aa r X i v : . [ m a t h . A T ] J u l FRAMINGS OF W g, ALEXANDER KUPERS AND OSCAR RANDAL-WILLIAMS
Abstract.
We compute the set of framings of W g, = D n gS n × S n , up tohomotopy and diffeomorphism relative to the boundary. Contents
1. Introduction 12. Recollections and generalities 23. Counting framings relative to a point 84. Arithmetic groups 105. Comparing stabilisers and the proof of Theorem A 146. θ -structures on W g, Introduction
Closed surfaces do not admit a framing unless they have genus 1, but surfaces of anygenus with non-empty boundary do admit framings and there has been recent interestin understanding the set of such framings up to homotopy and diffeomorphism and,relatedly, the stabilisers of framings with respect to the action of the mapping classgroup [RW14, Kaw18, CS20b, CS20a, PCS20].The analogues in higher dimensions of genus g surfaces with one boundary componentare the 2 n -manifolds W g, := D n gS n × S n , which play a distinguished role in the study of diffeomorphism groups of 2 n -manifoldsvia homological stability [GRW18, GRW17]. These manifolds also admit framings, andall framings of W g, induce the same homotopy class of framing of T W g, | ∂W g, . Fixingonce and for all a framing ℓ ∂ of T W g, | ∂W g, in this homotopy class, in our work onTorelli groups and diffeomorphism groups of disks [KRW19, KRW20] we have needed tostudy the moduli spaces of framed manifolds diffeomorphic to W g, and with boundarycondition ℓ ∂ . The set of path components of this space is the set Str fr ∂ ( W g, ) of homotopyclasses of framings of W g, extending ℓ ∂ , modulo the action of the diffeomorphism group.In that work we could get away with qualitative information about this set of pathcomponents: that it is finite. In this note we precisely determine it. Theorem A.
Let g ≥ and n ≥ . The action of the mapping class group π (Diff ∂ ( W g, )) on the set Str fr ∂ ( W g, ) of homotopy classes of framings extending ℓ ∂ has(i) two orbits if n = 1 , , or n ≡ ;(ii) one orbit if n = 1 , , and n .If n > then in fact these hold for g ≥ . Date : July 2, 2020.2010
Mathematics Subject Classification.
While Theorem A is quite simple to formulate, our main interest is not so muchin this statement but rather in related results concerning the stabilisers of framingswith respect to the action of the mapping class group π (Diff ∂ ( W g, )), especially indimensions 2 n ≥
6. These results require substantial background to formulate, and weleave them to the body of the text: the main results in this direction are Propositions3.3 and 3.5, and Corollary 5.2. In Section 6 we explain how highly-connected tangentialstructures can be analysed similarly.
Remark . The exceptions in Theorem A are covered by the following:(i) The cases g = 0 are as in Table 2. (This uses that framings of D d up to homotopyand diffeomorphism, relative to a fixed boundary condition, are in bijection with π d (O( d )), and that the diffeomorphisms of the disc act trivially on them by Lemma2.2.)(ii) The case n = 1 and g = 1 is given in [Kaw18, Theorem 3.12]; it is rather compli-cated.The case n = 1 and g ≥ Acknowledgements.
AK is supported by NSF grant DMS-1803766. ORW was par-tially supported by the ERC under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No. 756444), and by a Philip Leverhulme Prizefrom the Leverhulme Trust.2.
Recollections and generalities
In this section we recall some notation and results from [KRW19], in particular Section8 of that paper, specialized to framings.2.1.
The mapping class group.
The mapping class group of W g, is defined asΓ g := π (Diff ∂ ( W g, )) = π ( B Diff ∂ ( W g, )) , where Diff ∂ ( W g, ) is the topological group of diffeomorphisms fixing a neighborhood ofthe boundary pointwise, in the C ∞ -topology. Let us write H n := H n ( W g, ; Z ) , which has an action of the mapping class group. In high dimensions we will use ananalysis of Γ g due to Kreck [Kre79] (see that paper for details about the definitions ofthe homomorphisms between the terms): Theorem 2.1 (Kreck) . For n ≥ , the mapping class group Γ g := π (Diff ∂ ( W g, )) isdescribed by the pair of extensions −→ I g −→ Γ g −→ G ′ g −→ , −→ Θ n +1 −→ I g −→ Hom( H n , Sπ n (SO( n ))) −→ , with Sπ n (SO( n )) as in Table 1 and G ′ g := Sp g ( Z ) if n is 3 or 7, Sp q g ( Z ) if n is odd but not 3 or 7, O g,g ( Z ) if n is even,where Sp q g ( Z ) ≤ Sp g ( Z ) denotes the proper subgroup of symplectic matrices whichpreserve the standard quadratic refinement µ . RAMINGS OF W g, Table 1.
The abelian groups Sπ n (SO( n )) for n ≥
1, with the excep-tions that Sπ (SO(1)) = 0, Sπ (SO(2)) = 0 and Sπ (SO(6)) = 0. n (mod 8) 0 1 2 3 4 5 6 7 Sπ n (SO( n )) ( Z / Z / Z / Z Z / Z / Z In particular, the homomorphism Γ g → G ′ g ⊂ GL g ( Z ) is given by sending a diffeo-morphism to the induced automorphism of H n . Part of Kreck’s theorem is the analysisof precisely which automorphisms of H n ( W g, ; Z ) arise from diffeomorphisms: they mustpreserve the intersection form, and for n = 3 , n = 1 , g −→ G ′ g := ( Sp g ( Z ) if n is 1,O g,g ( Z ) if n is 2,and these are still surjective: the first is folklore, the second is [Wal64, Theorem 2].The subgroup Θ n +1 ≤ Γ g corresponds to π (Diff ∂ ( D n )). We will make use of thefollowing well-known fact about diffeomorphisms of the disc. Lemma 2.2.
For n ≥ the derivative map Θ n +1 = π (Diff ∂ ( D n )) −→ π (Bun ∂ ( T D n )) = π n (SO(2 n )) . is trivial.Proof. This is equivalent to the well-known statement that the tangent bundle of a homo-topy sphere is isomorphic to the tangent bundle of the standard sphere, see e.g. [RP80,§1]. Another argument is given in [KRW19, Lemma 8.15] (take g = 0), using [BL74]. (cid:3) Framings.
