [Wal62b], specialised to the case of our interest-the iterated connected sums W = ♯ (S n × S n ) in dimension 2n ≥ 6.
The intersection form λ : H ( )⊗H ( ) → Z on the middle cohomology H ( ) ≔ H n (W ; Z) is a nondegenerate (−1) n -symmetric bilinear form. We use Poincaré duality to identify H ( ) with π n (W ; Z) H n (W ) and a result of Haefliger [Hae61] to represent classes in π n (W ) by embedded spheres e : S n ֒→ W , unique up to isotopy as long as n ≥ 4. As W is stably parallelisable, the normal bundle of such e is stably trivial and hence gives a class q([e]) ∈ ker π n (BSO(n)) → π n (BSO(n + 1) , where Λ n is the image of the usual map π n (SO(n + 1)) → π n (S n ) Z (see e.g.
so the automorphism group of the quadratic form can be identified as In the theory of theta-functions, the finite index subgroup Sp q 2 (Z) ≤ Sp 2 (Z) is known as the theta group; it is the stabiliser of the standard theta-characteristic with respect to the transitive Sp 2 (Z)-action on the set of even characteristics (see e.g. [Wei91]). Using this description, it is straightforward to compute its index in Sp 2 (Z) to be 2 2 −1 + 2 −1 .

Kreck's extensions. To recall Kreck's extensions [Kre79, Prop. 3] describing Γ n
, 1 for n ≥ 3, note that an orientation-preserving diffeomorphism of W induces an automorphism of the quadratic form (H ( ), λ, q). This provides a morphism Γ n , 1 → G , which Kreck proved to be surjective using work of Wall [Wal63]. 2 This explains the first extension The second extension describes the Torelli subgroup T n , 1 ⊂ Γ n , 1 and has the form where Sπ n SO(n) denotes the image of the morphism S : π n SO(n) → π n SO(n + 1) induced by the usual inclusion SO(n) ⊂ SO(n + 1). 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 Θ 2n+1 → Γ n , 1 in (1.2) has image in T n , 1 , which explains first map in the extension (1.5). To define the second one, we canonically identify H ( ) ⊗ Sπ n SO(n) with Hom(H n (W ; Z), Sπ n SO(n)) using that H n (W ; Z) is free and note that for a given isotopy class [ϕ] ∈ T n and a class [e] ∈ H n (W ; Z) represented by an embedded sphere e : S n ֒→ W , the embedding ϕ • e is isotopic to e, so we can assume that ϕ fixes e pointwise by the isotopy extension theorem. The derivative of ϕ thus induces an automorphism of the once stabilised normal bundle ϑ (e) ⊕ ε, which after choosing a triviali- is independent of all choices and actually lies in the subgroup Sπ n SO(n) ⊂ π n SO(n + 1) (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 Θ 2n+1 → Γ n , 1 is injective, so gives rise to an extension 0 and agrees with the extension induced by taking path components of the chain of inclusions . The action of Γ n , 1/2 on H ( ) preserves the quadratic form as Γ n , 1 → Γ n , 1/2 is surjective by Lemma 1.2, so this action yields an extension which, via the isomorphism T n , 1 /Θ 2n+1 H ( ) ⊗ Sπ n SO(n) induced by (1.5), corresponds to the quotient of the extension (1.4) by Θ 2n+1 , using Γ n , 1 /Θ 2n+1 Γ n , 1/2 once more. Proof. In view of the commutative diagram for all ϕ ∈ Γ n , 1 and ψ ∈ T n , 1 . Unwrapping the definition of ρ, the image of p(ϕ) · ρ(ψ ) on a homology class in H n (W , 1 ; Z) is given by the automorphism where e is an embedded sphere which represents the homology class and is pointwise fixed by ϕ • ψ • ϕ −1 and F is any choice of framing of ϑ (ϕ −1 • e) ⊕ ε. Choosing the framing to compute the image of [e] ∈ H n (W , 1 ; Z) under ρ(ϕ • ψ • ϕ −1 ), 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 of group extensions for h ≤ , 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 h ≤ . The situation for the extension (1.7) is similar: and two morphisms of G h -modules, an inclusion H (h) → s * H ( ) and a projection s * H ( ) → H (h). These morphisms express the extension (1.7) for genus h ≤ as being obtained from that for genus by pulling back along s : G h → G followed by forming the extension pushout along s * H ( ) → H (h). They also induce a morphism of the form

T T A
This section serves to resolve the extension problem Our approach is in parts inspired by work of Crowley [Cro11], who identified this extension in the case n = 3, 7.
The group Γ n , 1/2 acts on the set of equivalence classes of stable framings F : TW , 1 ⊕ ε k ε 2n+k for k ≫ 0 that extend the standard stable framing on T D 2n−1 , 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 [W , 1 , SO] * , the action of Γ n , 1/2 on a fixed choice of stable framing F as above yields a function where the first isomorphism is induced by π n (−) and the Hurewicz isomorphism, and the second one by the universal coefficient theorem. This function is a 1-cocycle (or crossed homomorphism) for the canonical action of Γ n , 1/2 on H ( ) ⊗ π n SO (cf. [Cro11,Prop. 3.1]) and as this action factors through the map p : Γ n , 1/2 → G , we obtain a morphism of the form (s F , p) : Γ n , 1/2 → (H ( ) ⊗ π n SO) ⋊ G , which is independent of F up to conjugation in the target by a straightforward check. This induces a morphism from (2.1) to the trivial extension of G by the G -module H ( ) ⊗ π n SO, The left vertical map is induced by the natural map Sπ n SO(n) → π n SO originating from the inclusion SO(n) ⊂ SO and is an isomorphism for n 1, 3, 7 odd as a consequence of the following lemma whose proof is standard (see e.g. [Lev85, §1B)]).
Lemma 2.1. For n odd, the morphism Sπ n SO(n) → π n SO induced by the inclusion SO(n) ⊂ SO is an isomorphism for n 1, 3, 7 odd. For n = 1, 3, 7, it is injective with cokernel Z/2.
As a result, the diagram (2.2) induces a splitting of (2.1) for n 3, 7 odd, since all vertical maps are isomorphisms. This proves the cases n 3, 7 of the following reformulation of Theorem A (see Section 1.3). We postpone the proof of the cases n = 3, 7 to Section 3.6.
