aa r X i v : . [ m a t h . A T ] F e b A HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I
MANUEL KRANNICHAbstract. We construct a zig-zag from the space of pseudoisotopies of a closed 2 n -discto the once looped algebraic K -theory space of the integers and show that the maps in-volved are p -locally ( n − ) -connected for n > p . The proof uses thecomputation of the stable homology of the moduli space of high-dimensional handlebodiesdue to Botvinnik–Perlmutter and is independent of the classical approach to pseudoiso-topy theory based on Igusa’s stability theorem and work of Waldhausen. Combined witha result of Randal-Williams, one consequence of this identification is a calculation of therational homotopy groups of BDiff ∂ ( D n + ) in degrees up to 2 n − The homotopy type of the group of pseudoisotopies , or concordance diffeomorphisms ,C ( M ) = { ϕ : M × I (cid:27) −→ M × I | ϕ | M ×{ }∪ ∂ M × I = id } , of a smooth compact d -dimensional manifold M in the smooth topology has been an objectof interest to geometric topologists for many years, not least because of its intimate con-nection to algebraic K -theory already visible on the level of path components. Building onCerf’s proof that C ( M ) is connected if M is simply connected and d ≥ π C ( M ) of isotopy classes of concordances inhigh dimensions by relating it to the lower algebraic K -groups of the integral group ring Z [ π M ] of the fundamental group of M . Beyond its components, the homotopy type ofthe space of concordances C ( M ) and its relation to K -theory has so far been studied intwo steps: deep work of Igusa [Igu88] shows that the stabilisation map C ( M ) −→ C ( M × I ) induced by crossing with an interval is min ( d − , d − ) -connected, so in this range up toabout a third of the dimension, one may consider the stable concordance space colim k C ( M × I k ) instead, which in turn admits a complete description in terms of Waldhausen’s gen-eralised algebraic K -theory for spaces by Waldhausen, Jahren, and Rognes’ foundational stable parametrised h -cobordism theorem [WJR13].In this work, we focus on the case M = D n of a closed disc of even dimension andstudy its space of concordances via a new route—independent of the classical approach—which does not involve stabilising the dimension and is, vaguely speaking, homological instead of homotopical ; we shall elaborate on this at a later point. Our main result relatesthe delooped concordance space BC ( D n ) to the once looped algebraic K -theory space ofthe integers Ω ∞ + K ( Z ) in a range up to approximately the dimension, p -locally for primes p that are large with respect to the dimension and the degree. Theorem A.
For n > , there exists a zig-zag BC ( D n ) −→ · ←− Ω ∞ + K ( Z ) whose maps are p -locally min ( n − , p − − n ) -connected for primes p .Remark. The result we prove is slightly stronger than stated here (see Theorem 5.1) andimplies for instance that π n − BC ( D n ) ⊗ Q surjects onto K n − ( Z ) ⊗ Q as long as n > Mathematics Subject Classification. See also Igusa’s corrections in [Igu84].
When combined with Borel’s work on the stable rational cohomology of arithmeticgroups [Bor74], Theorem A provides an isomorphism π ∗ BC ( D n ) ⊗ Q (cid:27) K ∗ + ( Z ) ⊗ Q (cid:27) ( Q if ∗ ≡ ( mod 4 ) < ∗ < n − , and an epimorphism π ∗ BC ( D n ) ⊗ Q → K ∗ + ( Z ) ⊗ Q in degree 2 n − n >
3, whichgoes significantly beyond the range that was previously accessible by relying on Igusa’sstability result and shows for instance that BC ( D ) is nontrivial, even rationally. Giventhat the K -groups K ∗ ( Z ) are known to contain p -torsion for comparatively large primeswith respect to the degree due to contributions from Bernoulli numbers, Theorem A alsoexhibits many new torsion elements in π ∗ BC ( D n ) , such as one of order 691 in π BC ( D n ) as long as n >
12 resulting from the fact that K ( Z ) is cyclic of that order.In the remainder of this introduction, we explain more direct applications of Theorem Aand conclude by indicating some of the ideas that go into its proof. Diffeomorphisms and concordances of odd discs.
Restricting a concordance to themoving part of its boundary induces a homotopy fibre sequenceDiff ∂ ( D d + ) −→ C ( D d ) −→ Diff ∂ ( D d ) that compares the group of concordances C ( D d ) of a d -disc to its group of its diffeo-morphisms Diff ∂ ( D d ) fixing the boundary pointwise. By a result of Randal-Williams[RW17, Thm 4.1] based on Morlet’s lemma of disjunction and work of Berglund and Mad-sen [BM20] (a combination which incidentally inspired parts of our strategy to prove The-orem A), the space BDiff ∂ ( D n ) is rationally ( n − ) -connected, so the delooped mapsBDiff ∂ ( D n + ) −→ BC ( D n ) and BC ( D n + ) −→ BDiff ∂ ( D n + ) are rationally ( n − ) -connected as well, resulting in the following corollary of Theorem A. Corollary B.
There exist isomorphisms π ∗ BDiff ∂ ( D n + ) ⊗ Q (cid:27) π ∗ BC ( D n + ) ⊗ Q (cid:27) K ∗ + ( Z ) ⊗ Q in degrees ∗ < n − and epimorphisms π ∗ BDiff ∂ ( D n + ) ⊗ Q −→ K ∗ + ( Z ) ⊗ Q and π ∗ BC ( D n + ) ⊗ Q −→ K ∗ + ( Z ) ⊗ Q in degree n − .Remark. (i) The range in Corollary B is nearly optimal: Watanabe [Wat09] has shown that π n − BDiff ∂ ( D n + )⊗ Q is nontrivial for many even values of n , for which K n − ( Z )⊗ Q vanishes by Borel’s work. Alternatively, one may use Weiss’ work on Pontrya-gin classes [Wei15] to deduce that π n + BDiff ∂ ( D n + ) ⊗ Q is nontrivial for large n , which leads to a slightly weaker upper bound for the range in Corollary B.(ii) A combination of the strengthening of Theorem A mentioned earlier with forth-coming work of Kupers and Randal-Williams [KRW20] improves the range ofCorollary B by one degree from 2 n − n − n /
3, Corollary B was previously known as a result of a classical computation dueto Farrell and Hsiang [FH78] based on Waldhausen’s approach to pseudoisotopytheory (of which the proof of Corollary B is independent).
HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 3
Homeomorphisms of Euclidean spaces.
By a well-known theorem of Morlet and en-hancements due to Burghelea–Lashof [BL77], there are equivalencesBDiff ∂ ( D d ) ≃ Ω d Top ( d )/ O ( d ) andBC ( D d ) ≃ Ω d hofib (cid:16) Top ( d )/ O ( d ) → Top ( d + )/ O ( d + ) (cid:17) for d ≥ d -disc to thehomotopy fibre Top ( d )/ O ( d ) of the map BO ( d ) → BTop ( d ) that classifies the inclusionof the orthogonal group O ( d ) into the topological group Top ( d ) of homeomorphisms of R d , and to its stabilisation map Top ( d )/ O ( d ) → Top ( d + )/ O ( d + ) induced by takingproducts with the real line. Theorem A and Corollary B can thus be reformulated in termsof these equivalent spaces and result in particular in the following corollary, using the factthat the groups π ∗ Top ( d )/ O ( d ) are finite for ∗ < d + d , Corollary C.
There exists an isomorphism π ∗ Top ( n + )/ O ( n + ) ⊗ Q (cid:27) ( for ∗ ≤ n + K ∗ + n + ( Z ) ⊗ Q for n + < ∗ < n − and an epimorphism π ∗ Top ( n + )/ O ( n + ) ⊗ Q → K ∗ + n + ( Z ) ⊗ Q in degree n − .Remark. As per item (ii) of the previous remark, this range can be improved by one degree.
Idea of proof.