Tangential structure on 2 n -dimensional manifolds can be equivalentlydescribed as GL n ( R )-spaces, or as fibrations over B O(2 n ) [GRW19, Section 4.5]. Inparticular, framings can be described by the GL n ( R )-space given by Θ fr = GL n ( R )with action given by right multiplication, or by the fibration θ fr : E O(2 n ) → B O(2 n ).Though we used the description in terms of the fibration θ fr in [KRW19, KRW20], herewe find the former description more convenient.A framing on W g, is a map of GL n ( R )-spaces ℓ : Fr( T W g, ) → Θ fr , with Fr( T W g, )the frame bundle of the tangent bundle of W g, , which is a principal GL n ( R )-bundle.We shall fix a boundary condition ℓ ∂ : Fr( T W g, | ∂W g, ) → Θ fr and only consider those θ -structures which extend this boundary condition: the space of framings of W g, ex-tending ℓ ∂ is defined to be the spaceBun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ )of GL n ( R )-equivariant maps Fr( T W g, ) → Θ fr extending ℓ ∂ . We writeStr fr ∂ ( W g, ) := π (Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ))for its set of path components.The manifold W g, indeed admits a framing: viewing it as the boundary connect-sumof g copies of the plumbing of S n × D n with itself, it is enough to note that S n × D n may be framed. The following is a special case of [KRW19, Lemma 8.5]: Lemma 2.3. (i) Up to homotopy there is a unique orientation preserving boundary condition ℓ ∂ which extends to a framing ℓ on all of W g, . ALEXANDER KUPERS AND OSCAR RANDAL-WILLIAMS (ii) For such a boundary condition there is a homeomorphism
Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) ≃ map ∂ ( W g, , SO(2 n )) , depending on a choice of reference framing τ satisfying this boundary condition. A choice of reference framing τ therefore induces, by Lemma 2.3 (ii), a bijectionStr fr ∂ ( W g, ) ∼ −→ π (map ∂ ( W g, , SO(2 n ))) , though one must use it carefully as it depends on the choice of framing τ .The advantage of using GL n ( R )-spaces as a model for tangential structures, ratherthan fibrations over BO (2 n ), is that the homotopy equivalence in Lemma 2.3 (ii) canbe upgraded to a homeomorphismBun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) ∼ = −→ map ∂ ( W g, , GL n ( R )) . (1)The group Diff ∂ ( W g, ) acts on the space of framings Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) by takingderivatives. In particular it acts through the topological monoid Bun ∂ (Fr( T W g, )) ofGL n ( R )-maps Fr( T W g, ) → Fr(
T W g, ) that are the identity on the boundary. Usingthe reference framing τ , the argument in [KRW19, Section 4] provides a homeomorphismof topological monoidsBun ∂ ( T W g, ) ∼ = −→ map ∂ ( W g, , W g, × GL n ( R )) (2)under which composition of bundle maps corresponds to the operation( f, λ ) ⊛ ( g, ρ ) := ( f ◦ g, ( λ ◦ g ) · ρ ) , with ◦ denoting composition of maps and · denoting pointwise multiplication. Theright action of Bun ∂ (Fr( T W g, )) on Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) by precomposition thencorresponds tomap ∂ ( W g, , GL n ( R )) × map ∂ ( W g, , W g, × GL n ( R )) −→ map ∂ ( W g, , GL n ( R ))( h, ( f, λ )) ( h ◦ f ) · λ, under the homeomorphisms (1) and (2), where here · denotes the multiplication inGL n ( R ). We write h ⊛ ( f, λ ) for this operation.Since ∂W g, → W g, is 0-connected, all maps in the right hand side of (2) haveimage in W g, × GL +2 n ( R ), with GL +2 n ( R ) ≤ GL n ( R ) the path-component of orientation-preserving invertible matrices. As the inclusion SO(2 n ) ֒ → GL +2 n ( R ) is a homotopyequivalence, we phrase later computations in terms of the homotopy groups of SO(2 n )rather than GL +2 n ( R ).2.3. The moduli space of framed manifolds.
The moduli space of framed manifoldsdiffeomorphic to W g, relative to the boundary , mentioned in the introduction, is definedto be the homotopy quotient B Diff fr ∂ ( W g, ; ℓ ∂ ) := Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) (cid:12) Diff ∂ ( W g, ) . (3)There is a bijection π ( B Diff fr ∂ ( W g, ; ℓ ∂ )) ∼ = ←− Str fr ∂ ( W g, ) / Γ g , and this is the set which Theorem A proposes to describe.For [ ℓ ] ∈ Str fr ∂ ( W g, ) we shall writeΓ fr , [ ℓ ] g := Stab Γ g ([ ℓ ])for its stabiliser.The framed mapping class group of a framing ℓ ∈ Bun ∂ (Fr( T W g, ) , Θ fr ; ℓ ∂ ) is definedas ˇΓ fr ,ℓg := π ( B Diff fr ∂ ( W g, ; ℓ ∂ ) , ℓ ) , RAMINGS OF W g, and the long exact sequence for the homotopy orbits (3) gives a surjection ˇΓ fr ,ℓg → Γ fr , [ ℓ ] g .We write G fr , [ ℓ ] g := im(Γ fr , [ ℓ ] g → Γ g → G ′ g ) . Relaxing the boundary condition.
It will be helpful to relax the condition thatframings agree with ℓ ∂ on all of the boundary and only ask that they agree at a point.Fix a point ∗ ∈ ∂W g, . Then we let Str fr ∗ ( W g, ) denote the homotopy classes offramings on W g, which agree with ℓ ∂ at ∗ ∈ ∂W g, . Using the vanishing of Whiteheadbrackets in SO(2 n ), as in [KRW19, Section 8.2.2] we obtain a short exact sequence0 Str fr ∂ ( D n ) Str fr ∂ ( W g, ) Str fr ∗ ( W g, ) 00 π n (SO(2 n )) π (map ∂ ( W g, , SO(2 n ))) Hom( H n , π n (SO(2 n ))) 0 . ∼ = (cid:8) ∼ = ∼ = (cid:8) Here the vertical isomorphisms are induced by τ , and this is in fact a short exactsequence of groups using the group structure coming from pointwise multiplication inSO(2 n ). The groups π n (SO(2 n )) are well-known by Bott periodicity, and the groups π n (SO(2 n )) were computed by Kervaire [Ker60] (see Table 2). Table 2.
The groups π n (SO(2 n )) for n ≥
1, with the exceptions that π (SO(2)) = 0 and π (SO(6)) = 0. n (mod 4) 0 1 2 3 π n (SO(2 n )) ( Z / Z / Z / Z / g . The induced right actions on the bottom short exact sequence are asfollows. It acts on the middle term via the derivative map Γ g → π (Bun ∂ (Fr( T W g, ))),the identification Bun ∂ (Fr( T W g, )) ∼ = map ∂ ( W g, , W g, × GL n ( R )) from (2) given by τ , and the action ⊛ described above. It acts on the right-hand term via the derivativemap composed with the map Bun ∂ (Fr( T W g, )) → Bun / ∂ (Fr( T W g, )) which relaxesthe boundary composition, followed by the analogueBun / ∂ (Fr( T W g, )) −→ map / ∂ ( W g, , W g, × GL n ( R ))of (2) induced by τ , followed by the formula α ⊛ ( B, β ) = α ◦ B + β written in terms of π (map / ∂ ( W g, , W g, × GL n ( R ))) ∼ = GL( H n ) ⋉ Hom( H n , π n (SO(2 n ))) . The Γ g - and Str fr ∂ ( D n )-actions on Str fr ∂ ( W g, ) commute because the Str fr ∂ ( D n )-actionis by changing the framings in a small disc near the boundary, and diffeomorphisms inΓ g can be changed by an isotopy so that they fix such a disc.We write [[ ℓ ]] ∈ Str fr ∗ ( W g, ) for the class of a framing ℓ , and letΓ fr , [[ ℓ ]] g := Stab Γ g ([[ ℓ ]]) . We define G fr , [[ ℓ ]] g := im(Γ fr , [[ ℓ ]] g → Γ g → G ′ g ) . Remark . In [KRW19] we defined G fr , [[ ℓ ]] g as the image of the stabiliser Λ fr , [[ ℓ ]] g :=Stab Λ g ([[ ℓ ]]) in G ′ g , where Λ g := Γ g / Θ n +1 . The group Λ g acts on Str fr ∗ ( W g, ) becauseΘ n +1 consist of diffeomorphisms supported in a small disc near the boundary, andwhen the boundary condition has been relaxed near this ball the derivatives of suchdiffeomorphisms are homotopic to the identity (in fact, by Lemma 2.2 the subgroupΘ n +1 already acts trivially on Str fr ∂ ( W g, )). Then Λ fr , [[ ℓ ]] g = Γ fr , [[ ℓ ]] g / Θ n +1 , so the images ALEXANDER KUPERS AND OSCAR RANDAL-WILLIAMS of Λ fr , [[ ℓ ]] g and Γ fr , [[ ℓ ]] g in G ′ g are equal. The group Λ fr , [[ ℓ ]] g can also be interpreted in termsof self-embeddings, c.f. [KRW19, Section 8.5.2].2.5. Quadratic refinements.
The group H n = H n ( W g, ; Z ) is equipped with the in-tersection form λ : H n ⊗ H n → Z , which is ( − n -symmetric. A function µ : H n −→ ( Z if n is even Z / n is oddis called a quadratic refinement of ( H n , λ ) if it satisfies µ ( a · x ) = a µ ( x ) µ ( x + y ) = µ ( x ) + µ ( y ) + λ ( x, y ) , (4)for a ∈ Z and x, y ∈ H n , where in the latter λ ( x, y ) is reduced modulo 2 if n is odd. If n iseven then these properties imply that µ ( x ) = λ ( x, x ), so ( H n , λ ) has a unique quadraticrefinement and it carries no further information: we shall therefore now suppose that n is odd, and write Quad( H n , λ ) for the set of quadratic refinements.If µ and µ ′ are quadratic refinements of ( H n , λ ) then µ ′ − µ : H n → Z / H n , λ ) forms a Hom( H n , Z / e , f , e , f , . . . , e g , f g for H n , a quadratic refinement µ is uniquely and freely determinedby the quadratic property (4) and the values µ ( e ) , . . . , µ ( f g ) ∈ Z / Classification of quadratic refinements.