Even though the extension does not split for n = 3, 7, the morphism (2.2) is still injective by Lemma 2.1 and thus expresses the extension in question as a subextension of the trivial extension of G by the G -module H ( ) ⊗ π n SO. Crowley [Cro11,Cor. 3.5] gave an algebraic description of this subextension and concluded that it splits if and only if = 1. 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 H 2 (G ; H ( ) ⊗ Sπ n SO(n)) for ≫ 0. For n = 3, 7, we have G Sp 2 (Z) with its usual action on H ( ) ⊗ Sπ n SO(n) Z 2 . Using work of Djament [Dja12, Thm 1], one can compute H 2 (Sp 2 (Z); Z 2 ) Z/2 for ≫ 0, so there is only one nontrivial extension of G by H ( ) ⊗ Sπ n SO(n), which must be the one in consideration because of the second part of Theorem 2.2. Note that this line of argument gives a geometric proof for the following useful fact on the twisted cohomology of Sp 2 (Z) 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 H 2 (Sp 2 (Z); We close this section by relating the abelianisation of Γ n , 1/2 to that of G . The latter is content of Lemma A.1.
Corollary 2.4. For n ≥ 3 odd, the morphism is split surjective and has the coinvariants (H ( ) ⊗ Sπ n SO(n)) G as its kernel, which vanish for ≥ 2. For = 1 it vanishes if and only if n ≡ 5 (mod 8) or n = 3, 7, and has order 2 otherwise.
Proof. The claim regarding the coinvariants follows from Lemma A.2 and Table 1. Since they vanish for ≥ 2, the remaining statement follows from the exact sequence .1), combined with the fact that this extension splits for = 1 by Theorem 2.2.

S , , T B
By Lemma 1.2, the extension 3 is central and is as such classified by a class in H 2 (Γ n , 1/2 ; Θ 2n+1 ) with Γ n , 1/2 acting trivially on Θ 2n+1 . In this section, we identify this extension class in terms of the algebraic description of Γ n , 1/2 provided in the previous section, leading to a proof of our main result Theorem B. Our approach is partially based on ideas of Galatius-Randal-Williams [GRW16, Sect. 7], who determined the extension for n ≡ 5 (mod 8) and ≥ 5 up to automorphisms of Θ 2n+1 .
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 k ≥ 1 and ξ : X → τ >4k−1 BO be a map to the (4k − 1)st connected cover of BO with a liftξ : A → τ >4k BO over a subspace A ⊂ X along the canonical map τ >4k BO → τ >4k−1 BO. Such data has a relative Pontryagin class p k (ξ ,ξ ) ∈ H 4k (X , A; Z) given as the pullback along the map (ξ ,ξ ) : . We suppress the liftξ from the notation whenever there is no source of confusing. For us, X = M will usually be a compact oriented 8k-manifold and A = ∂M its boundary, in which case we can evaluate χ 2 (ξ ,ξ ) ∈ H 8k (M, ∂M; Z) against the relative fundamental class [M, ∂M] to obtain a number χ 2 (ξ ,ξ ) ∈ Z. The following two sources of manifolds are relevant for us.
(i) For a compact oriented n-connected (2n + 2)-manifold whose boundary is a homotopy sphere, there is a (up to homotopy) unique lift M → τ >n BO of the stable oriented normal bundle. On the boundary ∂M, this lifts uniquely further to τ >n+1 BO, so we obtain a canonical class χ(M) ∈ H n+1 (M, ∂M; Z) and a characteristic number The standard framing of D 2n−1 induces a trivialisation of stable vertical tangent bundleT π E : E → BSO over the subbundle B ×D 2n−1 ⊂ ∂E, which extends uniquely to a τ >n+1 BO-structure on T π E | ∂E by obstruction theory. Using that W is n-parallelisable, another application of obstruction theory shows that the induced τ >n BO-structure on T π E | ∂E extends uniquely to a τ >n BOstructure on T π E, so the above discussion provides a class χ(T π E) ∈ H n+1 (E, ∂E; Z), and, assuming B is an oriented closed surface, a number χ 2 (T π E) ∈ Z.
3.2. Highly connected almost closed manifolds. As a consequence of Theorem 3.12, we shall see that (W , 1 , D 2n−1 )-bundles over surfaces are closely connected to n-connected almost closed (2n + 2)-manifolds. These manifolds were classified by Wall [Wal62b], which we now recall for n ≥ 3 in a form tailored to later applications, partly following [KR20, Sect. 2]. A compact manifold M is almost closed if its boundary is a homotopy sphere. We write A τ >n d for the abelian group of almost closed oriented n-connected d-manifolds up to oriented n-connected bordism restricting to an h-cobordism on the boundary. Recall that Ω τ >n d denotes the bordism group of closed d-manifolds M equipped with a τ >n BO-structure on their stable normal bundle M → BO, i.e. a lift M → τ >n BO to the n-connected cover. By classical surgery, the group Ω τ >n d is canonically isomorphic to the bordism group of closed oriented n-connected d-manifolds up to n-connected bordism as long as d ≥ 2n + 1, so we will use both descriptions interchangeably. There is an exact sequence Wall [Wal67,p. 293] in which the two outer morphisms are the obvious ones, noting that homotopy d-spheres n-connected for n < d. The morphisms Ω τ >n 2n+2 → A τ >n 2n+2 and ∂ : A τ >n 2n+2 → Θ 2n+1 are given by cutting out an embedded disc and by assigning to an almost closed manifold its boundary, respectively. By surgery theory, the subgroup of homotopy (2n + 1)-spheres bounding n-connected manifolds contains the cyclic subgroup bP 2n+2 ⊂ Θ 2n+2 of homotopy (2n + 1)-spheres bounding parallelisable manifolds, so the right end of (3.1) receives canonical a map from Kervaire-Milnor's exact sequence [KM63], which in particular induces a morphism coker( ) 2n+1 → Ω τ >n 2n+1 , concretely given by representing a class in coker( ) 2n+1 by a stably framed manifold and restricting its stable framing to a τ >n BO-structure.
3.2.1. Wall's classification. For our purposes, Wall's computation [Wal62b,Wal67] of A τ >n 2n+2 is for n ≥ 3 odd is most conveniently stated in terms of two particular almost closed n-connected (2n + 2)-manifolds, namely (i) Milnor's E 8 -plumbing P, arising from plumbing together 8 copies of the disc bundle of the tangent bundle of the standard (n + 1)-sphere such that the intersection form of P agrees with the E 8 -form (see e.g.[Bro72, Ch. V.2]), and (ii) the manifold Q, obtained from plumbing together two copies of a linear D n+1 -bundle over the (n + 1)-sphere representing a generator of Sπ n SO(n).
The following can be derived from Wall's work, as explained for instance in [KR20, Thm 2.1].
Theorem 3.2 (Wall). For n ≥ 3 odd, the bordism group A τ >n 2n+2 satisfies The first summand is generated by the class of P in all cases but n = 3, 7 in which it is generated by HP 2 and OP 2 . The second summand for n 5 (mod 8) is generated by Q.