Instead of sketching the proof of Theorem A, we outline a strategy toachieve a seemingly different task: relating the p -local homology of BC ( D n ) to K -theoryin a range of degrees. This should, however, convey the main ideas; the actual proof ofTheorem A uses a similar strategy to construct a zig-zag between BC ( D n ) and Ω ∞ + K ( Z ) that consists of maps that are p -local homology isomorphisms in a range and then arguesthat the maps are actually p -locally highly connected. In the sketch that follows, we allowourselves to be somewhat vague; full details shall be given in the body of this work.The root of the proof of Theorem A is to consider the odd-dimensional disc D n + asthe 0th member of a whole family of manifolds—the high-dimensional handlebodies V д ≔ ♮ д D n + × S n . Comparing the groups of diffeomorphisms Diff D n ( V д ) that pointwise fix a chosen disc D n ⊂ ∂ V д in the boundary to the corresponding block diffeomorphism groups yieldshomotopy fibre sequences of the form g Diff D n ( V д )/ Diff D n ( V д ) −→ BDiff D n ( V д ) −→ B g Diff D n ( V д ) , one for each д . Varying д , these fibre sequences are connected by stabilisation maps in-duced by extending (block) diffeomorphisms along the inclusion V д ⊂ V д + by the identity,and Morlet’s lemma of disjunction ensures that the map between homotopy fibres g Diff D n ( V д )/ Diff D n ( V д ) −→ g Diff D n ( V д + )/ Diff D n ( V д + ) is highly connected. It is not hard to see that the space BC ( D n ) of interest is equivalent tothis fibre for д =
0, so to access BC ( D n ) in a range, we may as well study the homotopyfibre of the sequence obtained from the previous one by taking homotopy colimits,(1) g Diff D n ( V ∞ )/ Diff D n ( V ∞ ) −→ BDiff D n ( V ∞ ) −→ B g Diff D n ( V ∞ ) . By work of Botvinnik and Perlmutter [BP17], the homology of BDiff D n ( V ∞ ) has a surpris-ingly simple description in homotopy theoretical terms, so to compute the homology ofthe fibre of (1) and hence that of BC ( D n ) in a range, one might try to compute the ho-mology of B g Diff D n ( V ∞ ) and analyse the Serre spectral sequence of (1). This is essentiallywhat we do, and it involves several steps of which some might be of independent interest: MANUEL KRANNICH (i) In Section 2, we use surgery theory to express the space of block diffeomorphismsof a general manifold triad satisfying a π - π -condition p -locally for large primes interms of its homotopy automorphisms covered by certain bundle data. A similarresult in the rational setting which inspired ours but applies to another class oftriads was obtained by Berglund and Madsen [BM20] (see also Remark 2.3).(ii) Section 3 serves to compute variants of the mapping class group π Diff D n ( V д ) upto extensions in terms of automorphisms groups of the integral homology of V д .(iii) In Section 4, we calculate the p -local homotopy and homology groups of the spaceof homotopy automorphisms hAut D n ( V д , W д , ) of V д that fix D n and restrict toa homotopy automorphism of the complement of the boundary as a module overthe group π hAut D n ( V д , W д , ) in a range of degrees. This uses some pieces of theapparatus of rational homotopy theory, as well as an ad-hoc p -local generalisationwe provide along the way.(iv) Acting on the n th homology group H n ( V д ; Z ) (cid:27) Z д induces a map of the formB g Diff D n ( V ∞ ) −→ BGL ∞ ( Z ) + ≃ Ω ∞ K ( Z ) , a variant of which we show in Section 5 with the help of all previous steps to bea p -local homology isomorphism in a range of degrees. Remark.
The requirement in Botvinnik and Perlmutter’s calculation of the stable homol-ogy of BDiff D n ( V д ) that the dimension be greater than 7 (see [BP17, Rem. 1.1]) is responsi-ble for the same assumption in Theorem A. An extension of their work to the case 2 n + = Outlook.
In a sequel to this work [Kra20], we take a different approach and study spacesof concordances C ( M ) without restriction on the p -torsion.As a byproduct, the setup of [Kra20] will also make apparent that the zig-zag estab-lished in Theorem A is compatible with the iterated stabilisation map C ( D n ) → C ( D n × I ) ≃ C ( D n + ) (see Remark 5.10). When combined with work of Goodwillie, this re-finement of Theorem A leads to a proof that the concordance stabilisation map C ( M ) → C ( M × I ) of any 2-connected compact smooth manifold M of dimension d > ( d − ) -connected. This is optimal up to at most two degrees and implies for instance thatthe second derivative in the sense of orthogonal calculus [Wei95, Wei98] of the functorthat sends a vector space V to BTop ( V ) is rationally bounded from below. Acknowledgements.
My thanks go to Oscar Randal-Williams for several valuable dis-cussions. I was partially supported by O. Randal-Williams’ Philip Leverhulme Prize fromthe Leverhulme Trust and the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreement No. 756444).
Contents
1. Preliminaries 52. Surgery theory and spaces of block diffeomorphisms 113. High-dimensional handlebodies and their mapping classes 164. Relative homotopy automorphisms of handlebodies 255. The proof of Theorem A 32Appendix A. Stable tangential bundle maps are stable bundle maps 38References 41
HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 5
1. Preliminaries
We start off with a lemma on semi-simplicial actions and a short recollection on nilpo-tent spaces for later reference, followed by foundational material on various types auto-morphisms of manifolds with bundle data. Primarily, this serves us to set up a convenienttheory of block automorphism spaces with tangential structures.1.1.
Semi-simplicial monoids and their actions.
The homotopy quotient of a semi-simplicial set X • semi-simplicially acted upon by a semi-simplicial monoid M • from theright is the semi-simplicial space X • (cid:12) M • whose space of p -simplices is defined as thebar-construction B ( X p , M p , ∗) , with face maps induced by the face maps of M • and X • . For X • = ∗ • the semi-simplicial point, i.e. ∗ p a singleton for all p , we abbreviate X • (cid:12) M • byB M • . The unique semi-simplicial map X • → ∗ • induces a natural map X • (cid:12) M • → B M • which is well-known to geometrically realise to a quasi-fibration with fibre the realisationof X • if M • is a group-like simplicial monoid acting simplicially on a simplicial set X • . Toexplain a generalisation of this fact for semi -simplicial M • and X • , we denote the geometricrealisation by |−| and consider the natural zig-zag | X • | × | M • | ←− | X • × M • | −→ | X • | whose left map is induced by the projections and the right map by the action. If the un-derlying semi-simplicial sets of M • and X • admit degeneracies , i.e. if they can be enhancedto simplicial sets, then the left arrow is an equivalence (see e.g. [ERW19, Thm 7.2]), so acontractible choice of a homotopy inverse yields an action map µ : | X • | × | M • | −→ | X • | .In this situation, we say that M • acts on X • by equivalences if µ (− , m ) : | X • |→ | X • | is anequivalence for all m ∈ | M • | , and this turns out to be sufficient to conclude that the naturalmap X • (cid:12) M • → B M • realises to a quasi-fibration. Lemma 1.1.
For a semi-simplicial monoid M • acting on a semi-simplicial set X • such that M • and X • admit degeneracies and the action of M • on X • is by equivalences, the sequence X • −→ X • (cid:12) M • −→ B M • induces a quasi-fibration on geometric realisations.Remark . (i) In all situations we shall encounter, the condition that M • and X • admit degen-eracies is ensured by them being Kan (every semi-simplicial Kan complex admitsdegeneracies [Kan70]) and the condition that M • acts on X • by equivalences by M • being group-like, i.e. the monoid of components π | M • | having inverses. Notethat Lemma 1.1 neither requires the degeneracies of M • to be compatible with itsmonoid structure nor those of X • with the action.(ii) The condition that M • and X • admit degeneracies is crucial, even if M • = G • is a simplicial group. For an instructive example, consider the semi-simplicialset X • = G ≤ • which agrees with G in degree 0 and is empty otherwise, semi-simplicially acted upon by G • via right translations. In this case, the realisation | X • (cid:12) G • | is contractible, but | G ≤ • | ≃ G is rarely equivalent to Ω | B G • | ≃ | G • | . Proof of Lemma 1.1.
The sequence in question is the geometric realisation of a sequence(2) X • −→ B (cid:4) ( X • , M • , ∗) −→ B (cid:4) M • of simplicial semi-simplicial sets, where the simplicial (bar-)direction is indicated by thesquare (cid:4) and the semi-simplicial direction by the bullet • ; the fibre X • is constant in the (cid:4) -direction. Since the realisation of a simplicial set is canonically equivalent to the reali-sation of its underlying semi-simplicial set [ERW19, Lem. 1.7] and the realisation of a bi-semi-simplicial sets is independent of which direction one realises first [ERW19, p. 2106],we may rely on a result of Segal [ERW19, Thm 2.12] and its simplification explained in[ERW19, Lem. 2.11] to reduce the assertion to showing that the commutative squares MANUEL KRANNICH | B p ( X • , M • , ∗)| | B p − ( X • , M • , ∗)|| B p M • | | B p − M • | d p d p and | B ( X • , M • , ∗)| | B ( X • , M • , ∗)|| B M • | | B M • | d d obtained from the simplicial structure of B (cid:4) ( X • , M • , ∗) → B (cid:4) ( M • ) by realising the • -directionare homotopy cartesian for p ≥
0. For the left square, this follows directly from the defini-tion of the bar-construction together with the above mentioned fact that the realisation ofthe product of two semi-simplicial sets that admit degeneracies is canonically equivalentto the product of their realisations. The right hand square is homotopy cartesian if theshear map sh : | X • × M • | → | X • | × | M • | induced by the action on the first coordinate andby the projection on the second is an equivalence. This map fits into a triangle | X • × M • | | X • | × | M • || M • | sh whose diagonal maps are induced by the projection to M • . This triangle is commutativeand the map induced by taking diagonal homotopy fibres at a point m ∈ | M • | agrees upto equivalence with the action map µ (− , m ) : | X • | → | X • | , which is an equivalence byassumption, so the shear map is an equivalence as well and the claim follows. (cid:3) Nilpotent spaces.
A space is nilpotent if it is path connected and its fundamentalgroup is nilpotent and acts nilpotently on all higher homotopy groups. Such spaces havean unambiguous p -localisation at a prime p , which on homology and homotopy groups(including the fundamental group) has the expected effect of p -localisation in the algebraicsense [MP12, Thm 6.1.2]. Localisations are defined in terms of a universal property [MP12,Def. 5.2.3], which ensures that they are unique and functorial up to homotopy. A mapbetween nilpotent spaces is p -locally k -connected for a prime p and k ≥ p -localisations is k -connected in the usual sense. Lemma 1.3.
For a map X → Y between nilpotent spaces, the following are equivalent.(i) The map X → Y is p -locally k -connected.(ii) The induced map on p -local homotopy groups π ∗ X ⊗ Z ( p ) −→ π ∗ Y ⊗ Z ( p ) is anisomorphism for ∗ < k and surjective for ∗ = k .(iii) The induced map on p -local homology groups H ∗ ( X ; Z ( p ) ) −→ H ∗ ( Y ; Z ( p ) ) is anisomorphism for ∗ < k and surjective for ∗ = k .Proof. The above mentioned fact that p -localisation of nilpotent spaces commutes withtaking homotopy groups shows that the first two items are equivalent. The equivalencebetween the second two follows for k = k ≥ p -localisation of the map in question. (cid:3) At several points in this work, we will make use of the fact that nilpotent spaces behavewell with respect to taking homotopy fibres (see e.g. [MP12, Prop. 4.4.1] for a proof).