The action of Sp g ( Z ) on Quad( H n , λ ) iswell-known [Arf41] to have two orbits, distinguished by the Arf invariant
Arf( µ ) = g X i =1 µ ( e i ) µ ( f i ) ∈ Z / , where e , f , e , f , . . . , e g , f g ∈ H n is a symplectic basis.Let us introduce some further notation for specific quadratic forms. Let H (0) denotethe module Z { e, f } with anti-symmetric form determined by λ ( e, f ) = 1 and quadraticrefinement determined by µ ( e ) = 0 = µ ( f ). Let H (1) denote the module Z { e, f } withthe same anti-symmetric form but quadratic refinement determined by µ ( e ) = 1 = µ ( f ).These quadratic forms have Arf invariant 0 and 1 respectively. Since the Arf-invariant isadditive under orthogonal sum, is valued in Z /
2, and is a complete invariant of quadraticforms, there is an isomorphism H (1) ⊕ ∼ = H (0) ⊕ of quadratic forms. (In the proof ofLemma 4.2 we will make an explicit choice of such an isomorphism.) Example . There are 2 g − + 2 g − quadratic refinements of Arf invariant 0. The standard quadratic refinement µ is that of H (0) ⊕ g , determined by µ ( e ) = µ ( f ) = · · · = µ ( e g ) = µ ( f g ) = 0 . The group Sp q g ( Z ) in the statement of Theorem 2.1 is the stabiliser of µ for the actionof Sp g ( Z ) on the set of quadratic refinements. Example . Similarly, there are 2 g − − g − quadratic refinements of Arf invariant 1.For concreteness, we take H (1) ⊕ H (0) ⊕ g − as the standard choice, and write Sp a g ( Z ) ≤ Sp g ( Z ) for its stabiliser.2.5.2. A quadratic form.
Suppose that n >
1. The group π n (Fr( T W g, )) may be in-terpreted via Hirsch–Smale theory as the set I fr n ( W g, ) of regular homotopy classes ofimmersions j : S n × D n W g, . Using the map π n (Fr( T W g, )) → π n ( W g, ) = H n andthe intersection form λ , this has a (degenerate) ( − n -symmetric bilinear form λ fr . Thishas a quadratic refinement µ fr given by µ fr ([ j ]) = { self-intersections of j | S n ×{ } } (mod 2) . RAMINGS OF W g, This construction is due to Wall [Wal99, Theorem 5.2]; see [GRW18, Definition 5.2] fora discussion specific to the manifolds W g, .As the manifolds W g, admit a framing there is a splittable short exact sequence0 −→ π n (SO(2 n )) i −→ π n (Fr( T W g, )) −→ π n ( W g, ) → . Although the function µ fr is quadratic, the composition µ fr ◦ i : π n (SO(2 n )) → Z / i ( π n (SO(2 n ))) is the radical of the bilinear form λ fr ). Lemma 2.7.
The map µ fr ◦ i : π n (SO(2 n )) → Z / is onto if and only if n = 3 , .Proof. Recall the Whitney “figure eight” immersion S n D n , which has one doublepoint and has normal bundle isomorphic to T S n .If n = 3 , T S n is trivial, so the Whitney immersion may be extended to animmersion j : S n × D n D n ⊂ W g, which satisfies µ ([ j ]) = 1, and so the composition π n (SO(2 n )) = π n (Fr( D n )) i −→ π n (Fr( T W g, )) µ fr −→ Z / n = 3 ,
7, suppose for a contradiction that j : S n × D n D n ⊂ W g, has µ fr ([ j ]) = 1.We may form the ambient connect-sum inside D n of the immersion j | S n ×{ } with adisjoint copy of the Whitney immersion, giving an immersion j ′ : S n D n havingan even number of double points and having normal bundle T S n . Using the Whitneytrick we can eliminate the double points to obtain an embedding j ′′ , still having normalbundle T S n : as T S n is a non-trivial bundle for n = 3 , (cid:3) It follows that for n = 3 , µ fr descends to a function µ : H n → Z /
2, aquadratic refinement of ( H n , λ ). As the standard symplectic basis of H n may be rep-resented by embedded normally-framed spheres, in the notation of the previous sectionthis gives the quadratic form H (0) ⊕ g . Diffeomorphisms of W g, must also preserve thisquadratic refinement: this accounts for the fact that G ′ g = Sp q g ( Z ) in these dimensionsin Theorem 2.1.2.5.3. Quadratic refinements from framings.
For n odd a framing ℓ : W g, → Fr(
T W g, )may be used to define µ ℓ : H n = π n ( W g, ) ℓ ∗ −→ π n (Fr( T W g, )) µ fr −→ Z / , which is a quadratic refinement of ( H n , λ ). This construction provides a Γ g -equivariantfunction Φ : Str fr ∗ ( W g, ) −→ Quad( H n , λ ) , where the action of Γ g on Quad( H n , λ ) is via G ′ g . Lemma 2.8.
For n = 3 , the function Φ is surjective.Proof. Choose a framing ℓ , and let µ ′ ∈ Quad( H n , λ ). The function µ ′ − µ ℓ is a homo-morphism L : H n → Z /
2, and as H n is a free Z -module and µ fr ◦ i is surjective by Lemma2.7, we may choose a homomorphism δ : H n → π n (SO(2 n )) such that µ fr ◦ i ◦ δ = L . Butthen if the framing ℓ is changed using δ to get a new framing δ · ℓ , we have µ δ · ℓ = µ ℓ + L ,so µ ′ = µ δ · ℓ . Thus Φ is a surjection. (cid:3) ALEXANDER KUPERS AND OSCAR RANDAL-WILLIAMS
Orbits and stabilisers.
Our proof of Theorem A will be in terms of the sequence0 → Γ fr , [ ℓ ] g → Γ fr , [[ ℓ ]] g f ℓ −→ Str fr ∂ ( D n ) −· [ ℓ ] −→ Str fr ∂ ( W g, ) / Γ g → Str fr ∗ ( W g, ) / Γ g → {∗} (5)which is exact in the sense of groups and pointed sets. In particular, f ℓ is a grouphomomorphism. This sequence comes from the long exact sequence on homotopy groupsfor the principal Str fr ∂ ( D n )-bundleStr fr ∂ ( D n ) −→ Str fr ∂ ( W g, ) // Γ g −→ Str fr ∗ ( W g, ) // Γ g , and the orbit-stabiliser theorem.The strategy of our argument will be as follows. We will estimate the size of im( f ℓ ) =Γ fr , [[ ℓ ]] g / Γ fr , [ ℓ ] g from below using the surjectionΓ fr , [[ ℓ ]] g / Γ fr , [ ℓ ] g −→ G fr , [[ ℓ ]] g /G fr , [ ℓ ] g . We will identify the group G fr , [[ ℓ ]] g as the automorphism group of a certain quadratic form(in fact we will have G fr , [[ ℓ ]] g = G ′ g for n = 1 , ,
7, and a mild variant for n = 1 , , G fr , [ ℓ ] g has trivial abelianisation, so that G fr , [[ ℓ ]] g /G fr , [ ℓ ] g has order at least 4. We thenuse Str fr ∂ ( D n ) = π n (SO(2 n )) and consult Table 2.For n π n (SO(2 n )) has order 4, so it follows that the map f ℓ is a surjection, and so Γ fr , [ ℓ ] g has index precisely 4 in Γ fr , [[ ℓ ]] g . Thus G fr , [[ ℓ ]] g /G fr , [ ℓ ] g hasorder precisely 4, from which we will deduce that G fr , [ ℓ ] g is precisely the commutatorsubgroup of G fr , [[ ℓ ]] g .For n ≡ π n (SO(2 n )) has ordder 8, and the argument is a littlemore complicated. We will show that the image of f ℓ has index 2 in Str fr ∂ ( D n ) = π n (SO(2 n )) = ( Z / .3. Counting framings relative to a point
In this section we determine Str fr ∗ ( W g, ) / Γ g . Proposition 3.1.