From a consultation of Table 1, one sees that the group Sπ n SO(n) vanishes for n ≡ 5 (mod 8), so Q ∈ A τ >n 2n+2 is trivial in this case, which shows that the subgroup bA 2n+2 is for all n ≥ 3 odd generated by the boundaries In the cases n ≡ 1 (mod 8) in which Q defines a Z/2-summand, its boundary Σ Q is trivial by a result of Schultz   For n ≥ 3 odd, the Milnor sphere Σ P ∈ Θ 2n+1 is well-known to be nontrival and to generate the cyclic subgroup bP 2n+2 ⊂ Θ 2n+1 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 Theorem 3.3 has the following corollary.
Combining the previous results with the diagram (3.2), we obtain the following result, which we already mentioned in the introduction.
3.2.2. Invariants. It follows from Theorem 3.2 and Theorem 3.3 that the boundary of an nconnected almost closed (2n + 2)-manifold M is determined by at most two integral bordism invariants of M. Concretely, we consider the signature sgn : A τ >n 2n+2 → Z and for n ≡ 3 (mod 4) the characteristic number χ 2 : A τ >n 2n+2 → Z, explained in Example 3.1. As discussed for example in [KR20, Sect. 2.1], these functionals evaluate to (3.3) sgn(P) = 8 sgn(Q) = 0 χ 2 (P) = 0 χ 2 (Q) = 8 for n = 3, 7 2 otherwise , and on the closed manifolds HP 2 and OP 2 to which results in the following formula for boundary spheres of highly connected manifolds when combined with the discussion above.
Proposition 3.9. For n ≥ 3 odd, the boundary ∂M ∈ Θ 2n+1 of an almost closed oriented nconnected (2n + 2)-manifold M satisfies 3.2.3. The minimal signature. As in the introduction, we denote by σ ′ n the minimal positive signature of a smooth closed n-connected (2n + 2)-manifold. This satisfies σ ′ n = 1 for n = 1, 3, 7 as witnessed by CP 2 , HP 2 , and OP 2 , and in all other cases, it can be expressed in terms of the subgroup bA 2n+2 ⊂ Θ 2n+1 as follows.
Lemma 3.10. For n ≥ 3 odd, the quotient bA 2n+2 / Σ Q is a cyclic group generated by the class of Σ P . It is trivial if n = 3, 7 and of order σ ′ n /8 otherwise. Proof. For n = 3, 7, the claim is a direct consequence of Theorem 3.2 and Lemma 3.4 and in the case n 3, 7, it follows from taking vertical cokernels in the commutative diagram 8 · Z 0 sgn sgn with exact rows, obtained from a combination of Theorem 3.2 with (3.1) and (3.3).
(i) The natural map MSO → HZ is 4-connected, so pushing forward fundamental classes induces an isomorphism Ω SO * (X ) → H * (X ; Z) for * ≤ 3 and any space X , and (ii) the 1-truncation of a connected space X (in particular the natural map BG → Bπ 0 G for a topological group G) induces a surjection H 2 (X ; Z) ։ H 2 (K(π 1 X , 1); Z), whose kernel agrees with the image of the Hurewicz homomorphism π 2 X → H 2 (X ; Z). The key geometric ingredient to identify the differential d 2 is the following result.
Theorem 3.12. Let n ≥ 3 be odd and π : E → S a (W , 1 , D 2n−1 )-bundle over an oriented closed surface S. There exists a class E ′ ∈ A τ >n 2n+2 such that (i) its boundary ∂E ′ ∈ Θ 2n+1 is the image of the class [π ] ∈ H 2 (BDiff ∂/2 (W , 1 ); Z) under Proof. By the isotopy extension theorem, the restriction map to the moving part of the boundary Diff ∂/2 (W , 1 ) → Diff ∂ (D 2n−1 ) is a fibration. As its image is contained in the component of the identity (see the proof of Lemma 1.2), this fibration induces the upper row in a map of fibre sequences , 1/2 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 ΩDiff ∂ (D 2n−1 ) → Diff ∂ (D 2n ) followed by taking components, which one checks by looping the fibre sequences and using that ΩDiff id ∂ (D 2n−1 ) −→ Diff ∂ (W , 1 ) is given by "twisting" a collar [0, 1] × S 2n−1 ⊂ W , 1 , meaning that it sends a smooth loop γ ∈ ΩDiff id ∂ (D 2n−1 ) ⊂ ΩDiff id ∂ (S 2n−1 ) to the diffeomorphism that is the identity outside the collar and is given by (t, x) → (t, γ (t) · x) on the collar. Now consider the square (3.7) obtained from delooping (3.6) once to the right and using H 2 (B 2 Θ 2n+1 ) Θ 2n+1 . By transgression, the bottom horizontal arrow agrees with the differential d 2 in the statement. Combining this with the Hurewicz theorem, the square (3.7) provides a factorisation H 2 (BDiff ∂/2 (W , 1 ); Z) → H 2 (BDiff id ∂ (D 2n−1 ); Z) π 2 (BDiff id ∂ (D 2n−1 )) −→ Θ 2n+1 of the map in the first part of the statement, which thus has the following geometric description: a smooth (W , 1 , D 2n−1 )-bundle π : E → S represents a class [π ] ∈ H 2 (BDiff ∂/2 (W , 1 ); Z) and its image under the first map in the composition is the class [π + ] ∈ H 2 (BDiff id ∂ (D 2n−1 ); Z) of its (∂W , 1 , D 2n−1 )-bundle π + : E + → S of boundaries, which in turn maps under the inverse of the Hurewicz homomorphism to a (∂W , 1 , D 2n−1 )-bundle π − : E − → S 2 over the 2-sphere that is bordant, as a bundle, to π − . That is, there exists a (∂W , 1 , D 2n−1 )-bundleπ :Ē → K over an oriented bordism K between S and S 2 that restricts to π + over S and to π − over S 2 . We claim that the image of the (∂W , 1 , D 2n−1 )-bundle π − under the final map in the composition is the homotopy sphere Σ π ∈ Θ 2n+1 obtained by doing surgery on the total space E − along the trivialised subbundle D 2n−1 × S 2 ⊂ E − . This is most easily seen by thinking of a class in π k BDiff ∂ (D d ) as a smooth bundle D d → P → D k together with a trivialisation φ : D d × ∂D k P | ∂D k and a trivialised ∂D k -subbundle ψ : ∂D d × D k ֒→ P such that φ and ψ agree on +1 ) induced by the Gromoll map is given by sending such a bundle p : P → D k D k−1 × D 1 to the (D d × D 1 )-bundle (pr 1 • p) : P → D k−1 , and the isomorphism π 1 BDiff ∂ (D d ) Θ d +1 is given by assigning to a disc bundle D d → P → D 1 the manifold P ∪ ∂D d ×D 1 ∪D d ×∂D 1 D d × D 1 . Deccomposing the sphere into half-discs S n = D n + ∪ D n − , we see from this description that the composition π k BDiff ∂ (D d ) → Θ d +k of the iterated Gromoll map with the isomorphism π 1 BDiff ∂ (D d +k−1 ) Θ d +k maps a class represented by an . This in particular implies the claim we made above in the case k = 2 and d = 2n − 1.