Lemma 1.4.
Let π : E → B be a fibration, e ∈ E a point, and E e ⊂ E and F e ⊂ π − ( π ( e )) the respective path components containing e . If E e is nilpotent, then F e is nilpotent as well. HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 7
Block spaces.
As is customary, we denote the standard p -simplex by ∆ p ⊂ R p + andidentify its faces σ : ∆ q → ∆ p with their images. A p -block space is a space X togetherwith a map π : X → ∆ p . For a q -face σ ⊂ ∆ p , the preimage X σ ≔ π − ( σ ) becomesa q -block space by pulling back π along σ . A block map between block p -spaces is amap f : X → Y between underlying spaces such that f ( X σ ) ⊂ Y σ for all faces σ ⊂ ∆ p and a block homotopy equivalence is a block map f : X → Y such that the induced map f σ : X σ → Y σ is a homotopy equivalence for all faces σ ⊂ ∆ p . Spaces of the form ∆ p × M are implicitly considered as p -block spaces via the projection. For i = , . . . , p , the map c i : [ , ] × ∆ p − −→ ∆ p ( s , t , . . . , t p − ) 7−→ (( − s ) t , . . . , ( − s ) t i − , s , ( − s ) t i , . . . , ( − s ) t p − ) induces by restriction for 0 < ϵ ≤ c i , ϵ : [ , ε ) × ∆ p − −→ c i ([ , ϵ ) × ∆ p − ) ≕ ∆ pi , ϵ onto an open neighborhood ∆ pi , ϵ of the i th codimension 1 face ∆ pi ⊂ ∆ p . A block map f : ∆ p × M → ∆ p × N for spaces M and N is collared if there exists an ε > f ( ∆ pi , ϵ × M ) ⊂ ∆ pi , ε × N and(ii) ( c − i , ϵ × id N ) ◦ f | ∆ pi , ϵ × M ◦ ( c i , ϵ × id M ) = f ∆ pi × id [ , ϵ ) are satisfied for i = , . . . , p .1.4. Diffeomorphisms.
For a compact smooth d -manifold W and two compact subman-ifolds M , N ⊂ W , we denote by g Diff M ( W , N ) • the semi-simplicial group of block diffeo-morphisms whose p -simplices consists of all diffeomorphisms of ∆ p × W that are collaredblock maps, fix ∆ p × N setwise, and fix a neighborhood of ∆ p × M pointwise. The semi-simplicial structure is induced by restricting diffeomorphisms of ∆ p × M to σ × M for faces σ ⊂ ∆ p . Whenever one of the submanifolds is empty, we omit it from the notation andin the case M = ∂ W , we write Diff ∂ ( W , N ) instead of Diff M ( W , N ) . Making use of thecollaring condition, it is not hard to see that g Diff M ( W , N ) • satisfies the Kan property. The semi-simplicial subgroup of diffeomorphisms
Diff M ( W , N ) • ⊂ g Diff M ( W , N ) • is defined by requiring the diffeomorphisms of ∆ p × M to commute with the projection tothe simplex ∆ p instead of just preserving its faces. This semi-simplicial subgroup agreeswith the (collared and smooth) singular set of the topological group Diff M ( W , N ) of dif-feomorphisms by which we mean the set of 0-simplices Diff M ( W , N ) equipped with thesmooth Whitney topology, so there is a canonical weak equivalence | Diff M ( W , N ) • | → Diff M ( W , N ) and we shall not distinguish between these spaces.1.5. Homotopy automorphisms.
The p -simplices of the semi-simplicial monoid of blockhomotopy automorphisms (cid:157) hAut M ( W , N ) • are the block homotopy automorphisms of ∆ p × W that fix ∆ p × M pointwise and restrict to homotopy automorphisms of ∆ p × N . As fordiffeomorphisms, insisting that the homotopy equivalences of ∆ p × M be over ∆ p definesa sub semi-simplicial monoid of homotopy automorphisms (3) hAut M ( W , N ) • ⊂ (cid:157) hAut M ( W , N ) • , which agrees with the singular set of the space hAut M ( W , N ) obtained by equipping theset of homotopy equivalences hAut M ( W , N ) with the compact open topology, so also | hAut M ( W , N ) • | and hAut M ( W , N ) are canonically equivalent. An aspect which distin-guishes the situation for homotopy automorphisms from that for diffeomorphisms is that In [BLR75, App. A §3 a], the authors attempt to construct degeneracy maps for certain semi-simplicial setsof block embeddings , which would in particular enhance g Diff ∂ ( W ) • to a simplicial group and thus imply that itsatisfies the Kan property as every simplicial group does. However, the argument in [BLR75] is flawed: on twoof the faces, the proposed degeneracy maps violate the collaring condition described on p. 116 loc. cit. MANUEL KRANNICH the inclusion (3) of Kan complexes induces an equivalence on geometric realisation, whichone can see from the combinatorial description of their homotopy groups together withthe contractibility of hAut ∂ ∆ p ( ∆ p ) .1.6. Bundle maps, unstably. A bundle map between two vector bundles ξ → X and ν → Y over finite CW complexes X and Y is a commutative square of the form ξ νX Y ϕ ¯ ϕ whose induced maps on vertical fibres are linear isomorphisms. Of course the underlyingmap of spaces ¯ ϕ can be recovered from ϕ , so we often omit it. Given a subcomplex A ⊂ X and a bundle map ℓ : ξ | A → ν defined on the restriction of ξ to A , the semi-simplicial setof block bundle maps g Bun A ( ξ , ν ; ℓ ) • has as its p -simplices the bundle maps ∆ p × ξ → ∆ p × ν that agree with id ∆ p × ℓ on ∆ p × ξ | A and whose underlying map ∆ p × X → ∆ p × Y is ablock map. As before, the semi-simplicial structure is induced by restriction to subspacesof the form σ × X for faces σ ⊂ ∆ p . Insisting that the underlying map between base spacesbe over ∆ p defines the sub semi-simplicial set Bun A ( ξ , ν ; ℓ ) • ⊂ g Bun A ( ξ , ν ; ℓ ) • of bundle maps , which agrees with the singular set of the space Bun A ( ξ , ν ; ℓ ) obtained byequipping the set Bun A ( ξ , ν ; ℓ ) of bundle maps ξ → ν relative to ℓ with the compact-open topology. If ξ = ν and ℓ = inc is the inclusion, then the semi-simplicial sets of (block)bundle maps g Bun A ( ξ , ξ ; inc ) • and Bun A ( ξ , ξ ; inc ) • are semi-simplicial monoids under com-position, and they act by precomposition on g Bun A ( ξ , ν ; ℓ ) • respectively Bun A ( ξ , ν ; ℓ ) • for any bundle ν and bundle map ℓ : ξ | A → ν . For a subcomplex C ⊂ X , we denote by (cid:157) hAut A ( ξ , C ) • ⊂ g Bun A ( ξ , ξ ; inc ) • the submonoid of those block bundle maps whose un-derlying selfmap of ∆ p × X is a homotopy equivalence that restricts to an equivalence of ∆ p × C . The submonoid hAut A ( ξ , C ) • ⊂ Bun A ( ξ ; ξ , inc ) • is defined analogously.Introducing yet another variant of bundle maps, we define the semi-simplicial set of tangential block bundle maps g Bun A ( ξ , ν ; ℓ ) τ • as follows: writing τ M for the tangent bundleof a manifold M and ε for the trivial line bundle, the p -simplices of g Bun A ( ξ , ν ; ℓ ) τ • arethe bundle maps φ : τ ∆ p × ξ → τ ∆ p × ν that agree with id τ ∆ p × ℓ on τ ∆ p × ξ | A , satisfy φ ( τ ∆ pi × ξ ) ⊂ τ ∆ pi × ν for 0 ≤ i ≤ p , and make the diagram(4) τ ∆ p | ∆ pi × ξ τ ∆ p | ∆ pi × ν ( τ ∆ pi ⊕ ε ) × ξ ( τ ∆ pi ⊕ ε ) × ν φ commute, where the bottom horizontal map is given by the restriction of φ on τ ∆ pi × ξ and the identity on ε , and the vertical maps are induced by the canonical trivialisation of τ [ , ] and the derivative of the map c i defined in Section 1.3. Requiring the underlying map ∆ p × X → ∆ p × Y to be over ∆ p defines the sub semi-simplicial set(5) Bun A ( ξ , ν ; ℓ ) τ • ⊂ g Bun A ( ξ , ν ; ℓ ) τ • of tangential bundle maps . As before, we have a chain of semi-simplicial monoidshAut A ( ξ , C ) τ • ⊂ (cid:157) hAut A ( ξ , C ) τ • ⊂ g Bun A ( ξ , ξ ; inc ) τ which are defined in the same way as for non-tangential (block) bundle maps. Note thatthere is a canonical map(6) Bun A ( ξ , ν ; ℓ ) • −→ Bun A ( ξ , ν ; ℓ ) τ • HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 9 given by extending a bundle map ∆ p × ξ → ∆ p × ν over ∆ p to τ ∆ p × ξ → τ ∆ p × ν by theidentity, but there is no analogous map between the block variants. This map is not anequivalence, but we shall see in the next paragraph that it becomes one after stabilisation .1.7. Bundle maps, stably. A stable vector bundle is a sequence of vector bundles ψ = { ψ k → B k } k ≥ l for some l ≥