Suppose n ≥ and g ≥ , then · If n = 1 , , , then Str fr ∗ ( W g, ) / Γ g consists of a single element. · If n = 1 , , , then Str fr ∗ ( W g, ) / Γ g consists of two elements.If n > then these in fact hold for g ≥ . The cases n ≥ . In this case Theorem 2.1 is available, and we will study theaction of I g ≤ Γ g on Str fr ∗ ( W g, ). Derivatives of elements of Θ n +1 ≤ I g are bundle mapssupported in a small disc which can be taken to be near the boundary: as in Remark2.4 these act trivially on Str fr ∗ ( W g, ). Thus the action of I g on Hom( H n , π n (SO(2 n )))factors over I g → I g / Θ n +1 = Hom( H n , Sπ n (SO( n ))). This lands in the subgroupHom( H n , π n (SO(2 n ))) ⊂ GL( H n ) ⋉ Hom( H n , π n (SO(2 n )))of π (map / ∂ ( W g, , W g, × SO(2 n ))) by applying the homomorphism Sπ n ( O ( n )) → π n (SO(2 n )) to the target. This homomorphism was studied by Levine; Theorem 1.4 of[Lev85] and Table 1 imply: Lemma 3.2.
For n ≥ the stabilisation Sπ n (SO( n )) → π n (SO(2 n )) is:(i) surjective with kernel Z / when n is even,(ii) an isomorphism when n is odd but not or ,(iii) injective with cokernel Z / when n = 3 , . We conclude that:
Proposition 3.3.
Suppose n ≥ and consider the action of I g on Str fr ∗ ( W g, ) . RAMINGS OF W g, (i) When n = 3 , , this action has a single orbit.(ii) When n = 3 , , the set of orbits is in bijection with Hom( H n , Z / .In either case the stabiliser I fr , [[ ℓ ]] g of any [[ ℓ ]] ∈ Str fr ∗ ( W g, ) satisfies I fr , [[ ℓ ]] g / Θ n +1 ∼ = ( if n is odd Hom( H n , Z / if n is even . Proof.
When n = 3 ,
7, Lemma 3.2 says that I g surjects on to Hom( H n , π n (SO(2 n ))).The action on Hom( H n , π n (SO(2 n ))) is through addition, which is therefore transitive.When n = 3 , I g maps to Hom( H n , π n (SO(2 n ))) with cokernelHom( H n , Z / Sπ n ( O ( n )) → π n (SO(2 n )) if 0 if n is odd and Z / n iseven. (cid:3) This proves Proposition 3.1 when n ≥ n = 3 ,
7, as Str fr ∗ ( W g, ) /I g alreadyconsists of a single point, so Str fr ∗ ( W g, ) / Γ g does too.To finish the argument in the cases n = 3 ,
7, we use quadratic refinements to give amore invariant description of Str fr ∗ ( W g, ) /I g . Recall that in Section 2.5.3 we describeda Γ g -equivariant function Φ : Str fr ∗ ( W g, ) → Quad( H n , λ ). Lemma 3.4.
For n = 3 , the induced function Str fr ∗ ( W g, ) /I g −→ Quad( H n , λ ) is a bijection.Proof. By Lemma 2.8 it is surjective. The target has 2 g elements as it is a Hom( H n , Z / g elements by Proposition 3.3 (ii), so it is a bijection. (cid:3) Referring to the discussion in Section 2.5.1, it follows from the theorem of Arf thatStr fr ∗ ( W g, ) / Γ g consists of two elements, distinguished by the Arf invariants of theirassociated quadratic forms. This completes the proof of Proposition 3.1 in the cases n = 3 , The case n = 2 . When n = 2 we have π n (SO(2 n )) = 0 so Str fr ∗ ( W g, ) consists ofa single point, so has a single Γ g -orbit too.3.3. The case n = 1 . As Str fr ∂ ( D ) = π (SO(2)) = 0, the orbit-stabiliser sequencegives a bijection Str fr ∗ ( W g, ) / Γ g ∼ = Str fr ∂ ( W g, ) / Γ g . For g ≥ r = 0) as having two elements, whichproves Proposition 3.1 in this case.In fact the analogue of Lemma 3.4 also holds in this case. A framing of W g, deter-mines a Spin structure, which via a construction of Johnson [Joh80] gives a quadraticrefinement of the symplectic form ( H ( W g, ; Z ) , λ ). This construction yields a surjectivemap Str fr ∗ ( W g, ) −→ Quad( H , λ )which as long as g ≥ I g . This has been shown in [CS20b, Proposition 5.1]. Itmay also be seen using the methods of [RW14, Sections 2.3, 2.4] and the fact that theTorelli group is generated by Dehn twists along separating curves and bounding pairsof curves. The group G fr , [[ ℓ ]] g . The results of the previous sections combine to give the fol-lowing complete description of the group G fr , [[ ℓ ]] g . Proposition 3.5.
For n ≥ , and g ≥ if n = 1 , we have G fr , [[ ℓ ]] g = Sp q or a g ( Z ) if n is 1, 3 or 7, and ℓ has Arf invariant or , Sp q g ( Z ) if n is odd but not 1, 3 or 7, O g,g ( Z ) if n is even.Proof. If n = 2 then as in Section 3.2 there is only one framing relative to a point, soΓ fr , [[ ℓ ]] g = Γ g and hence G fr , [[ ℓ ]] g = G ′ g .For n = 2 the group G fr , [[ ℓ ]] g is the stabiliser of the class of ℓ in Str fr ∗ ( W g, ) /I g underthe residual Γ g /I g = G ′ g -action. If n = 1 , , G ′ g itself. If n = 1 , , H n , λ )and so G fr , [[ ℓ ]] g is the stabiliser of the quadratic form determined by ℓ . As there are twoorbits of quadratic forms, distingushed by their Arf invariant, this stabiliser is conjugateto Sp q g ( Z ) if the Arf invariant is 0, and to Sp a g ( Z ) if the Arf invariant is 1. (cid:3) For 2 n ≥ −→ I fr , [[ ℓ ]] g −→ Γ fr , [[ ℓ ]] g −→ G fr , [[ ℓ ]] g −→ −→ Θ n +1 −→ I fr , [[ ℓ ]] g −→ ( n is oddHom( H n , Z /
2) if n is even −→ . With Proposition 3.5 this gives a description of Γ fr , [[ ℓ ]] g analogous to the theorem of Kreck(Theorem 2.1). 4. Arithmetic groups
Abelianisations of some arithmetic groups.
By stabilising by direct sum witha hyperbolic form, of Arf invariant 0 in the case of quadratic structures, we have stablegroups Sp ∞ ( Z ) , Sp q ∞ ( Z ) , Sp a ∞ ( Z ) , O ∞ , ∞ ( Z ) . Furthermore, as H (0) ⊕ ∼ = H (1) ⊕ there is a direct system of groups containing both { Sp q g ( Z ) } g ≥ and { Sp a g ( Z ) } g ≥ cofinally, so Sp q ∞ ( Z ) ∼ = Sp a ∞ ( Z ). The abelianisations ofthese stable groups are well-known: H (Sp ∞ ( Z )) = 0 ,H (Sp q ∞ ( Z )) = H (Sp a ∞ ( Z )) = Z / ,H (O ∞ , ∞ ( Z )) = ( Z / . These are collected from the literature in [GRW16, Proposition 2.2]. Such automorphismgroups of quadratic forms over Z enjoy homological stability: in the generality neededhere this may be found in [Fri17, Theorem 3.25], but for some of these groups it wasknown much earlier. We shall only need to know that the abelianisation of the g = 1group surjects onto to the abelianisation of the g = ∞ one, but we give completeinformation about their abelianisations for all g in Table 3. Remark . It may be helpful to alert the reader that [Kre79, page 645] states incorrectlythat O , ( Z ) = Z /
4, a mistake going back to [Sat69].
RAMINGS OF W g, Table 3.