As a consequence of this description of the morphism in consideration, the image Σ π ∈ Θ 2n+1 of the class [π ] comes equipped with a nullbordism, namely where W is the trace of the performed surgery. Omitting the trivialised D 2n−1 -subbundles, the situation can be summarised schematically as follows A choice of a stable framing of K induces stable framings on S and S 2 and thus a stable isomorphism T E T π E ⊕ π * T S s T π E using which the canonical τ >n BO-structure on T π E and the τ >n+1 BO-one on T π E | E + (see Example 3.1) induce a τ >n BO-structure on T E and a τ >n+1 BO-structure on T E | E + s T E + . With these choices, we have χ(T π E) = χ(T E,T E + ). By construction, the restriction of this τ >n+1 BO-structure to T E + | S ×D 2n−1 s T S agrees with the τ >n+1 BO-structure on T S obtained from the stable framing of K, so we obtain a τ >n+1 BOstructure on TĒ | E + ∪K ×D 2n−1 , which by obstruction theory extends to one on T (Ē ∪ E − W ): the relative Serre spectral sequence shows that H * (Ē, E + ∪ K × D 2n−1 ) vanishes for * ≤ 2n − 2 and thus that and π 2 SO = 0. The restriction of this τ >n+1 BO-structure on TĒ | E + ∪K ×D 2n−1 to a τ >n BO and the canonical τ >n BO-structure on T E T π E (see Example 3.1) assemble to a τ >n BO-structure on N . By construction, the canonical restriction map (using excision) sends χ(T π E) = χ(T E,T E + ) to χ(T N ,T Σ π ), so we conclude χ 2 (T π E) = χ 2 (T N ,T Σ π ). To finish the proof, note that the τ >n BO-structure on T N allows us to do surgery away from the boundary on N to obtain an n-connected manifold E ′ , which gives a class in A τ >n 2n+2 as aimed for: ∂E ′ = Σ π holds by construction, χ 2 (E ′ ) = χ 2 (T N ,T Σ π ) = χ 2 (T π E) by the bordism invariance of Pontryagin numbers (see Example 3.1), and sgn(E ′ ) = sgn(N ) = sgn(E) by the additivity and bordism invariance of the signature.
Combining the previous result with Proposition 3.9, we conclude that the composition sends a homology class [π ] represented by a bundle π : E → S to a certain linear combination of Σ P and Σ Q whose coefficients involve the invariants sgn(E) and χ 2 (T π E). In the following two subsections, we shall see that these functionals sgn : H 2 (BDiff ∂/2 (W , 1 ); Z) −→ Z and χ 2 : H 2 (BDiff ∂/2 (W , 1 ); Z) −→ Z factor through the composition and have a more algebraic description in terms of H ( ) ⊗ π n SO) ⋊ G . This uses the morphism Remark 3.13. As Sp 2 (Z) is perfect for ≥ 3 (see Lemma A.1), the morphism (3.8) determines a unique cohomology class sgn ∈ H 2 (Sp 2 (Z); Z). 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 (4k + 2)manifold M, the action of its group of diffeomorphisms on the middle cohomology induces a morphism Diff + (M) → Sp 2 (Z) for 2 = rk(H 2k+1 (M)) and the resulting composition 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.
We proceed by computing the image of the signature morphism (3.8) and of its pullback to the theta-subgroup Sp q 2 (Z) ⊂ Sp 2 (Z) as defined in Section 1.2.
Lemma 3.15. The signature morphism satisfies Proof. The signatures realised by classes in H 2 (Sp 2 (Z); Z) are well-known (see e.g. [BCRR20, Lem 6.5, Thm 6.6 (vi)]). To prove that the signature of classes in H 2 (Sp q 2 (Z)) is divisible by 8, recall from Sections 1.2 and 1.3 that for n odd the morphism Diff ∂/2 (W , 1 ) → Sp 2 (Z) lands in the subgroup Sp q 2 (Z) ⊂ Sp 2 (Z) as long as n 1, 3, 7, so we have a composition which maps the class of a bundle π : E → S by Lemma 3.14 to sgn(E). The latter agrees by Theorem 3.12 with the signature of an almost closed n-connected (2n+2)-manifold, so it is divisible by 8 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 Corollary 2.4. As the signature morphism vanishes on H 2 (Sp 2 (Z); Z) by the first part, it certainly vanishes on H 2 (Sp q 2 (Z); Z). Consequently, by the compatibility of the signature with the inclusion Sp 2 (Z) ⊂ Sp 2 +2 (Z), the remaining claim follows from constructing a class in H 2 (Sp q 2 (Z); Z) of signature 8 for = 2. Using H 2 (Sp 4 (Z); Z) Z ⊕ Z/2 (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 H 2 (Sp q 4 (Z); Z) in the torsion free quotient H 2 (Sp 4 (Z); Z) free Z containing 2. That it contains 10 is ensured by transfer, since the index of Sp q 4 (Z) ⊂ Sp 4 (Z) is 10 (see Section 1.2). As H 1 (Sp 4 (Z); Z) and H 1 (Sp q 4 (Z); Z) are 2-torsion by Lemma A.1, it therefore suffices to show that H 2 (Sp q 4 (Z); F 5 ) → H 2 (Sp 4 (Z); F 5 ) is nontrivial, for which we consider the level 2 congruence subgroup Sp 4 (Z, 2) ⊂ Sp 4 (Z), i.e. the kernel of the reduction map Sp 4 (Z) → Sp 4 (Z/2), which is surjective (see e.g. [NS64, Thm 1] for an elementary proof). From the explicit description of Sp q 2 (Z) presented in Section 1.2, one sees that it contains the congruence subgroup Sp 4 (Z, 2). As a result, it is enough to prove that H 2 (Sp 4 (Z, 2); F 5 ) → H 2 (Sp 4 (Z); F 5 ) is nontrivial, which follows from an application of the Serre spectral sequence of the extension 0 −→ Sp 4 (Z, 2) −→ Sp 4 (Z) −→ Sp 4 (Z/2) −→ 0, using that H 1 (Sp 4 (Z, 2); which is compatible with natural inclusion Z 2 ⋊ Sp 2 (Z) ⊂ Z 2 +2 ⋊ Sp 2 +2 (Z) and takes for n ≡ 3 (mod 4) part in a composition where the first morphism is induced by acting on a stable framing F of W , 1 as in Section 2. A priori, this requires three choices: a stable framing, a generator π n SO Z, and a symplectic basis H ( ) Z 2 . 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 (W , 1 , D 2n−1 )-bundles explained in Example 3.1.