0, where ψ k is k -dimensional, together with structure maps ψ k ⊕ ε → ψ k + for k ≥ l . Given a d -dimensional vector bundle ξ , its stabilisation is thestable vector bundle ξ s with ξ sd + k = ξ ⊕ ϵ k and the identity as structure maps. For a d -dimensional vector bundle ξ → X , a stable vector bundle ψ , and a bundle map ℓ : ξ | A ⊕ ε k → ψ d + k for some k , the semi-simplicial set of stable bundle maps is the colimitBun A ( ξ s , ψ ; ℓ ) • ≔ colim m ≥ k Bun A ( ξ ⊕ ε m , ψ d + m ; ℓ ) • over the stabilisation mapsBun A ( ξ ⊕ ε m , ψ d + m ; ℓ ) • −→ Bun A ( ξ ⊕ ε m + , ψ d + m + ; ℓ ) • given by adding a trivial line bundle followed by postcomposition with the structure map ψ d + m ⊕ ε → ψ d + m + . Analogously, we define stable tangential bundle maps as the colimitBun A ( ξ s , ψ ; ℓ ) τ • ≔ colim m ≥ k Bun A ( ξ ⊕ ε m , ψ d + m ; ℓ ) τ • . As in Section 1.6, there are semi-simplicial sub-monoids hAut A ( ξ s ; C ) • ⊂ Bun A ( ξ s , ξ s ; inc ) • and hAut A ( ξ s ; C ) τ • ⊂ Bun A ( ξ s , ξ s ; inc ) τ • , and also block variants of these semi-simplicialsets, defined by adding appropriate tildes. As the extension map (6) is compatible with thestabilisation maps, it gives rise to maps of the form(7) Bun A ( ξ s , ψ ; ℓ ) • −→ Bun A ( ξ s , ψ ; ℓ ) τ • and hAut A ( ξ s ; C ) • −→ hAut A ( ξ s ; C ) τ • which we show in Lemma A.4 to be equivalences.1.8. Tangential structures. A d -dimensional tangential structure is a fibration θ : B d → BO ( d ) whose target is the base of the universal d -dimensional vector bundle. The semi-simplicial set of θ -structures on a d -dimensional vector bundle ξ is the semi-simplicial setof bundle maps Bun A ( ξ , θ ∗ γ d ; ℓ ) from ξ to the pullback θ ∗ γ d of the universal bundle γ d → BO ( d ) along θ , relative to a fixed bundle map ℓ : ξ | A → θ ∗ d γ d . We denote the homotopyquotient (in the sense of Section 1.1) of the action of hAut A ( ξ , C ) • ⊂ Bun A ( ξ ; ξ , inc ) • onthe semi-simplicial set of θ -structures byBhAut θA ( ξ , C ; ℓ ) • ≔ Bun A ( ξ , θ ∗ γ d ; ℓ ) • (cid:12) hAut A ( ξ , C ) • . Remark . Note that BhAut θA ( ξ , C ; ℓ ) • is in many cases empty or disconnected, so despitethe suggestive notation, it is in general not the classifying space of any kind of group ormonoid, (semi-)simplicial or topological.A stable tangential structure is a fibration of the form Ξ : B → BO, which induces a d -dimensional tangential structure Ξ d : B d → BO ( d ) for any d ≥ Ξ alongthe stabilisation map BO ( d ) → BO. Note, however, that not all d -dimensional tangentialstructures arise this way, for instance the tangential structure EO ( d ) → BO ( d ) encodingunstable framings does not. A stable tangential structure in the sense above defines astable vector bundle Ξ ∗ γ = { Ξ ∗ d γ d } d ≥ whose structure maps are induced by the canonicalbundle map γ d ⊕ ε → γ d + covering the usual stabilisation map BO ( d ) → BO ( d + ) . Givena stable bundle map ℓ : ξ s | A → Ξ ∗ γ , we call Bun A ( ξ s , Ξ ∗ γ ; ℓ ) the semi-simplicial set of stable θ -structures on ξ . Analogous to the unstable case, we abbreviate(8) BhAut Ξ A ( ξ s , C ; ℓ ) • ≔ Bun A ( ξ s , Ξ ∗ γ ; ℓ ) • (cid:12) hAut A ( ξ s , C ) • . Tangential and or block variants of the previous definitions are defined by using the re-spective variants of bundle maps and adding appropriate tildes and or τ -superscripts.It follows from Lemma 1.1 that the canonical sequence of semi-simplicial spaces(9) Bun A ( ξ , θ ∗ γ d ; ℓ ) • −→ BhAut θA ( ξ , C ; ℓ ) • −→ BhAut A ( ξ , C ) • realises to a quasi-fibration, because Bun A ( ξ , θ ∗ γ d ; ℓ ) • and hAut A ( ξ , C ) • admit degenera-cies since they satisfy the Kan property (they agree with the singular complex of a space,see Section 1.6) and the action is by equivalences as hAut A ( ξ , C ) • is group-like. The sameargument applies to the stable analogue of this sequence involving (8) and, using Corol-lary A.2, also to its variants involving tangential and or block bundle maps. By definitionof the universal bundle, the semi-simplicial set Bun A ( ξ , θ ∗ γ d ; ℓ ) • is contractible in theuniversal case θ = id, so the second map in (9) is an equivalence for this particular choiceof θ . The analogous statement holds in the stable case as well and, as a consequence ofLemmas A.3 and A.4, also for the tangential and or block variants.1.9. The derivative maps.
Assigning a diffeomorphism φ : ∆ p × W → ∆ p × W over ∆ p its fibrewise derivative ∆ p × τ W → ∆ p × τ W induces a canonical semi-simplicial map(10) Diff M ( W , N ) • −→ hAut M ( τ W , N ) • , which we call the derivative map . Furthermore, the notion of a tangential bundle map isdesigned exactly so that there is a block derivative map (11) g Diff M ( W , N ) • −→ (cid:157) hAut M ( τ sW , N ) τ • . given by assigning a block diffeomorphism ∆ p × W → ∆ p × W its derivative τ ∆ p × τ W → τ ∆ p × τ W , which indeed makes the square (4) commute as φ is assumed to be collared asdefined in Section 1.3. Remark . (i) A different model of the block derivative map (11) was already considered byBerglund and Madsen in their prominent study of the rational homotopy type ofspaces of block diffeomorphisms of manifolds with certain boundary conditions[BM20] (see Section 4.3 loc. cit. and Remark 2.3 below).(ii) Lemmas A.3 and A.4 provide a canonical equivalence (cid:157) hAut M ( τ sW , N ) τ • ≃ hAut M ( τ sW , N ) • , so the delooped space B (cid:157) hAut M ( τ sW , N ) τ classifies fibrations π W : E → B with fibre W together with the following data:(1) maps of fibrations π M → π W ← π N over the identity whose induced mapson fibres is equivalent to the system of inclusions M ⊂ W ⊃ N ,(2) a trivialisation of π M , and(3) a stable vector bundle E → BO over the total space of π W whose restrictionto each fibre agrees with the stable tangent bundle of W .From this point of view, the block derivative (11) comes as no surprise: it is rem-iniscent of the fact that a block bundle has an underlying fibration and a stablevertical tangent bundle by [ERW14]. However, somewhat curiously, the block de-rivative map (11) obviously factors over the variant (cid:157) hAut M ( τ W , N ) τ • involving the unstable tangent bundle of W , giving rise to an unstable block derivative map B g Diff M ( W , N ) −→ B (cid:157) hAut M ( τ W , N ) τ • . The target of this map is neither equivalent to BhAut M ( τ sW , N ) • nor BhAut M ( τ W , N ) • ,and it would be interesting to have a good description of what it classifies.We denote the submonoids of the path components hit by the derivative maps byhAut (cid:27) M ( τ W , N ) • ⊂ hAut M ( τ W , N ) • and (cid:157) hAut (cid:27) M ( τ sW , N ) τ • ⊂ (cid:157) hAut M ( τ sW , N ) τ • and add the same (cid:27) -superscript to (8) and its tangential block variant to indicate when wetake homotopy quotients by these submonoids instead of the full monoids. DefiningB g Diff Ξ M ( W , N ; ℓ ) • ≔ g Bun M ( τ sW , Ξ ∗ γ ; ℓ ) τ • (cid:12) g Diff M ( W , N ) • HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 11 for a stable tangential structure Ξ and a Ξ -structure ℓ on τ sW | M and, more commonly,BDiff θM ( W , N ; ℓ ) • ≔ Bun M ( τ W , θ ∗ γ d ; ℓ ) • (cid:12) Diff M ( W , N ) • in the unstable case, the two derivative maps fit into a a commutative squareBDiff Ξ d M ( W , N ; ℓ ) • BhAut Ξ d , (cid:27) M ( τ W , N ; ℓ ) • B g Diff Ξ M ( W , N ; ℓ ) • B (cid:157) hAut Ξ , (cid:27) M ( τ sW , N ; ℓ ) τ • whose vertical maps are induced by the canonical compositionBun A ( τ W , Ξ ∗ d γ d ; ℓ ) • → Bun A ( τ sW , Ξ ∗ γ ; ℓ ) • → Bun A ( τ sW , Ξ ∗ γ ; ℓ ) τ • ⊂ g Bun A ( τ sW , Ξ ∗ γ ; ℓ ) τ • . All maps in this composition are equivalences, the first one by an exercise in obstructiontheory and the second map as well as the final inclusion as a result of Lemmas A.3 and A.4.By an application of Lemma 1.1, the composition of equivalences just discussed agreeswith the induced map on horizontal homotopy fibres of the commutative diagram(12) BDiff Ξ d M ( W , N ; ℓ ) • BDiff M ( W , N ) • B g Diff Ξ M ( W , N ; ℓ ) • B g Diff M ( W , N ) • induced by forgetting tangential structures, so this square is homotopy cartesian.