The first homology groups of Sp g ( Z ), Sp q g ( Z ), Sp a g ( Z ), and O g,g ( Z ). g ≥ H (Sp g ( Z )) Z / Z / H (Sp q g ( Z )) Z / ⊕ Z Z / ⊕ Z / Z / H (Sp a g ( Z )) Z / Z / Z / H (O g,g ( Z )) ( Z / ( Z / ( Z / The (quadratic) symplectic groups.
The first homology group of Sp g ( Z ) andSq q g ( Z ) in low genus is tabulated in [Kra19, Lemma A.1], and that of Sp a g ( Z ) hasrecently been calculated by Sierra [Sie] and will appear in his forthcoming CambridgePhD thesis. They are as shown in Table 3. Lemma 4.2.
Each of the stabilisation maps H (Sp ( Z )) → H (Sp ∞ ( Z )) , H (Sp q ( Z )) → H (Sp q ∞ ( Z )) , and H (Sp a ( Z )) → H (Sp a ∞ ( Z )) is surjective.Proof. In the first case there is nothing to show as the stable homology is trivial. Thesecond case is [Kra19, Lemma A.1 (iii)].In the third case observe that as in Example 2.6 the number of quadratic refinementsof ( H n ( W , ; Z ) , λ ) having Arf invariant 1 is 2 − − − = 1, so we have Sp a ( Z ) =Sp ( Z ) = SL ( Z ). The first homology of this group is Z /
12 and an element of order 4is represented by the matrix S := (cid:20) −
11 0 (cid:21) . Stabilising H (1) by taking the direct sum with another copy of H (1), we get a stabilisa-tion map Sp a ( Z ) → Sp q ( Z ). We shall compute its image under the stable abelianisationmap λ : Sp q ( Z ) → Z / H (1) ⊕ H (1) ∼ = H (0) ⊕ H (0). Let e , f , e , f be the standard “hyperbolic” basis of H (1) ⊕ H (1). Making explicit thisisomorphism amounts to finding a basis ˜ e , ˜ f , ˜ e , ˜ f of H (1) ⊕ H (1) satisfying · λ (˜ e i , ˜ e j ) = 0 = λ ( ˜ f i , ˜ f j ), · λ (˜ e i , ˜ f j ) = δ ij , and · µ (˜ e i ) = µ ( ˜ f i ) = 0.The choice˜ e = e + e ˜ f = f + e + e ˜ e = e − f + f ˜ f = − f + f will do. Writing S ⊕ [ ] in terms of the basis { ˜ e , ˜ e , ˜ f , ˜ f } gives the matrix˜ S = − − − −
12 0 − − . Johnson and Millson give an explicit formula for the homomorphism λ : Sp q ( Z ) → Z / { , − , i, − i } [JM90, pages 147-148] (their conventions are the reason for reorder-ing our basis). Evaluated on ˜ S we are in the case “2 (ii)”, and λ ( ˜ S ) = i − ǫ (1 · − ( − ·
1) = i , which has order 4. (cid:3) The orthogonal groups.
We provide the analogue of the results of the last sectionfor the groups O g,g ( Z ). Lemma 4.3.
For g = 2 the first homology groups of O g,g ( Z ) are as in Table 3. Thestabilisation map H (O , ( Z )) → H (O ∞ , ∞ ( Z )) is a surjection. Proof.
We shall use results of Hahn–O’Meara: [HO89, Theorem 9.2.8] says the kernelof the map O g,g ( Z ) → ( Z / given by the determinant and spinor norm is equal to thesubgroup generated by elementary matrices for g ≥
2, and [HO89, 5.3.8] says that thissubgroup is perfect for g ≥
3. It follows that ( Z / is the abelianisation for g ≥ g = 1 it is easy to verify that the determinant and spinor norm map O , ( Z ) → ( Z / is an isomorphism of groups, and the claim about the stabilisation map followsfrom this. (cid:3) It remains to describe the case g = 2. Lemma 4.4. H (O , ( Z )) ∼ = ( Z / .Proof. We claim that O ′ , ( Z ) := ker(det ⊕ spin : O , ( Z ) → ( Z / ) is isomorphic to thegroup SL ( Z ) × SL ( Z ) / (cid:10)(cid:0)(cid:2) − − (cid:3) , (cid:2) − − (cid:3)(cid:1)(cid:11) . By [HO89, Theorem 7.2.21] there is an extension1 −→ Z × −→ Spin , ( Z ) −→ O ′ , ( Z ) −→ , and by [HO89, p. 434] there is an exceptional isomorphismSpin , ( Z ) ∼ = SL ( Z ) × SL ( Z ) . We can make this explicit as follows. Consider the set M , ( Z ) of (2 × Z , equipped with the bilinear form given by h X, Y i = tr( X Ω Y t Ω t ) , Ω = (cid:20) −
11 0 (cid:21) . Explicitly it is given by (cid:28)(cid:20) a bc d (cid:21) , (cid:20) a ′ b ′ c ′ d ′ (cid:21)(cid:29) = ad ′ − bc ′ − cb ′ + da ′ , so is symmetric and even. This formula also makes it clear that e = (cid:20) (cid:21) , f = (cid:20) (cid:21) , e = (cid:20) (cid:21) , f = (cid:20) − (cid:21) provides a hyperbolic basis for ( M , ( Z ) , h− , −i ). There is a left action of SL ( Z ) × SL ( Z )on M , ( Z ) by ( A, B ) · X = AXB − , and one may check that this action preserves theform h− , −i , using that A − = Ω A t Ω t for A ∈ SL ( Z ). This describes the compositionSL ( Z ) × SL ( Z ) ∼ = Spin , ( Z ) −→ O , ( Z ) . We see that Z × → Spin , ( Z ) is given by (cid:0)(cid:2) − − (cid:3) , (cid:2) − − (cid:3)(cid:1) , which establishes theclaim, and also that the elements ([ ] , [ ]) and ([ ] , [ ]) in SL ( Z ) × SL ( Z ) mapto T := −
10 1 0 00 1 1 00 0 0 1 and T := − respectively,with respect to the basis ( e , f , e , f ).We now calculate as follows. It is well known that H (SL ( Z )) = Z /
12 is generated by T := [ ], and minus the identity matrix represents the element of order 2 in this group,so H (O ′ , ( Z )) has a presentation as an abelian group by h T , T | T , T , T + T ) i ,The group O , ( Z ) = (cid:10)(cid:2) − − (cid:3) , [ ] (cid:11) may be included in O , ( Z ) by stabilisation,and is mapped isomorphically to ( Z / by the determinant and spinor norm. Thus theouter action of ( Z / on O ′ , ( Z ) may be described by conjugating by (the stabilisationsof) these two matrices. Conjugating by (cid:2) − − (cid:3) ⊕ [ ] acts as T T − T T − W g, and conjugating by [ ] ⊕ [ ] acts as T T − T T − . Thus the coinvariants of the ( Z / -action on H (O ′ , ( Z )) are given by the abeliangroup presentation h T , T | T , T , T + T ) , T , T , T + T i which simplifies to give Z /
2. The argument is completed by considering the Leray–Hochschild–Serre spectral sequence for 1 → O ′ , ( Z ) → O , ( Z ) → ( Z / →
1, which issplit. (cid:3)
A vanishing result.
The previous section computed the abelianisation of thearithmetic groups G ′ g and G fr , [[ ℓ ]] g arising in the discussion in Section 2. Here we computethe abelianisation of the remaining arithmetic group G fr , [ ℓ ] g , at least for n = 1 , g tends to infinity. The strategy is quite different: the group G fr , [ ℓ ] g arises asa quotient of the framed mapping class group ˇΓ fr ,ℓg , and the proof is based on geometricconsiderations of framed fibre bundles. Lemma 4.5. H ( G fr , [ ℓ ] ∞ ) = 0 for n = 1 , . It will follow by combining Corollary 5.2, Proposition 3.5, and Table 3 that thisvanishing result does not hold for n = 1 , Proof.