Proof. The relative Serre spectral sequence of where f denotes the composition By the compatibility of the Serre spectral sequence with the cup-product and after identifying H 1 (S; f * H ( )) with H 1 (π 1 (S; * ); H ( ) ⊗ Sπ n SO(n)), it suffices to show that the second isomorphism sends χ(T π E) ∈ H n+1 (E, ∂E; Z) up to signs to the class represented by the cocycle involving the choice of stable framing F : TW , 1 ⊕ ε k ε 2n+k as in Section 2. As a first step, we describe this isomorphism more explicitly: Note that H n+1 (E, ∂E; π n SO) H n+1 (E; π n SO) as H * (∂E; π n SO) is trivial for * = n, n + 1. Unwinding the construction of the Serre spectral sequence using a skeletal filtration of S, one sees that after fixing an identification W , 1 π −1 ( * ) ⊂ E, the image of a class x ∈ H n+1 (E, π n SO) under the isomorphism in question is represented by the cocycle π 1 (S; * ) → H ( ) ⊗ π n SO which maps a loop ω : ([0, 1], {0, 1}) → (S, * ) to the class obtained from a choice of lift x ∈ H n+1 (E, π −1 ( * ); π n SO) by pulling it back along where the second morphism is induced by pulling back the bundle π : E → S along ω and the first morphism is the unique (up to homotopy) trivialisation ω * E [0, 1] × W , 1 relative to W , 1 × {0} of the pullback bundle over [0, 1]; here we used the canonical isomorphism H n+1 (W , 1 × [0, 1],W , 1 × {0, 1}; π n SO) H ( ) ⊗ π n SO.
Recall from Example 3.1 that the class χ(T π E) ∈ H n+1 (E; π n SO) is the primary obstruction to extending the canonical τ >n BO-structure on T π E to a τ >n+1 BO-structure. The choice of framing F induces such an extension on the fibre W , 1 π −1 ( * ) ⊂ E and thus induces a lift of χ(T π E) to a relative class in H n+1 (E, π −1 ( * ); π n SO). This uses that the τ >n BO structure on W , 1 induced by the framing agrees with the restriction of the τ >n BO-structure on T π E to π −1 ( * ) W , 1 by obstruction theory. Using the above description, we see that the image of χ(T π E) under the isomorphism in question is thus represented by the cocyle π 1 (S; * ) → H ( ) ⊗ π n SO that sends a loop ω to the primary obstruction to solving the lifting problem (3.12) in H n (W , 1 ; π n SO) H ( ) ⊗ π n SO, since this obstruction agrees with the corresponding ob- To see that the cocyle we just described agrees with (3.11), note that the function s F : Γ n , 1/2 → H ( ) ⊗ π n SO [W , 1 , τ ≤n SO] * induced by acting on the stable framing F arises as the connecting map π 1 (BDiff τ >n+1 ∂/2 (W , 1 ); [F ]) → π 0 Map * (W , 1 , τ ≤n SO) of the fibration ∂/2 (W , 1 ) is the space that classifies (W , 1 , D 2n−1 )-bundles with a τ >n+1 BO-structure on the vertical tangent bundle extending the given τ >n+1 BO-structure on the restriction to the trivial D 2n−1 -subbundle induced by the standard framing of D 2n−1 ; here we identified the space of τ >n+1 BO-structures of W , 1 relative to D 2n−1 with the mapping space Map * (W , 1 , τ ≤n SO) by using the choice of stable framing F , which also induces the basepoint [F ] ∈ BDiff τ >n+1 ∂/2 (W , 1 ). This shows that the value of (3.11) on a loop [ω] ∈ π 1 (S; * ) is given by the component in [W , 1 , τ ≤n SO] * H ( ) ⊗ π n SO obtained by evaluating a choice of path-lift at the end point. Such a path-lift precisely classifies a lift as in (3.12) relative to the subspace W , 1 × {0} ⊂ W , 1 × {0, 1}, so the claim follows from the second description of the obstruction to solving the lifting problem (3.12) mentioned above.
(i) The image of the composition induced by the inclusion Z 2 ⊂ Z 2 ⋊Sp q 2 (Z) ⊂ Z 2 ⋊Sp 2 (Z) contains 2 · Z and agrees with 2 · Z for = 1.
(iii) For n = 3, 7, we have Proof. By the compatibility of χ 2 with the stabilisation maps, it suffices to show the first part for = 1, which follows from checking that the image of a generator in H 2 (Z 2 ; Z) Z is mapped to ±2 under χ 2 by chasing through the definition. As the signature morphism pulls back from Sp 2 (Z), the second part follows from the first part and Lemma 3.15 by showing that the image of χ 2 is for Sp q 2 (Z) always divisible by 2 · Z and for Sp 2 (Z) divisible by 2 if and only if = 1. In the case of Sp q 2 (Z), this can be shown "geometrically" as in the proof of Lemma 3.15: choose n ≡ 3 (mod 4), n 3, 7 and consider the composition 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 2, which in turn follows from a combination of Proposition 3.18, Theorem 3.12 and the fact that in these dimensions, an n-connected almost closed (2n + 2)manifold E ′ satisfies χ 2 (E ′ ) ∈ 2 · Z by a combination of Theorem 3.2 and (3.3). For Sp 2 (Z), we argue as follows: in the case = 1, we show that the morphism H 2 (Z 2 ; Z) → H 2 (Z 2 ⋊Sp 2 (Z); Z) is surjective, which will exhibit the claimed divisibility as a consequence of (i). It follows from Lemma A.3 that the group H 1 (Sp 2 (Z); Z 2 ) vanishes and as Sp 2 (Z) = SL 2 (Z), we also have H 2 (Sp 2 (Z); Z) = 0 3 , so an application of the Serre spectral sequence to Z 2 ⋊ Sp 2 (Z) shows the claimed surjectivity. This leaves us with proving that χ 2 is not divisible by 2 for ≥ 2 for which we use that there is class [f : π 1 S → Sp 2 (Z)] ∈ H 2 (Sp 2 (Z); Z) of signature 4 by Lemma 3.15, so the form −, − f cannot be even and hence there is a 1-cocycle : π 1 S → Z 2 for which [ ], [ ] f is odd, which means that the image χ 2 ([ , f ]) of the class [( , f )] ∈ H 2 (Z 2 ⋊ Sp 2 (Z); Z) induced by the morphism ( , f ) : π 1 S → Z 2 ⋊ Sp 2 (Z) is odd. For the last part, note that the argument we gave for the divisibility in the second part for Sp q 2 (Z) shows for n = 3, 7 that the image of the composition in (iii) is contained in 8 · Z, since χ 2 (E ′ ) − sgn(E ′ ) is divisible by 8 if n = 3, 7. Hence, to finish the proof, it suffices to establish the existence of a class in H 2 (Γ n , 1/2 ; Z) for n = 3, 7 on which the composition evaluates to 8. To this end, we consider the square induced by the embedding (s F , p) of the extension describing Γ n , 1/2 into the trivial extension of Sp 2 (Z) by Z 2 ⊗ π n SO (see Section 2). By the first part, there is a class [f ] ∈ H 2 (Z 2 ⊗ π n SO; Z) with χ 2 ([f ]) = 2 and trivial signature, since the signature morphism pulls back from Sp 2 (Z). As a result of Lemma 2.1, the cokernel of the left vertical map in the square is 4-torsion if n = 3, 7, so 4 · [f ] lifts to H 2 (Z 2 ⊗ Sπ n SO(n); Z) and provides a class as desired.