2. Surgery theory and spaces of block diffeomorphisms
We use surgery theory to give a partial p -local description of the space B g Diff Ξ ∂ W ( W , ∂ W ) of block diffeomorphisms with tangential structures in terms of homotopy automorphismswith bundle data for manifold triads ( W ; ∂ W , ∂ W ) of dimension d ≥ π - π -condition. As a point of notation, we refer to the geometric realisation of any of thesemi-simplicial sets or spaces of the previous sections by omitting their • -subscripts.2.1. A reminder of surgery theory. A d -dimensional manifold triad is a triple W = ( W ; ∂ W , ∂ W ) consisting of a compact connected smooth d -manifold W (possibly withcorners) and connected (possibly empty) submanifolds ∂ W ⊂ W and ∂ W ⊂ W such that ∂ W = ∂ W ∪ ∂ W and ∂ ( ∂ W ) = ∂ W ∩ ∂ W = ∂ ( ∂ W ) . A diffeomorphism or (simple) homotopy equivalence ( W ; ∂ W , ∂ W ) → ( W ′ ; ∂ W ′ , ∂ W ) between two manifold triads is a diffeomorphism or (simple) homotopy equivalence W → W ′ which restricts to an automorphism of this kind between ∂ W and ∂ W ′ , between ∂ W and ∂ W ′ , and between their intersections. At times, we omit ∂ W and ∂ W fromthe notation and abbreviate a triad ( W ; ∂ W , ∂ W ) simply by W . The smooth structureset S( W ) of a triad W (see e.g. [Wal99, Ch. 10]) is the collection of equivalence classes ofsimple homotopy equivalences of triads N → W that restrict to a diffeomorphism ∂ N → ∂ W , where two such equivalences N → W and N ′ → W are considered equivalent ifthere exists a diffeomorphism of triads N → N ′ that makes the triangle of triads N WN ′ ≃ (cid:27) ≃ homotopy commute relative to ∂ N . The structure set S( W ) is canonically based; theidentity serves as basepoint. The main tool to access S( W ) is the surgery exact sequence (13) . . . S( W × D k ) N( W × D k ) L( W × D k ) S( W × D k − ) . . . L( W ) S( W ) N( W ) L( W ) which relates the structure sets S( W × D k ) of the triads W × D k = ( W × D k ; ∂ W × D k ∪ W × ∂ D k ; ∂ W × D k ) to the sets of normal invariants N( W × D k ) and the L -groups L( W × D k ) . Assuming d ≥ L( W × D ) where it continuesas an exact sequence of based sets (see e.g. [Wal99, Ch. 10]). The similarity with the longexact sequence of homotopy groups induced by a fibration is no coincidence, Quinn’s surgery fibration [Qui70a, Qui70b] is a homotopy fibration of based spaces(14) ˜ S ( W ) −→ N ( W ) −→ L ( W ) that induces (13) on homotopy groups (see also [Wal99, Ch. 17A], or [Nic82] for a detailedaccount in the topological category). We refrain from describing (13) or (14) in detail; allwe shall need to know are a few basic properties, which we explain in the following.2.1.1. The block structure space.
Assuming d ≥
6, an application of the s -cobordism theo-rem results in a preferred equivalence(15) (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) ≃ ˜ S ( W ) id between the homotopy fibre of the canonical map B g Diff ∂ W ( W ) → B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) and the basepoint component ˜ S ( W ) id ⊂ ˜ S ( W ) of the block structure space (cf. [Wal99,Ch. 17A] or [BM13, p. 33-34]). Here (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ⊂ (cid:157) hAut ∂ W ( W , ∂ W ) are the components in the image of the canonical map g Diff ∂ W ( W ) → (cid:157) hAut ∂ W ( W , ∂ W ) .Note that a diffeomorphism of W that fixes ∂ W pointwise automatically preserves ∂ W setwise, since ∂ W is the complement of the interior of ∂ W ⊂ ∂ W . On homotopy groups,the equivalence (15) can be described as follows: using the combinatorial description ofthe relative homotopy groups of a semi-simplicial Kan pair, a class in π k ( (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) ; id ) (cid:27) π k ( (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) , g Diff ∂ W ( W ) ; id ) is represented by a simple homotopy equivalence of triads W × D k → W × D k which isthe identity on ∂ W × D k and restricts to a diffeomorphism on W × ∂ D k , so it defines aclass in the structure set S( W × D k ) (cid:27) π k ( ˜ S ( W ) ; id ) .2.1.2. The space of normal invariants.
The space of normal invariants N ( W ) admits a pre-ferred homotopy equivalence to the pointed mapping space Maps ∗ ( W / ∂ W , G / O ) basedat the constant map, where G / O is the homotopy fibre of the canonical map BO → BGwitnessing the fact that a stable vector bundle has an underlying stable spherical fibra-tion (see e.g. [Qui70a] or [Wal99, Ch. 10, 17A]). This map is one of infinite loop spaces,so its homotopy fibre G / O is an infinite loop space and hence so is the mapping spaceMaps ∗ ( W / ∂ W , G / O ) . On homotopy groups, the composition˜ S ( W ) → N ( W ) ≃ Maps ∗ ( W / ∂ W , G / O ) → Maps ∗ ( W / ∂ W , BO ) has the following geometric description (see e.g. [Wal99, p. 113-114]): given a class in thestructure set π k ( ˜ S ( W ) ; ∗) (cid:27) S( W × D k ) represented by a simple homotopy equivalence φ : N → W × D k , choose a homotopy inverse ˜ φ : W × D k → N of triads that agrees with HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 13 ( φ | ∂ N ) − on ∂ ( W × D k ) . Writing ν s and τ s for the stable normal respectively tangentbundle of a manifold, the stable vector bundle ( ˜ φ ∗ ν sN ) ⊕ τ sW × D k on W × D k comes with atrivialisation on the subspace ∂ ( W × D k ) = ∂ W × D k ∪ W × ∂ D k by making use of the diffeomorphism ˜ φ | ∂ ( W × D k ) , and hence gives rise to a class [( ˜ φ ∗ ν sN ) ⊕ τ sW × D k ] ∈ π k ( Maps ∗ ( W / ∂ W , BO ) ; ∗) . The L -theory space. The L -theory space L ( W ) is an infinite loop space as well (seee.g. [Nic82, Prop. 2.2.2]), and its homotopy groups are canonically isomorphic to Wall’s quadratic L -groups (see e.g. [Nic82, Prop. 2.2.4]). We shall not need to know much aboutthese groups, except that π k ( L ( W ) ; ∗) (cid:27) L( W × D k ) vanishes if W satisfies the π - π -condition , i.e. if ∂ W is nonempty and the inclusion ∂ W ⊂ W is an isomorphism on funda-mental groups. This is a consequence of the exact sequence of L -groups of a triad (or moregenerally, n -ad) described for instance in [Wal99, Thm 3.1]. In other words, under theseassumptions, the L -theory space L ( W ) is weakly contractible, so (14) induces a preferredequivalence ˜ S ( W ) ≃ N ( W ) —an instance of the so-called π - π -theorem .2.2. The block derivative map of a triad.
With these basics of space-level surgery the-ory in mind, we now turn towards studying connectivity properties of the block derivativemap of Section 1.9 B g Diff ∂ W ( W ) −→ B (cid:157) hAut (cid:27) ∂ W ( τ sW , ∂ W ) τ and its tangential enhancements, beginning with a technical but useful lemma (see Sec-tion 1.2 for a recollection on nilpotent spaces). Lemma 2.1.
For a manifold triad ( W ; ∂ W , ∂ W ) of dimension d ≥ , the homotopy fibre (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) is a nilpotent space.Proof. By definition of the subspace (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ⊂ (cid:157) hAut ∂ W ( W , ∂ W ) , the mapB g Diff ∂ W ( W ) → BhAut (cid:27) ∂ W ( W , ∂ W ) is surjective on fundamental groups, so its homo-topy fibre is connected. To see that it is nilpotent, we use that it is homotopy equivalentto the identity component ˜ S ( W ) id of the block structure space (see Section 2.1.1), whichis itself equivalent to a component of the homotopy fibre of the surgery obstruction map N ( W ) → L ( W ) of (14). Given this description of the space in consideration, the claimfollows from an application of Lemma 1.4, using the fact that, being an infinite loop space,the space of normal invariants N ( W ) is nilpotent (see Section 2.1.2). (cid:3) To state the main result of this section, some notation is in order. For a stable tangentialstructure Ξ : B → BO and a Ξ -structure ℓ : τ sW → Ξ ∗ γ (see Section 1.8), we write ℓ forthe restriction of ℓ to τ sW | ∂ W and denote byB g Diff Ξ ∂ W ( W ; ℓ ) ℓ and B (cid:157) hAut Ξ , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) τ ℓ the components that correspond to ℓ under the canonical bijection π B g Diff Ξ ∂ W ( W ; ℓ ) (cid:27) π B (cid:157) hAut Ξ , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) τ (cid:27) π g Bun ∂ W ( τ sW , Ξ ∗ γ ; ℓ ) τ / π g Diff ∂ W ( W ) resulting from the discussion in Sections 1.8 and 1.9. Recall that a manifold triad W = ( W ; ∂ W , ∂ W ) satisfies the π - π -condition if ∂ W is nonempty and the inclusion ∂ W ⊂ W induces an isomorphism on fundamental groups. Theorem 2.2.