Recall that we write ˇΓ fr ,ℓg := π ( B Diff fr ∂ ( W g, ; ℓ ∂ ) , ℓ ), so that Γ fr , [ ℓ ] g is the imageof the forgetful map ˇΓ fr ,ℓg → Γ g . The compositionˇΓ fr ,ℓ = H (ˇΓ fr ,ℓ ) −→ H (ˇΓ fr ,ℓg ) −→ H (Γ fr , [ ℓ ] g ) −→ H ( G fr , [ ℓ ] g ) (6)is zero, as diffeomorphisms supported inside a disc act trivially on the middle homologyof W g, . The two rightmost maps are surjective, as the underlying maps of groups aresurjective.By an application of [GRW17, Theorem 1.5] there is a map H (ˇΓ fr ,ℓg ) = H ( B Diff fr ∂ ( W g, ; ℓ ∂ ) ℓ ) −→ H (Ω ∞ S − n ) = π s n +1 = π n +1 ( G )which is an isomorphism in the limit g → ∞ (this formulation is valid even if 2 n = 4, asit does not rely on homological stability). Considering the target as framed cobordism,this map is given by the Pontrjagin–Thom construction.If n = 2 then we use that π s = 0, so that colim g →∞ H (ˇΓ fr ,ℓg ) = 0 surjects ontocolim g →∞ H ( G fr , [ ℓ ] g ).For n > H (ˇΓ fr ,ℓ ) → colim g →∞ H (ˇΓ fr ,ℓg ) is surjective, which with thefact that the composition (6) is zero gives the result. To do so we use smoothing theoryto obtain an identification ˇΓ fr ,ℓ = π n +1 (Top(2 n )), and it is a matter of interpretingthe Pontrjagin–Thom construction to see that ˇΓ fr ,ℓ → colim g →∞ H (ˇΓ fr ,ℓg ) = π s n +1 = π n +1 ( G ) agrees with the natural composition π n +1 (Top(2 n )) −→ π n +1 (Top) −→ π n +1 ( G ) . The homotopy groups π i ( G/ Top) are identified with the simply-connected surgery ob-struction groups L i ( Z ) [KS77, p. 274], which vanish in odd degrees so it follows that theright map is surjective; thus we shall be done if we show that the left map is surjective.To show this we will instead show that the connecting map ∂ : π n +1 (Top , Top(2 n )) −→ π n (Top(2 n ))is injective. From [KS77, page 246], it follows that the map π n +1 (O , O(2 n )) −→ π n +1 (Top , Top(2 n )) is an isomorphism, and as in [GRW16, Lemma 5.2] by work of Paechter [Pae56] we have π n +1 (O , O(2 n )) = ( Z / n odd( Z / n even . We therefore consider the diagram π n +1 (Top(2 n ) / O(2 n )) π (Diff ∂ ( D n )) π n +1 (O , O(2 n )) π n (O(2 n )) π n (O) π n +1 (Top , Top(2 n )) π n (Top(2 n )) . ∼ = ∼ = The top vertical map is zero: it corresponds to the map sending a diffeomorphism toits derivative, which is trivial by Lemma 2.2. Thus the bottom middle vertical map isinjective. On the other hand we know the groups π n (O(2 n )) from Table 2, and thegroups π n (O) from Bott periodicity, so we may simply check that π n +1 (O , O(2 n )) → π n (O(2 n )) must be injective for n >
3. It follows from commutativity of the squarethat the bottom map is injective, as required. (cid:3) Comparing stabilisers and the proof of Theorem A
In this section we wish to analyse the exact sequence0 −→ Γ fr , [ ℓ ] g −→ Γ fr , [[ ℓ ]] g f ℓ −→ Str fr ∂ ( D n ) ∼ = π n (SO(2 n )) , (7)coming from (5), as follows. Proposition 5.1.
For any n > the map f ℓ factors as Γ fr , [[ ℓ ]] g −→ G fr , [[ ℓ ]] g −→ H ( G fr , [[ ℓ ]] g ) −→ H ( G fr , [[ ℓ ]] ∞ ) h ℓ −→ π n (SO(2 n )) for some h ℓ , where the first three maps are the natural quotient, abelianisation, andstabilisation maps.(i) If n = 1 , then π n (SO(2 n )) = 0 (so the map f ℓ is surjective).(ii) If n and n = 1 , then h ℓ is an isomorphism.(iii) If n ≡ then h ℓ is injective with image of index 2 in π n (SO(2 n )) ∼ =( Z / . In Remark 6.4 we will determine more precisely the index 2 subgroup in part (iii).
Proof. If n = 1 , π n (SO(2 n )) = 0 so the claims are vacuous.By naturality of the sequence (7) under boundary connect-sum, the connecting map f ℓ factors over (the abelianisation of) its stabilisation, i.e. asΓ fr , [[ ℓ ]] g −→ Γ fr , [[ ℓ ]] ∞ −→ H (Γ fr , [[ ℓ ]] ∞ ) g ℓ −→ π n (SO(2 n )) . To prove the first part we must show that this map g ℓ factors as H (Γ fr , [[ ℓ ]] ∞ ) −→ H ( G fr , [[ ℓ ]] ∞ ) h ℓ −→ π n (SO(2 n ))for some (unique, as the first map is surjective) map h ℓ .If n = 2 then by [Kre79, Theorem 1] the map Γ fr , [[ ℓ ]] g → G fr , [[ ℓ ]] g has kernel consist-ing of isotopy classes of diffeomorphisms which are pseudoisotopic to the identity, andby [Qui86, Theorem 1.4] such a diffeomorphism becomes isotopic to the identity aftersufficiently-many stabilisations by S × S . Thus these maps become isomorphisms ofgroups in the colimit, and so in particular H (Γ fr , [[ ℓ ]] ∞ ) → H ( G fr , [[ ℓ ]] ∞ ) is an isomorphism,so g ℓ factors as desired. RAMINGS OF W g, Suppose now that n >
3. By stabilising if necessary we may suppose that g is large.By Lemma 2.2 the subgroup Θ n +1 ≤ Γ g acts trivially on the set of framings relative tothe boundary, so Θ n +1 ≤ Γ fr , [ ℓ ] g ≤ Γ fr , [[ ℓ ]] g and is therefore annihilated by f ℓ . Now f ℓ isa homomorphism to an abelian group, so factors over H (Γ fr , [[ ℓ ]] g / Θ n +1 ). To calculatethe latter group, we use the extension0 −→ ( n oddHom( H n , Z / n even −→ Γ fr , [[ ℓ ]] g / Θ n +1 −→ G fr , [[ ℓ ]] g −→ · · · −→ "( n oddHom( H n , Z / n even G fr , [[ ℓ ]] g −→ H (Γ fr , [[ ℓ ]] g / Θ n +1 ) −→ H ( G fr , [[ ℓ ]] g ) −→ H (Γ fr , [[ ℓ ]] g / Θ n +1 ) ∼ → H ( G fr , [[ ℓ ]] g ) from which the factorisation of g ℓ over some h ℓ follows.Before proving (ii) and (iii), we first show that for large enough g the group Γ fr , [[ ℓ ]] g / Γ fr , [ ℓ ] g has order at least 4. To see this we use the quotient map to G fr , [[ ℓ ]] g /G fr , [ ℓ ] g . By Lemma 4.5,for n = 1 , G fr , [ ℓ ] g has trivial abelianisation, so there is an induced surjection G fr , [[ ℓ ]] g /G fr , [ ℓ ] g −→ H ( G fr , [[ ℓ ]] g ) = ( Z / n odd( Z / n even . Thus Γ fr , [[ ℓ ]] g / Γ fr , [ ℓ ] g = im( f ℓ ) = im( h ℓ ) indeed has order at least 4.On the other hand H ( G fr , [[ ℓ ]] ∞ ) has order precisely 4, so h ℓ must be injective. If n n = 3 then π n (SO(2 n )) has order 4, so h ℓ must be an isomorphism. If n ≡ π n (SO(2 n )) has order 8, so h ℓ must be injective onto an index 2subgroup. This proves parts (ii) and (iii). (cid:3) This discussion allows us to describe the subgroups G fr , [ ℓ ] g ≤ G fr , [[ ℓ ]] g , where they havethe following interesting intrinsic descriptions. Corollary 5.2. If n = 1 , then G fr , [ ℓ ] g = ker( G fr , [[ ℓ ]] g → H ( G fr , [[ ℓ ]] ∞ )) .If n = 1 , then G fr , [ ℓ ] g = G fr , [[ ℓ ]] g .Proof. For n = 1 , fr , [[ ℓ ]] g −→ G fr , [[ ℓ ]] g −→ H ( G fr , [[ ℓ ]] ∞ )is the same as the kernel Γ fr , [ ℓ ] g of f ℓ by Proposition 5.1. Thus G fr , [[ ℓ ]] g → H ( G fr , [[ ℓ ]] ∞ ) haskernel G fr , [ ℓ ] g .If n = 1 , π n (SO(2 n )) = 0 so Γ fr , [ ℓ ] g = Γ fr , [[ ℓ ]] g and hence their images in G ′ g areequal too. (cid:3) Proof of Theorem A.