Similar to the construction of sgn 8 ∈ H 2 (Sp q 2 (Z); Z), we would like to lift the morphism χ 2 /2 : H 2 (Z 2 ⋊ Sp  (i) Define the class χ 2 2 ∈ H 2 (Z 2 ⋊ Sp q 2 (Z); Z) for ≫ 0 via the splitting (3.13) by declaring its image in the first summand to be trivial and to be χ 2 /2 in the second. For small , the class χ 2 2 is defined as the pullback of the class for ≫ 0.
Proof. As all cohomology classes involved are compatible with the stabilisation map Γ n , 1/2 → Γ n +1, 1/2 , it is sufficient to show the first part of the claim for ≫ 0 (see Section 1.4). We assume n 3, 7 first. Identifying Γ n , 1/2 with (H ( ) ⊗ π n SO) ⋊ G via the isomorphism (s F , p) of Section 2, the morphism a : Z/4 → (H ( ) ⊗ π n SO) ⋊ G of the previous section induces a morphism between the sequences of the universal coefficient theorem By the exactness of the rows and the vanishing of H 2 (Z/4; Z), it is sufficient to show that the extension class in consideration agrees with the classes in the statement when mapped to Hom(H 2 (Γ n , 1/2 ; Z), Θ 2n+1 ) and H 2 (Z/4, Θ 2n+1 ). Regarding the images in the Hom-term, it is enough to identify them after precomposition with the epimorphism so from the construction of sgn 8 and χ 2 2 together with Lemmas 3.15 and 3.19, we see that it suffices to show that H 2 (BDiff ∂/2 (W , 1 ); Z) → Θ 2n+1 induced by the extension class maps the class of a bundle π : E → S to sgn(E)/8 · Σ P if n ≡ 1 (mod 4) and to sgn(E)/8 · Σ P + χ 2 (E)/2 · Σ Q otherwise, which is a consequence of Theorem 3.12 combined with Proposition 3.9. By construction, the classes sgn 8 · Σ P and χ 2 2 · Σ Q vanish in H 2 (Z/4; Θ 2n+1 ), so the claim for n 3, 7 follows from showing that the extension class is trivial in H 2 (Z/4; Θ 2n+1 ), i.e. that the pullback of the extension to Z/4 splits, which is in turn equivalent to the existence of a lift Using the standard embedding W 1 = S n × S n ⊂ R n+1 × R n+1 , we consider the diffeomorphism S n × S n −→ S n × S n (x 1 , . . . , x n+1 , 1 , . . . , n+1 ) −→ (− 1 , . . . , n+1 , x 1 , . . . , x n+1 ), which is of order 4, maps to 0 −1 1 0 ∈ Sp q 2 (Z), and has constant differential, so it vanishes in H ( ) ⊗ π n SO. As the natural map Γ n 1, 1 → Γ n 1 is an isomorphism by Lemma 1.1, this diffeomorphism induces a lift as required for = 1, which in turn provides a lift for all ≥ 1 via the stabilisation map Γ n 1, 1 → Γ n , 1 . For n = 3, 7, the abelianisation of Γ n , 1/2 vanishes for ≫ 0 due to Corollary 2.4, so it suffices to identify the extension class with the classes in the statement in Hom(H 2 (Γ n , 1/2 ; Z), Θ 2n+1 ) which follows as in the case n 3, 7. Lemma 3.19 (iii) implies that the image of d 2 for n = 3, 7 is generated by Σ Q for all ≥ 1. For n 3, 7, the map Γ n , 1/2 → (H ( ) ⊗ π n SO) ⋊ G is an isomorphism (see Section 2), so Lemma 3.19 tells us that the image of d 2 for n ≡ 3 (mod 4) is generated by Σ P and Σ Q if ≥ 2 and by Σ Q if = 1. For n ≡ 1 (mod 4), it follows from Lemma 3.15 that the image of d 2 is generated by Σ P if ≥ 2 and that it is trivial for = 1. In sum, this implies the second part of the claim by Corollary 3.5, and also that the differential d 2 does not vanish for ≥ 2, so the extension is nontrivial in these cases. For n ≡ 3 (mod 4), the homotopy sphere Σ Q is nontrivial by Theorem 3.3, so d 2 does not vanish for = 1 either. Finally, in the case n ≡ 1 (mod 4), the extension is classified by sgn 8 · Σ P , which is trivial for = 1 by Lemma 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 Theorem 2.2.
Proof of Theorem 2.2 for n = 3, 7. We have G = Sp 2 (Z), so the case = 1 follows from the fact that H 2 (Sp 2 (Z); Z 2 ⊗ Sπ n SO(n)) vanishes by Lemma A.3. To prove the case ≥ 2, note that a hypothetical splitting s : Sp 2 (Z) → Γ n , 1/2 of the upper row of the commutative diagram induces a splitting (s F , p) • s of the lower row, which agrees with the canonical splitting of the lower row up to conjugation with Z 2 ⊗ π n SO, because such splittings up to conjugation are a torsor for H 1 (Sp 2 (Z); Z 2 ⊗ π n SO) which vanishes by Lemma A.3. Lemma 3.15 on the other hand ensures that there is a class [f ] ∈ H 2 (Sp 2 (Z); Z) with signature 4, so s * [f ] ∈ H 2 (Γ n , 1/2 ; Z) satisfies sgn(s * [f ]) = 4 and χ 2 (s * [f ]) = 0, which contradicts Lemma 3.19 (iii).

K '
The inclusion T n , 1 ⊂ Γ n , 1 of the Torelli group extends to a pullback diagram , 1/2 0 of extensions whose bottom row we identified in Theorem 3.22. We now apply this to obtain information about the top row, and moreover to compute the abelianisations of Γ n , 1 and T n , 1 in terms of the homotopy sphere Σ Q ∈ Θ 2n+2 , the subgroup bA 2n+2 ⊂ Θ 2n+2 of Section 3.2, and the abelianisation of Γ n , 1/2 as computed in Corollary 2.4.