Let d ≥ and W be a d -dimensional triad satisfying the π - π -condition. Fora stable tangential structure Ξ and a Ξ -structure ℓ on τ sW , the homotopy fibre of the map B g Diff Ξ ∂ W ( W ; ℓ ) ℓ −→ B (cid:157) hAut Ξ , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) τ ℓ is nilpotent and p -locally ( p − − k ) -connected for primes p , where k is the relative handledimension of the inclusion ∂ W ⊂ W .Remark . Theorem 2.2 is inspired by a similar result of Berglund and Madsen [BM20,Thm 1.1], which applies to a different class of triads, namely those satisfying ∂ W = ∂ W (cid:27) S d − . Another point in which their result differs from ours is that it is purely rational, anddoes in fact not seem to admit a p -local refinement analogous to Theorem 2.2. This isbecause p -torsion occurs for primes p that can be rather large with respect to the degree,originating from contributions of numerators of divided Bernoulli numbers to the homo-topy groups of the homotopy fibre Top / O of the canonical map BO → BTop.
Proof of Theorem 2.2.
Using Corollary A.2, an application of Lemma 1.1 to the horizontalarrows of the canonical square(16) B g Diff Ξ ∂ W ( W ; ℓ ) τ ℓ B g Diff ∂ W ( W ) B (cid:157) hAut Ξ , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) τ ℓ B (cid:157) hAut (cid:27) ∂ W ( τ sW , ∂ W ) τ , identifies their horizontal homotopy fibres with the union of components of the space g Bun ∂ W ( τ sW , Ξ ∗ γ ; ℓ ) given by the π g Diff ∂ W ( W ) -orbit of the Ξ -structure ℓ , so (16) is ho-motopy cartesian, which reduces the proof to the case Ξ = id in which both rows of thesquare are equivalences (see Section 1.8). To settle this remaining case, we consult themap of fibre sequences (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) B g Diff ∂ W ( W ) B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ (cid:157) hAut (cid:27) ∂ W ( τ sW , ∂ W ) τ B (cid:157) hAut (cid:27) ∂ W ( τ sW , ∂ W ) τ B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) , in order to see that the homotopy fibre in question agrees with the homotopy fibre ofthe left vertical map and is therefore nilpotent by Lemmas 1.4 and 2.1. The bottom righthorizontal map of the latter map of fibre sequences fits into a commutative diagram(17) BhAut (cid:27) ∂ W ( τ sW , ∂ W ) BhAut (cid:27) ∂ W ( τ sW , ∂ W ) τ B (cid:157) hAut (cid:27) ∂ W ( τ sW , ∂ W ) τ BhAut (cid:27) ∂ W ( W , ∂ W ) BhAut (cid:27) ∂ W ( W , ∂ W ) B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ≃ ≃≃ whose right horizontal maps are induced by inclusion and are equivalences as a conse-quence of Lemma A.3. The left upper horizontal map is the equivalence described in (7)and the left vertical arrow fits into a commutative square induced by inclusion(18) BhAut (cid:27) ∂ W ( τ sW , ∂ W ) BhAut ∂ W ( τ sW , ∂ W ) BhAut (cid:27) ∂ W ( W , ∂ W ) BhAut ∂ W ( W , ∂ W ) τ sW , where hAut ∂ W ( W , ∂ W ) τ sW are the components in the image of the maphAut ∂ W ( τ sW , ∂ W ) −→ hAut ∂ W ( W , ∂ W ) HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 15 that forgets the bundle maps. Before taking geometric realisation, this map is easily seento be a Kan fibration, so the homotopy fibre of the right vertical map in (18) is equivalent tothe classifying space of the gauge group of τ sW relative to ∂ W , which is in turn canonicallyequivalent to the space Maps ∂ W ( W , BO ) τ sW of maps homotopic to a choice of classifyingmap for τ sW relative to ∂ W . The horizontal arrows in (18) are 1-coconnected, so the sameholds for the induced map on vertical homotopy fibres. Combining this with the zig-zagof equivalences (17), we see that the left vertical map in the second diagram agrees, up tocanonical equivalence and postcomposition with a 1-coconnected map, with a map (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) −→ Maps ∂ W ( W , BO ) τ sW , so it suffices to show that this map between nilpotent spaces is p -locally ( p − − k ) -connected. This can be tested on homotopy groups, on which the map has the followingdescription: a class in π k ( g Diff ∂ W ( W ) , (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ; id ) is represented by a homotopyequivalence of triads φ : W × D k → W × D k that is the identity on D k × ∂ W and restrictsto a diffeomorphism on ∂ D k × W that is the identity on ∗ × M for a base point ∗ ∈ ∂ D k .The pullback φ ∗ τ sW × D k is a stable vector bundle over W × D k that agrees with τ sW × D k over D k × ∂ W ∪ {∗} × W and comes with a canonical identification φ ∗ τ sW × D k | ∂ D k × M (cid:27) τ sW × D k | ∂ D k × M given by the derivative of φ | ∂ D k × M , so it defines a class in the k th homotopygroup of Maps ∂ W ( W , BO ) τ sW based at a choice of classifying map for τ sW . Comparing thisdescription with that of the equivalence (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) ≃ ˜ S ( W ) id and the map ˜ S ( W ) id → Maps ∗ ( W / ∂ W , BO ) on homotopy groups explained in Sections 2.1.1and 2.1.2, we see that the diagram of nilpotent spaces (cid:157) hAut (cid:27) ∂ W ( W , ∂ W )/ g Diff ∂ W ( W ) Maps ∂ W ( W , BO ) τ sW ˜ S ( W ) id Maps ∗ ( W / ∂ W , G / O ) Maps ∗ ( W / ∂ W , BO ) , ≃ ι ◦((−)· ν sW )≃≃ commutes upon taking homotopy groups. Here the upper horizontal map is the one wejust discussed, the right vertical equivalence is induced by multiplication with the stablenormal bundle ν sW using the infinite loop space structure induced from that of BO followedby the involution on Maps ∂ W ( W , BO ) induced by the canonical involution on BO, theleft vertical equivalences are induced by the surgery fibration (see Sections 2.1.1 and 2.1.2),and the bottom horizontal is given by postcomposition with the canonical map G / O → BO. Note that, being an infinite loop space, the mapping space Maps ∗ ( W / ∂ W , BO ) issimple, so its homotopy groups at different base points are canonically identified. Theproof finishes by noting that the lower horizontal map in the diagram is p -locally ( p − − k ) -connected by obstruction theory, since W / ∂ W has no cohomology above degree k bythe assumption on the relative handle dimension of ∂ W ⊂ W and because G / O → BOis p -locally ( p − ) -connected for p , Z / p -torsion free in degrees ∗ < p − (cid:3) The stable tangential structure we shall be primarily interested in is the one encodingstable framings, which we denote by sfr : EO → BO. In this case, the p -local approxima-tion of the space of block diffeomorphisms with tangential structures provided by Theo-rem 2.2 can be further simplified in terms of the union of components (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ℓ ⊂ (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) given by the image of the canonical mapB (cid:157) hAut sfr , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) ℓ −→ B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) on fundamental groups based at a fixed stable framing ℓ of τ W . Loosely speaking, theseare the components of homotopy equivalences of triads that are homotopic to a diffeomor-phism preserving the component of the stable framing ℓ . Corollary 2.4.
Let d ≥ and W be a d -dimensional manifold triad satisfying the π - π -condition. For a stable framing ℓ of W , the homotopy fibre of the natural map B g Diff sfr ∂ W ( W ; ℓ ) ℓ −→ B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ℓ is nilpotent and p -locally ( p − − k ) -connected for primes p , where k is the relative handledimension of the inclusion ∂ W ⊂ W .Proof. Once we show that the mapB (cid:157) hAut sfr , (cid:27) ∂ W ( τ sW , ∂ W ; ℓ ) τ ℓ −→ B (cid:157) hAut (cid:27) ∂ W ( W , ∂ W ) ℓ is an equivalence, the statement is a consequence of Theorem 2.2. By construction, thehomotopy fibre of this map is connected, which, combined with (17) and (18) reduces theproof to showing that the mapBhAut sfr ∂ W ( τ sW , ∂ W ; ℓ ) ℓ −→ BhAut ∂ W ( W , ∂ W ) is an equivalence. Taking vertical homotopy fibres in the map of fibre sequencesBun ∂ W ( τ sW , sfr ∗ γ ; ℓ ) ℓ BhAut sfr ∂ W ( τ sW , ∂ W ; ℓ ) ℓ BhAut ∂ W ( τ sW , ∂ W )∗ BhAut ∂ W ( W , ∂ W ) BhAut ∂ W ( W , ∂ W ) , where Bun ∂ W ( τ sW , sfr ∗ γ ; ℓ ) ℓ ⊂ Bun ∂ W ( τ sW , sfr ∗ γ ; ℓ ) are the components of the ℓ -orbit ofthe π hAut ∂ W ( τ sW , ∂ W ) -action, we see that it suffices to show that the induced action ofthe loop space of the homotopy fibre of the right vertical map on Bun ∂ W ( τ sW , sfr ∗ γ ; ℓ ) ℓ isa torsor in the homotopical sense, i.e. that the map given by acting on ℓ is an equivalence.Since hAut ∂ W ( τ sW , ∂ W ) • → hAut ∂ W ( W , ∂ W ) • is a Kan fibration, this loop space iscanonically equivalent to the space of bundle self-maps of τ sW that cover the identity agreewith the identity on τ sW | ∂ W . Moreover, by construction of the top fibration, the inducedaction on Bun ∂ W ( τ sW , sfr ∗ γ ; ℓ ) ℓ is given by precomposition. Suitably modeled, this actionis simply transitive, so the assertion follows. (cid:3)
3. High-dimensional handlebodies and their mapping classes
This section serves to compute variants of the mapping class group of a high-dimensionalhandlebody up to extensions in terms of automorphisms of the integral homology.3.1.