The argument will be by analysing the sequence (5). Suppose that n ≥ g ≥
1. By the calculations in Section 4.1 the compositionΓ fr , [[ ℓ ]] g −→ G fr , [[ ℓ ]] g −→ H ( G fr , [[ ℓ ]] g ) −→ H ( G fr , [[ ℓ ]] ∞ )is surjective as long as g ≥
1. By Proposition 5.1 if n f ℓ is surjective; if n ≡ Z /
2. By Proposition 3.1 the setStr fr ∗ ( W g, ) / Γ g has a single element, unless n = 3 , fr ∂ ( W g, ) / Γ g has a single element unless n = 3 , n ≡ n = 1 and g ≥ (cid:3) θ -structures on W g, In [KRW19, Section 8] we more generally considered tangential structures whoseGL n ( R )-space Θ has the property that the homotopy quotient B := Θ (cid:12) GL n ( R ) is n -connected (in terms of the associated fibration θ : B → B O(2 n ) this means that B is n -connected). Our results about framings can be used to also classify such θ -structureson W g, , up to homotopy and diffeomorphisms. We will assume that n ≥ ℓ ∂ : Fr( T W g, | ∂W g, ) → Θ we let Bun ∂ (Fr( T W g, ) , Θ; ℓ ∂ )be the space of GL n ( R )-maps Fr( T W g, ) → Θ extending ℓ ∂ . Its set of path componentsis denoted Str θ∂ ( W g, ). The moduli space of W g, ’s with θ -structures is defined as B Diff θ∂ ( W g, ) := Bun ∂ (Fr( T W g, ) , Θ; ℓ ∂ ) (cid:12) Diff ∂ ( W g, ) . Its set of path components is the set of orbits Str θ∂ ( W g, ) / Γ g . This is what we shallcompute in this section, but we first make some definitions.The boundary condition ℓ ∂ singles out a path component Θ + of Θ. Since Θ (cid:12) GL n ( R )is n -connected, the map π n (SO(2 n )) → π n (Θ + ) is surjective. By Lemma 3.2 the map Sπ n (SO( n )) → π n (SO(2 n )) is surjective unless n = 3 ,
7, in which case it has cokernel Z /
2. This leads to two cases when n = 3 , Sπ n (SO( n )) → π n (SO(2 n )) → π n (Θ + ) is not surjective (and thus has index 2),(B) Sπ n (SO( n )) → π n (SO(2 n )) → π n (Θ + ) is surjective.We also define Cπ n (Θ + ) := coker (cid:16) H ( G fr , [[ τ ]] ∞ ) h τ −→ π n (SO(2 n )) → π n (Θ + ) (cid:17) , which seems at first sight to depend on the orbit of homotopy class [[ τ ]] of referenceframing τ relative to ∗ , but is in fact independent of this choice: by Proposition 5.1 themap h τ is surjective unless n ≡ τ ]] is uniqueby Proposition 3.1. Theorem 6.1.
Suppose that Θ (cid:12) GL n ( R ) is n -connected. Let g ≥ and n ≥ . Theaction of the mapping class group Γ g on the set Str θ∂ ( W g, ) of homotopy classes of θ -structure extending ℓ ∂ is in bijection with(i) Cπ n (Θ + ) if n = 3 , ;(ii) Cπ n (Θ + ) × Z / if n = 3 , and we are in case (A);(iii) π n (Θ + ) if n = 3 , and we are in case (B). We will explain the proof of this theorem in parallel with that of Theorem A. Thedefinitions and results of Sections 2.2, 2.3, and 2.4 go through for θ -structures. By[KRW19, Lemma 8.5], up to homotopy there is a unique orientation preserving boundarycondition ℓ ∂ which extends to a θ -structure on all of W g, , which we may take to be ℓ τ∂ coming from a reference framing τ . The reference framing induces a homeomorphismBun ∂ (Fr( T W g, ) , Θ; ℓ ∂ ) ∼ = map ∂ ( W g, , Θ) . Its path components will be denoted Str θ∂ ( W g, ). We can relax boundary conditions toget an exact sequence0 Str θ∂ ( D n ) Str θ∂ ( W g, ) Str θ ∗ ( W g, ) 00 π n (Θ + ) π (map ∂ ( W g, , Θ)) Hom( H n , π n (Θ + )) 0 . ∼ = (cid:8) ∼ = ∼ = (cid:8) The arguments for Proposition 3.1 go through for θ -structures, giving the following: Proposition 6.2.
Suppose n ≥ and g ≥ , then · If n = 3 , or we are in case (B), then Str θ ∗ ( W g, ) / Γ g consists of a single element. RAMINGS OF W g, · If n = 3 , and we are in case (A), then Str θ ∗ ( W g, ) / Γ g consists of two elements. These arguments also give information about the stabiliser Γ θ, [[ ℓ ]] g of [[ ℓ ]] ∈ Str θ ∗ ( W g, ),as well as its image G θ, [[ ℓ ]] g in G ′ g . The mapStr fr ∗ ( W g, ) −→ Str θ ∗ ( W g, )[[ ℓ fr ]] [[ ℓ ]]associating to a framing the induced θ -structure is surjective, as by assumption the map π n (SO(2 n )) → π n (Θ + ) is. If n = 2 it follows that Str θ ∗ ( W g, ) is a single point; if n = 3 , θ ∗ ( W g, ) /I g is a single point; if n = 3 ,
7, it follows as in the proof ofProposition 3.3 that Str θ ∗ ( W g, ) /I g is in bijection with Quad( H n , λ ) in case (A) and isa single point in case (B). Thus for n ≥ g ≥
1, we have G θ, [[ ℓ ]] g = Sp q or a g ( Z ) if n = 3 ,
7, we are in case (A), and ℓ has Arf invariant 0 or 1,Sp g ( Z ) if n = 3 , q g ( Z ) if n is odd but not 3 or 7,O g,g ( Z ) if n is even.Furthermore, if n = 3 , n = 3 , fr , [[ ℓ fr ]] g → Γ θ, [[ ℓ ]] g is an isomorphism.The analogue of the fundamental sequence (5) for θ -structures gives, for any framing ℓ fr , a commutative diagramΓ fr , [ ℓ fr ] g Γ fr , [[ ℓ fr ]] g Str fr ∂ ( D n ) Str fr ∂ ( W g, ) / Γ g Str fr ∗ ( W g, ) / Γ g Γ θ, [ ℓ ] g Γ θ, [[ ℓ ]] g Str θ∂ ( D n ) Str θ∂ ( W g, ) / Γ g Str θ ∗ ( W g, ) / Γ g . f ℓ fr ( ∗ ) −· [ ℓ fr ] f ℓ −· [ ℓ ] Proof of Theorem 6.1.
We proceed in three cases.
The cases n = 3 , . In this case the map indicated by ( ∗ ) is an isomorphism and f ℓ is determined by f ℓ fr . Using Proposition 5.1, we then identify the cokernel of f ℓ with Cπ n (Θ + ). As Str θ ∗ ( W g, ) / Γ g consists of a single element, we conclude thatStr θ∂ ( W g, ) / Γ g ∼ = Cπ n (Θ + ). Case (A).