(i) The kernel K of the morphism Θ 2n+1 → H 1 (Γ n , 1 ) is generated by Σ Q for = 1 and agrees with the subgroup bA 2n+2 for ≥ 2. The induced extension ≡ 3 (mod 4) and splits G -equivariantly for n ≡ 1 (mod 4). The image of its differential d 2 : H 2 (H ( ) ⊗ Sπ n SO(n); Z) → Θ 2n+1 is generated by Σ Q . (iii) The commutator subgroup of T n , 1 is generated by Σ Q and the resulting extension Proof. By the naturality of the Serre spectral sequence, the morphism of extension (4.1) induces a ladder of exact sequences from which we see that the kernel K in question agrees with the image of the differential d Γ 2 , which we described in Theorem 3.22. By the universal coefficient theorem, the pushforward of the extension class of the bottom row of (4.1) along the quotient map Θ 2n+1 → Θ 2n+1 /im(d Γ 2 ) is classified by a class in Ext(H 1 (Γ n , 1/2 ), Θ 2n+1 /im(d Γ 2 )), which also describes the exact sequence in (i). By a combination of Theorem 3.22 and Corollary 3.5, this class is trivial for n = 3, 7, and for n 3, 7 as long as ≥ 2. For = 1 and n 3, 7, this class agrees with the image of sgn 8 · Σ P in H 2 (Γ n , 1 ; Θ 2n+1 /im(d Γ 2 )) and therefore vanishes as sgn 8 ∈ H 2 (G ; Z) is trivial by Lemma 3.21. The extension class of the bottom row of (4.1), determined in Theorem 3.22, pulls back to the extension class of the top one. Since sgn 8 ∈ H 2 (Γ n , 1/2 ; Z) is pulled back from G by construction, it is trivial in H ( ) ⊗ Sπ n SO(n), so the extension in (ii) is trivial for n ≡ 1 (mod 4), classified by χ 2 2 · Σ Q for n ≡ 3 (mod 4) if n 3, 7, and by · Σ Q if n = 3, 7. From this, the claimed image of d T 2 follows from Lemma 3.19 and its proof. For n ≡ 3 (mod 4), the homotopy sphere Σ Q 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 Σ Q is trivial for n ≡ 1 (mod 4) by Theorem 3.3.
The diagram above shows that the commutator subgroup of T n , 1 agrees with the image of d T 2 , which we already showed to be generated by Σ Q . To construct a G -equivariant splitting as claimed, note that the quotient of the lower row of (4.1) by Σ Q pulls back from G by Theorem 3.22 because sgn 8 ∈ H 2 (Γ n , 1 ; Θ 2n+1 ) has this property. Consequently, there is a central 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 Lemma 1.1, Corollary 2.4, and Theorem 3.2 has Corollary C (ii) and Corollary D as a consequence. It also implies the following.

4.1.
A geometric splitting. The splittings of the abelianisations of Γ n , 1 and T n , 1 provided by Theorem 4.1 are of a rather abstract nature. Aiming towards splitting these sequences more geometrically, we consider the following construction.
A diffeomorphism ϕ ∈ Diff ∂ (W , 1 ) fixes a neighbourhood of the boundary pointwise, so its mapping torus T ϕ comes equipped with a canonical germ of a collar of its boundary S 1 × ∂W , 1 ⊂ T ϕ using which we obtain a closed oriented (n − 1)-connected (2n + 1)-manifoldT ϕ by gluing in D 2 × S 2n−1 . By obstruction theory and the fact that W is n-parallelisable, the stable normal bundleT ϕ → BO has a unique lift to τ >n BO → BO compatible with the lift on D 2 × S 2n−1 induced by its standard stable framing. This gives rise to a morphism which is compatible with the stabilisation map s : Γ n , 1 → Γ n , 1 since where S 1 × S n × S n carries the τ >n BO-structure induced by the standard stable framing, which bounds. Using this, it is straight-forward to see that the composition of the iterated stabilisation map with t agrees with the canonical epimorphism appearing in Wall's exact sequence (3.1), so its kernel agrees with the subgroup bA 2n+2 . Together with the first part of Theorem 4.1, we conclude that the dashed arrow in the commutative diagram is an isomorphism if and only if K = bA 2n+2 which is the case for ≥ 2, and for = 1 as long as n = 3, 7 since Σ Q generates bA 2n+2 for n = 3, 7 by Corollary 3.5. Consequently, in these cases, the morphism (4.2) t * ⊕ p * : H 1 (Γ n , 1 ) −։ Ω τ >n 2n+1 ⊕ H 1 (Γ n , 1/2 ) is an isomorphism, whereas its kernel for = 1 coincides with the quotient bA 2n+2 /Σ Q , which is nontrivial as long as n 3, 7, generated by Σ P , and whose order can be interpreted in terms of signatures (see Lemma 3.10). Since p * splits by Theorem 4.1 and the natural map coker( ) 2n+1 /[Σ Q ] → Ω τ >n 2n+1 is an isomorphism by Corollary 3.6, the morphism (4.2) splits for = 1 if and only if the natural map (4.3) 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 [Σ Q ] ∈ coker( ) 2n+1 is trivial, which is conjecturally always the case (see Conjecture 3.7) and known in many cases as a result of Theorem 3.3 (see also Remark 3.8).
The situation for H 1 (T n , 1 ) is similar. By Theorem 4.1, the morphism ρ * in the diagram 0 splits G -equivariantly and since the morphism t * is defined on H 1 (Γ n , 1 ), its restriction to H 1 (T n , 1 ) is G -equivariant when equipping Ω τ >n 2n+1 with the trivial action. By an analogous discussion to the one above, the kernel of the resulting morphism of G -modules (4.4) t * ⊕ ρ * : H 1 (T n , 1 ) −։ Ω τ >n 2n+1 ⊕ H ( ) ⊗ Sπ n SO(n) is trivial for n = 3, 7 and given by the quotient bA 2n+2 /Σ Q for n 3, 7. Moreover, this morphism splits if and only if it splits G -equivariantly, 4 which is precisely the case if the natural map (4.3) admits a splitting. We summarise this discussion in the following corollary, which has Corollary E as a consequence when combined with Lemma 3.10. (i) The morphism t * ⊕ p * : H 1 (Γ n , 1 ) −։ Ω τ >n 2n+1 ⊕ H 1 (Γ n , 1/2 ) is an isomorphism for ≥ 2. For = 1, it is an epimorphism and has kernel bA 2n+2 /Σ Q .