Automorphisms of handlebodies.
The proof of Theorem A relies on consideringa more general family of manifold than discs, the boundary connected sums V д ≔ ♮ д ( D n + × S n ) , and their boundaries as well as the manifolds obtained by cutting out an embedded disc D n ⊂ ∂ V д , which we denote by W д ≔ ∂ V д (cid:27) ♯ д ( S n × S n ) and W д , ≔ W д \ int ( D n ) . HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 17
This includes the case д =
0, that is V = D n + , W = S n , and W , = D n . Using thenotation introduced in Section 1, the restriction of diffeomorphisms and relative homotopyautomorphisms of V д to its boundary induces a commutative diagram(19) Diff ∂ ( V д ) Diff D n ( V д ) Diff ∂ ( W д , ) hAut ∂ ( V д ) hAut D n ( V д , W д , ) hAut ∂ ( W д , ) where the right horizontal maps are fibrations and the left maps the inclusions of the fibresover the identity. These fibrations need not be surjective and we denote their images byDiff ext ∂ ( W д , ) ⊂ Diff ∂ ( W д , ) and hAut ext ∂ ( W д , ) ⊂ hAut ∂ ( W д , ) . Furthermore, we write hAut (cid:27) ∂ ( V д ) , hAut (cid:27) D n ( V д , W д , ) and hAut (cid:27) ∂ ( W д , ) for the componentshit by the vertical maps. Block variants of all of the above automorphism spaces are de-fined in the same way.3.2. The mapping class group.
As a first step in our analysis of the mapping class group π Diff D n ( V д ) , we observe that we may equally study orientation-preserving diffeomor-phisms of V д that do not necessarily fix the embedded disc in the boundary D n ⊂ ∂ V д , ordiffeomorphisms that preserve a disc D n + ⊂ int ( V д ) in the interior set- or pointwise. Lemma 3.1.
For n ≥ and discs D n ⊂ ∂ V д and D n + ⊂ int ( V д ) , the canonical composi-tions π Diff D n + ( V д ) −→ π Diff + ( V д , D n + ) −→ π Diff + ( V д ) and π Diff D n ( V д ) −→ π Diff + ( V д , D n ) −→ π Diff + ( V д ) consist of isomorphisms.Proof. Up to isotopy, preserving a disc in the interior setwise is equivalent to preservinga point, so the group π Diff + ( V д , D n + ) agrees with the group of path components of thesubgroup Diff + ( V д , ∗) ⊂ Diff + ( V д ) of diffeomorphisms that fix the centre ∗ ∈ D n + ⊂ int ( V д ) . As V д is ( n − ) -connected, the long exact sequence of the fibration Diff + ( V д ) → int ( V д ) given by evaluating diffeomorphisms at ∗ implies that π Diff + ( V д , ∗) agrees with π Diff + ( V д ) , so the second map in the first composition in the statement is an isomorphism.Taking derivatives at ∗ yields a homotopy fibre sequence of the form(20) Diff D n + ( V д ) −→ Diff + ( V д , ∗) d −→ SO ( n + ) and hence an exact sequence π SO ( n + ) t −→ π Diff D n + ( V д ) −→ π Diff + ( V д , ∗) −→ . On the subgroup SO ( n ) ⊂ SO ( n + ) , the derivative map d has a section since V д admitsa smooth SO ( n ) -action with fixed point ∗ ∈ V д whose tangential representation agreeswith the restriction of the standard representation to the subgroup SO ( n ) : take the д -foldequivariant boundary connected sums of D n + × S n with SO ( n ) acting by rotating S n alongan axis. As SO ( n ) ⊂ SO ( n + ) is ( n − ) -connected, this section ensures that the long exactsequence on homotopy groups of (20) splits in degrees ∗ ≤ n − t is trivial for n ≥
2, so π Diff D n + ( V д ) (cid:27) π Diff + ( V д ) holdsas claimed. Replacing (20) by the fibration sequence Diff D n ( V д ) → Diff + ( V д , ∗) → SO ( n ) induced by taking the derivative at the centre ∗ ∈ D n ⊂ ∂ V д of the disc in the boundary,the proof of the claim regarding the maps in the second composition of the statementproceeds analogous to the first part of the proof. (cid:3) To obtain further information on the mapping class group π Diff D n ( V д ) , we considerits action on the n th integral homology of V д and its boundary ∂ V д = W д , abbreviated by H W д , ≔ H n ( W д , ; Z ) and H V д ≔ H n ( V д ; Z ) . This action preserves further algebraic structures, such as the intersection pairing λ : H W д , ⊗ H W д , −→ Z , which equips H W д , with a nondegenerate (− ) n -symmetric form by Poincaré duality. Inaddition, any automorphism of H W д , induced by a diffeomorphism of W д , has to preservethe function α : H W д , −→ π n BSO ( n ) given by representing a homology class by an embedded n -sphere and taking its normalbundle. Wall [Wal63a, Thm 2] has shown that λ and α satisfy the relations(i) λ ( x , x ) = ∂ n α ( x ) (ii) α ( x + y ) = α ( x ) + α ( y ) + λ ( x , y ) · τ S n as long as n ≥
3, where ∂ n : π n BSO ( n ) −→ π n − S n − (cid:27) Z is induced by the fibration S n − → BSO ( n − ) → BSO ( n ) and τ S n ∈ π n BSO ( n ) is the classrepresenting the tangent bundle of the n -sphere. In sum, we arrive at a morphism π Diff ( W д , ) −→ G д ≔ Aut ( H W д , , λ , α ) to the subgroup G д ⊂ GL ( H W д ) of automorphisms preserving λ and α , which is sur-jective by [Wal63b, Lem. 10]. However, we are interested in the mapping class group π Diff D n ( V д ) and not every automorphism in G д is realised by a diffeomorphism of W д , that extends to one of V д ; it would at least have to preserve the Lagrangian subspace K д ≔ ker (cid:0) H W д , → H V д (cid:1) , so there is a canonical map π Diff D n ( V д ) → G ext д to the subgroup G ext д ≔ { Φ ∈ G д | Φ ( K д ) ⊂ K д } of automorphisms that preserve this Lagrangian, given by acting on the homology of theboundary. Using the canonical isomorphism H W д , / K д (cid:27) H V д , the subgroup G ext д mapsfurther to GL ( H V д ) . The resulting composition(21) π Diff D n ( V д ) −→ G ext д −→ GL ( H V д ) agrees with action on the homology of V д and one may ask for the (co)kernel of the threemaps involved. Extending work of Wall [Wal63b, Wal65], we express the answer in The-orem 3.3 below in terms of an exact sequence of GL ( H V д ) -modules(22) 0 −→ H ∨ V д ⊗ Sπ n SO ( n ) −→ N д −→ M д −→ , where(1) Sπ n SO ( n ) ⊂ π n SO ( n + ) is the image of the stabilisation π n SO ( n ) → π n SO ( n + ) ,(2) M д ⊂ ( H V д ⊗ H V д ) ∨ is the submodule of bilinear forms µ ∈ ( H V д ⊗ H V д ) ∨ that are • (− ) n + -symmetric and satisfy • µ ( x , x ) ∈ im ( ∂ n : π n + SO ( n + ) → Z ) for x ∈ H V д , and(3) N д ⊂ M д ⊕ ( π n SO ( n + )) H Vд is the submodule of pairs ( µ , β ) of a bilinear form µ ∈ M д and a function β : H V д → π n SO ( n + ) that fulfil the conditions • µ ( x , x ) = ∂ n + β ( x ) for x ∈ H V д and • β ( x + y ) = β ( x ) + β ( y ) + µ ( x , y ) · τ S n + for x , y ∈ H V д ,all equipped with the evident GL ( H V д ) -action. Here and henceforth, we denote the integraldual of a G -module M by M ∨ . HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 19
Remark . The image of the map ∂ n + : π n SO ( n + ) → π n S n (cid:27) Z is generated by theorder of the tangent bundle τ S n ∈ π n − SO ( n ) , so we have (see [Lev85, §1B)])(23) im ( ∂ n + ) = n even Z for n = , , · Z otherwise , which exhibits the condition µ ( x , x ) ∈ im ( ∂ n + ) in (ii) as vacuous unless n , , , H V д (cid:27) Z д , the module M д can bedescribed equivalently in terms of (− ) n + -symmetric integral ( д × д ) -matrices, with evendiagonal entries if n , , , Theorem 3.3.