In this case n = 3 or 7 and Sπ n (SO( n )) → π n (SO(2 n )) → π n (Θ) not sur-jective. Then Str θ ∗ ( W g, ) / Γ g consists of two elements, distinguished by an Arf invariant.Choosing framings ℓ fr with Arf invariant 0 and 1 respectively, we get two commutativediagrams as above. In both cases, the map indicated by ( ∗ ) is an isomorphism, andas above we identify the cokernel of f ℓ with Cπ n (Θ + ). Thus we get a collection oforbits with Arf invariant 0 and another collection of orbits with Arf invariant 1, each inbijection with Cπ n (Θ + ). Case (B).
In this case n = 3 or 7 and Sπ n (SO( n )) → π n (SO(2 n )) → π n (Θ) surjective.The proof of Proposition 5.1 gives a factorisationΓ θ, [[ ℓ ]] g π n (Θ) G θ, [[ ℓ ]] g H ( G θ, [[ ℓ ]] g ) H ( G θ, [[ ℓ ]] ∞ ) f ℓ h ℓ with left map the quotient map, and bottom maps abelianisation followed by stabili-sation. As G θ, [[ ℓ ]] g = Sp g ( Z ), the bottom-right term vanishes by the computations in Section 4.1, so f ℓ = 0. Combining this with the fact that Str θ ∗ ( W g, ) / Γ g is a single pointin this case gives the claimed result. (cid:3) Example: stable framings.
Stable framings are trivialisations of the stable tan-gent bundle. In this case we take Θ = GL ∞ ( R ), made into a GL n ( R )-space bystabilisation. Then Θ + = GL + ∞ ( R ), which deformation retracts onto SO. The map π n (SO(2 n )) → π n (SO) induced by stabilisation is an isomorphism as long as n ≥
3, sowhen n = 3 , Lemma 6.3.
The map f ℓ : Γ sfr , [[ ℓ ]] g → π n (SO) is zero.Proof. The group π n (SO) vanishes unless n ≡ Z /
2. In this case we claim that h ℓ : H ( G sfr , [[ ℓ ]] ∞ ; Z ) → π n (SO) is zero. To see this notethat the composition H (ˇΓ sfr ,ℓ ∞ ; Z ) −→ H ( G sfr , [ ℓ ] ∞ ; Z ) −→ H ( G sfr , [[ ℓ ]] ∞ ; Z ) h ℓ −→ π n (SO)is zero by the analogue for stable framings of the exact sequence (7). But G sfr , [ ℓ ] g = G sfr , [[ ℓ ]] g = O g,g ( Z ) for n even as we have discussed above, and by Section 5.2 of [GRW16]the composition π (Σ − n SO / SO(2 n )) ∼ = H (ˇΓ sfr ,ℓ ∞ ; Z ) −→ π ( MT θ n ) ∼ = H (Γ ∞ ; Z ) −→ H (O ∞ , ∞ ( Z ); Z )is surjective. (cid:3) We conclude that there are, up to homotopy and diffeomorphism, two stable framingson W g, when n = 3 , n ≡ W g, arises from a unique framing. Remark . This identifies the index 2 subgroup hit by h ℓ in Proposition 5.1 (iii): it isthe kernel of the stabilisation map π n (SO(2 n )) → π n (SO). References [Arf41] C. Arf,
Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I , J.Reine Angew. Math. (1941), 148–167. 6[BL74] D. Burghelea and R. Lashof,
The homotopy type of the space of diffeomorphisms. I, II , Trans.Amer. Math. Soc. (1974), 1–36; ibid. 196 (1974), 37–50. 3[CS20a] A. Calderon and N. Salter,
Framed mapping class groups and the monodromy of strata ofabelian differentials , https://arxiv.org/abs/2002.02472 , 2020. 1[CS20b] , Relative homological representations of framed mapping class groups , https://arxiv.org/abs/2002.02471 , 2020. 1, 9[Fri17] N. Friedrich, Homological stability of automorphism groups of quadratic modules and mani-folds , Doc. Math. (2017), 1729–1774. 10[GRW16] S. Galatius and O. Randal-Williams, Abelian quotients of mapping class groups of highlyconnected manifolds , Math. Ann. (2016), no. 1-2, 857–879. 10, 14, 18[GRW17] ,
Homological stability for moduli spaces of high dimensional manifolds. II , Ann. ofMath. (2) (2017), no. 1, 127–204. 1, 13[GRW18] ,
Homological stability for moduli spaces of high dimensional manifolds. I , J. Amer.Math. Soc. (2018), no. 1, 215–264. 1, 7[GRW19] , Moduli spaces of manifolds: a user’s guide , Handbook of homotopy theory, Chapman& Hall/CRC, CRC Press, Boca Raton, FL, 2019, pp. 445–487. 3[HO89] A. J. Hahn and O.T. O’Meara,
The classical groups and K -theory , Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989, With a foreword by J. Dieudonné. 12[JM90] D. Johnson and J. Millson, Modular Lagrangians and the theta multiplier , Invent. Math. (1990), no. 1, 143–165. 11[Joh80] D. Johnson,
Spin structures and quadratic forms on surfaces , J. London Math. Soc. (2) (1980), no. 2, 365–373. 9[Kaw18] N. Kawazumi, The mapping class group orbits in the framings of compact surfaces , TheQuarterly Journal of Mathematics (2018), no. 4, 1287–1302. 1, 2 RAMINGS OF W g, [Ker59] M. A. Kervaire, Sur le fibré normal à une variété plongée dans l’espace euclidien , Bull. Soc.Math. France (1959), 397–401. 7[Ker60] , Some nonstable homotopy groups of Lie groups , Illinois J. Math. (1960), 161–169.5[Kra19] M. Krannich, Mapping class groups of highly connected (4 k + 2) -manifolds , http://arxiv.org/abs/1902.10097 , 2019. 11, 15[Kre79] M. Kreck, Isotopy classes of diffeomorphisms of ( k − -connected almost-parallelizable k -manifolds , Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978),Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663. 2, 10, 14[KRW19] A. Kupers and O. Randal-Williams, The cohomology of Torelli groups is algebraic , Forumof Mathematics, Sigma, to appear. https://arxiv.org/abs/1908.04724 , 2019. 1, 2, 3, 4, 5,6, 16[KRW20] ,
Diffeomorphisms of even-dimensional disks outside the pseudoisotopy stable range ,2020. 1, 3[KS77] R. C. Kirby and L. C. Siebenmann,
Foundational essays on topological manifolds, smoothings,and triangulations , Princeton University Press, Princeton, N.J.; University of Tokyo Press,Tokyo, 1977, Annals of Mathematics Studies, No. 88. 13[Lev85] J. P. Levine,
Lectures on groups of homotopy spheres , Algebraic and geometric topology (NewBrunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 62–95.8[Pae56] G. F. Paechter,
The groups π r ( V n, m ) . I , Quart. J. Math. Oxford Ser. (2) (1956), 249–268.14[PCS20] P. Portilla Cuadrado and N. Salter, Vanishing cycles, plane curve singularities, and framedmapping class groups , https://arxiv.org/abs/2004.01208 , 2020. 1[Qui86] F. Quinn, Isotopy of -manifolds , J. Differential Geom. (1986), no. 3, 343–372. 14[RP80] N. Ray and E. K. Pedersen, A fibration for
Diff Σ n , Topology Symposium, Siegen 1979 (Proc.Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980,pp. 165–171. 3[RW14] O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r -Spin and Pin surfaces , J. Topol. (2014), no. 1, 155–186. 1, 2, 9, 15[Sat69] H. Sato, Diffeomorphism group of S p × S q and exotic spheres , Quart. J. Math. Oxford Ser.(2) (1969), 255–276. 10[Sie] I. Sierra, personal communication. 11[Wal64] C. T. C. Wall, Diffeomorphisms of -manifolds , J. London Math. Soc. (1964), 131–140. 3[Wal99] , Surgery on compact manifolds , second ed., Mathematical Surveys and Monographs,vol. 69, American Mathematical Society, Providence, RI, 1999, Edited and with a forewordby A. A. Ranicki. 7
E-mail address : [email protected] Department of Mathematics, One Oxford Street, Cambridge MA, 02138, USA
E-mail address : [email protected]@dpmms.cam.ac.uk