Proof. The case n = 3, 7 and ≥ 2 is clear, since Theorem A shows that under this assumption even the quotient of the extension by the subgroup Θ 2n+1 ⊂ T n , 1 does not split. To deal with the other cases, we consider the morphism of extensions By the naturality of the Serre spectral sequence, we have a commutative square of the form H 2 (Γ n , 1/2 ; Z) Θ 2n+1 H 2 (G ; Z) H 1 (T n , 1 ; Z) G d 2 d 2 whose right vertical map has kernel generated by Σ Q by the computation of H 1 (T n , 1 ; Z) as a G -module in Theorem 4.1. Therefore, to finish the proof of the first claim, it suffices to show that the differential d 2 : H 2 (Γ n , 1/2 ; Z) → Θ 2n+1 /Σ Q is nontrivial for ≥ 2 and n 3, 7, which follows from Theorem 3.22 together with the fact that bA 2n+2 /Σ Q is nontrivial in these cases by Lemma 3.10 and Remark 3.11. Turning towards the second claim, we assume n 3, 7 and recall that the isomorphism (s F , p) : Γ n , 1/2 → (H ( ) ⊗ π n SO) ⋊ G induces a splitting of the right vertical map Γ n , 1/2 → G in the above diagram (see Section 2), so the claim follows from showing that the pullback of Γ n , 1 → Γ n , 1 along G ⊂ (H ( ) ⊗ π n SO) ⋊G Γ n , 1 splits for = 1, which is a consequence of Theorem B and Lemma 3.21.
Remark 4.5. Theorem 4.4 leaves open whether Γ n , 1 → G admits a splitting for = 1 in dimensions n = 3, 7. Krylov [Kry03, Thm 2.1] and Fried [Fri86,Sect. 2] showed that this can not be the case for n = 3 and we expect the same to hold for n = 7.

H
Our final result Corollary F is concerned with the morphism of extensions underlying work of Baues [Bau96, Thm 10.3], relating the mapping class group Γ n to the group π 0 hAut + (W ) of homotopy classes of orientation preserving homotopy equivalences. The left vertical morphism is induced by the restriction of the unstable homomorphism : π n SO(n + 1) → π 2n+1 S n+1 to the image of the stabilisation S : π n SO(n) → π n SO(n + 1) in the source and to the image of the suspension map S : π 2n S n → π 2n+1 S n+1 in the target, justified by the fact that the square π n SO(n) π n SO(n + 1) π 2n S n π 2n+1 S n+1 S S commutes up to sign (see [Tod62,Cor. 11.2]).
Proof of Corollary F. By Theorem A, the upper row of (5.1) splits for n 1, 3, 7 odd, so the first part of (i) is immediate. The second part follows from Theorem A as well if we show that the existence of a splitting of the lower row for n = 3, 7 is equivalent to one of the upper row. To this end, note that for n = 3, 7, we have isomorphisms Sπ n SO(n) Z and Sπ 2n S n = Tor(π 2n+1 S n+1 ) Z/d n for d 3 = 12 and d 7 = 120 with respect to which the -homomorphism : Sπ n SO(n) → Sπ n S n is given by reduction by d n (see e.g. [Tod62, Ch. XIV]). By Lemma A.3, the group H 2 (G ; H ( )) is annihilated by 2, so in particular by d n . Using this, the first claim follows from the long exact sequence on cohomology induced by the exact sequence 0 → H ( ) ⊗ Sπ n SO(n) To prove the second part, we consider the exact sequence (H ( ) ⊗ Sπ 2n S n ) G −→ H 1 (π 0 hAut + (W )) −→ H 1 (G ) −→ 0 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 Lemma A.2, exhibits H 1 (π 0 hAut + (W )) to be as asserted.

A A. L
This appendix contains various results on the low-degree (co)homology of the integral symplectic group Sp 2 (Z) and its theta subgroup Sp q 2 (Z) (see Section 1.2).
Proof. The fact that the abelianisation of Sp 2 (Z) = SL 2 (Z) is generated by 1 1 0 1 and of order 12 is well-known, and so is the isomorphism type of H 1 (Sp 2 (Z)) for ≥ 2 (see e.g. [BCRR20, Lem. A.1 (ii)]). The remaining claims regarding Sp 2 (Z) follows from showing that H 1 (Sp 2 (Z)) → H 1 (Sp 4 (Z)) is nontrivial, which can for instance be extracted from the proof of [GH98, Thm 2.1]: in their notation Γ 1 = Sp 4 (Z) and the map i ∞ : SL 2 (Z) → Γ 1 identifies with the stabilisation Sp 2 (Z) → Sp 4 (Z). The isomorphism type of H 1 (Sp q 2 (Z)) for ≥ 2 is determined in [End82, Thm 2] and for = 1 in [Wei91, Thm 1]. The first claim of (ii) follows from the main formula of [JM90] for ≥ 3 and from [Wei91, Cor. 2] for = 1, 2, which also gives the claimed generator of the Z-summand in H 1 (Sp q 2 (Z)). The proof of [End82, Thm 2] provides the asserted generator of the Z/2-summand in H 1 (Sp q 4 (Z)). The final claim follows from the first four items once we show that the image of 1 2 0 1 in H 1 (Sp q 4 (Z)) vanishes, which is another consequence of the formulas in [Wei91, Cor. 2].
Lemma A.2. The (co)invariants of the standard actions of Sp q 2 (Z) and O , (Z) on Z 2 ⊗ A for an abelian group A satisfy The same applies to the action of Sp 2 (Z), except that the (co)invariants also vanish for = 1.
Proof. We prove the claim for Sp Hom((Z 2 ) Sp (q) 2 (Z) , A), so the computation of the invariants is a consequence of that of the coinvariants. To settle the case ≥ 2 for both Sp q 2 (Z) and Sp 2 (Z), it thus suffices to prove that the coinvariants (Z 2 ⊗ A) Sp q 2 (Z) with respect to the smaller group vanish. We consider the matrices which can be seen to be contained in the subgroup Sp q 2 (Z) ⊂ Sp 2 (Z) using the description in Section 1.2; here I ∈ GL (Z) is the unit matrix and P σ ∈ GL (Z) the permutation matrix associated to σ ∈ Σ . It suffices to show that e i ⊗ a and f i ⊗ a are trivial in the coinvariants for 1 ≤ i ≤ and a ∈ A, where (e 1 , . . . , e , f 1 , . . . , f ) is the standard symplectic basis of Z 2 . Writing  [Wei91,p. 385]) to see that the surjection Z 2 ⊗A → A/2 induced by adding coordinates is invariant and surjective, and thus induces an isomorphism and the right hand side can easily be computed to be as claimed by applying the Mayer-Vietoris sequence to the presentation PSp 2 (Z) = S,T | S 2 = 1,T 3 = 1 and PSp q 2 (Z) = S, R | S 2 = 1 for S = 0 −1 1 0 , T = 0 −1 1 1 , and R = 1 2 0 1 , which is well-known for PSp 2 (Z) PSL 2 (Z) and appears e.g. in [Wei91,p. 385] for PSp q 2 (Z).