Let n ≥ .(i) The action of π Diff D n ( V д ) on H V д = H n ( V д ; Z ) gives rise to an extension −→ N д −→ π Diff D n ( V д ) −→ GL ( H V д ) −→ . (ii) The morphism G ext д → GL ( H V д ) fits into an extension of the form −→ M д −→ G ext д −→ GL ( H V д ) −→ . (iii) The action of π Diff D n ( V д ) on H W д , = H n ( W д , ; Z ) induces an extension −→ H ∨ V д ⊗ Sπ n SO ( n ) −→ π Diff D n ( V д ) −→ G ext д −→ . Moreover, the induced outer actions of these extensions is as specified above and the secondextension admits a preferred splitting.Remark . For a complete description of π Diff D n ( V д ) , one still needs to determine theextension problems of the first or third part of the theorem, which we do not pursue atthis point. Similar extensions by Kreck [Kre79] describing the closely related mappingclass group π Diff ∂ ( W д , ) for n ≥ n odd. Proof of Theorem 3.3.
We begin with three preparatory remarks.(1) Results of Wall we shall use rely on are phrased in terms of pseudoisotopy insteadof isotopy, but these notions agree in our situation by Cerf’s work [Cer70].(2) Justified by Lemma 3.1, we do not distinguish between seemingly different vari-ants of π Diff D n ( V д ) fixing various discs point- or setwise.(3) We identify K д canonically with the dual H V д ∨ as a G ext д -module via the isomor-phism induced by the form λ , and dually H V д with K ∨ д .Wall [Wal63b, Lem. 10] showed that the action π Diff D n ( V д ) → GL ( H V д ) is surjective andidentified its kernel with those isotopy classes of diffeomorphisms φ that are homotopic tothe identity. Moreover, in [Wal65, p. 298], he defined a complete obstruction ( µ φ , β φ ) ∈ N д for such a homotopically trivial diffeomorphism to be isotopic to the identity. By [Wal63b,Lem. 12–13], these obstructions are additive and exhaust N д , so the resulting function(24) ker (cid:0) π Diff ( V д ) → GL ( H V д ) (cid:1) −→ N д is an isomorphism of groups, which establishes the first of the three claims. To demon-strate the second, note that, as automorphisms in G ext д preserve the form λ , the composition G ext д −→ GL ( H V д ) ((−) − ) ∨ −→ GL ( K д ) agrees with the restriction to K д . In particular, this shows that the kernel of the first mapin this composition acts trivially on K д , so there is a canonical monomorphismker ( G ext д → GL ( H V д )) Ψ −→ Hom ( H V д , K д ) (cid:27) ( H V д ⊗ H V д ) ∨ defined by sending ϕ ∈ G ext д to the linear map Ψ ( ϕ ) : H V д → K д induced by the difference ( ϕ − id ) : H W д , → K д . This leaves us with identifying the image of Ψ with M д for whichit is helpful to note that a morphism f ∈ Hom ( H V д , K д ) lies in the subspace M д ⊂ ( H V д ⊗ H V д ) ∨ (cid:27) Hom ( H V д , K д ) if and only if f ∨ = (− ) n + f and λ ( f ( x ) , ˜ x ) ∈ im ( ∂ n + ) holds for all x ∈ H V д . Here˜ x ∈ H W д , is a choice of preimage of x under the projection H W д , → H V д , but the value λ ( f ( x ) , ˜ x ) is independent of this choice ˜ x since K д is Lagrangian. For elements of the form Ψ ( ϕ ) , the first property Ψ ( ϕ ) ∨ = (− ) n + Ψ ( ϕ ) follows from the fact that ϕ preserves theform λ . To see the second, we note that, since K д is Lagrangian, the function α is additiveon K д , so it vanishes on it since the images of the second S n -factors of the connected sum W д , (cid:27) ♯ д ( S n × S n )\ D n in V д = ♮ д ( D n + × S n ) induce a basis of K д and have trivial normalbundle. Using the fact that ϕ ∈ G ext д preserves α and property (ii), we compute α ( ˜ x ) = α (cid:0) Ψ ( ϕ )( x ) + ˜ x (cid:1) = α (cid:0) Ψ ( ϕ )( x ) (cid:1) + α ( ˜ x ) + λ (cid:0) Ψ ( ϕ )( x ) , ˜ x (cid:1) · τ S n = α ( ˜ x ) + λ (cid:0) Ψ ( ϕ )( x ) , ˜ x (cid:1) · τ S n and conclude that λ (cid:0) Ψ ( ϕ )( x ) , ˜ x (cid:1) · τ S n vanishes, so λ (cid:0) Ψ ( ϕ )( x ) , ˜ x (cid:1) ∈ im ( ∂ n + ) holds as claimed.This proves that the image of Ψ is contained in M д , and to show that it agrees with it, weconsider the commutative diagram0 N д π Diff ( V д ) GL ( H V д )
00 ker ( G ext д → GL ( H V д )) G ext д GL ( H V д ) M д Ψ whose vertical arrow N д → ker ( G ext д → GL ( H V д )) is induced by the isomorphism (24).By [Wal65, Lem. 24], the vertical composition in the diagram agrees with the projection N д → M д in (22), so it is surjective. Consequently, Ψ is surjective as well and hence anisomorphism, which proves the second claim of the statement. To show the third, weobserve that, given that Ψ is an isomorphism and N д → M д is surjective, the diagramimplies that π Diff ( V д ) → G ext д is surjective and moreover that its kernel agrees with thekernel of N д → M д , which is H ∨ V д ⊗ Sπ n SO ( n ) as claimed.We now identify the actions as asserted. For the second extension, one can argue asfollows: an automorphism φ ∈ GL ( H V д ) acts on ker ( G ext д → GL ( H V д )) by conjugatingwith a choice of lift e φ ∈ G ext д , so the isomorphism Ψ is equivariant simply because of theidentity ( e φ ( ϕ − id ) e φ − ) = ( e φϕ e φ − − id ) in Hom ( H V д , K д ) for all ϕ ∈ Hom ( H V д , K д ) . For thefirst extension, we use that the left vertical composition in the diagram above agrees withthe projection N д → M д , so it suffices to show that the composition(25) ker ( π Diff D n ( V д ) → GL ( H V д )) (cid:27) −→ N д −→ ( π n SO ( n + )) H Vд of (24) with the projection is equivariant. From Wall’s definition [Wal63b, p. 267] of theinvariant β φ of an element φ in the kernel, we see that this composition can be described asfollows: representing a homology class [ e ] ∈ H V д by an embedded sphere e : S n → V д , wecan alter φ by an isotopy such that it preserves e pointwise. In this case, the derivative of φ restricts an automorphism of the normal bundle ν ( e ) (cid:27) ε n + which induces an element β φ ([ e ]) ∈ π n SO ( n + ) . This uses a trivialisation of ν ( e ) , but the resulting element β φ ([ e ]) is independent of this choice. From this description, the claimed equivariance is straight-forward to check, and the identification of the action of the last sequence follows fromthat of the first two by chasing through the diagram obtained by extending the diagramabove by taking vertical kernels. HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 21
To see that the second extension splits, note that the first S n -factors in the connectedsum decomposition of W д , = ♯ д S n × S n \ int ( D n ) ⊂ ♮ д D n + × S n = V д induces a splittingof the canonical map and thus an isomorphism of the form K д ⊕ H V д (cid:27) H W д , . Usingthis splitting of the homology, we can define a morphism GL ( H V д ) → G ext д by assigning ϕ ∈ GL ( H V д ) the automorphism H W д , (cid:27) K д ⊕ H V д ( ϕ − ) ∨ ⊕ ϕ −−−−−−−→ K д ⊕ H V д (cid:27) H W д , , which clearly preserves K д . Moreover, its induced automorphism of H W д , agrees with ϕ by construction, so we obtain a splitting as desired. (cid:3) Remark . With respect to the basis H W д , (cid:27) Z д suggested by the connected sum de-composition W д , (cid:27) ♯ д ( S n × S n )\ int ( D n ) , the subgroup G ext д ⊂ GL ( H W д , ) (cid:27) GL д ( Z ) agrees with the group of block matrices of the form (cid:18) A M ( A − ) T (cid:19) with M ∈ M д , using the matrix description of M д explained in Remark 3.2. From this pointof view, the splitting GL д ( Z ) → G ext д described in the proof of Theorem 3.3 is the obviousone that sends a matrix A ∈ GL д ( Z ) to the block diagonal matrix with M = Stable framings.
Choosing the canonical map sfr : EO → BO as a stable tangentialstructure in the sense of Section 1.8, the space of sfr-structuresBun D n ( τ sV д , sfr ∗ γ ; ℓ ) is the space of stable framings of V д relative to a fixed stable framing ℓ : τ sV д | D n → sfr ∗ γ .We denote the stabiliser of the canonical action of π Diff D n ( V д ) on the set of components π Bun D n ( τ sV д , sfr ∗ γ ; ℓ ) and its image in G ext д by π Diff D n ( V д ) ℓ ⊂ π Diff D n ( V д ) and G ext д ,ℓ ⊂ G ext д . Note that π Diff D n ( V д ) ℓ agrees with the image of the canonical map BDiff sfr D n ( V д ) → BDiff D n ( V д ) on fundamental groups based at the point induced by the stable framing ℓ ,or equivalently, with the kernel of the crossed homomorphism(26) π Diff D n ( V д ) −→ H ∨ V д ⊗ π n SOgiven by acting on ℓ . This uses the identification H ∨ V д ⊗ π n SO (cid:27) Hom ( H V д , π n SO ) (cid:27) π Maps D n ( V д , SO ) (cid:27) π Bun D n ( τ sV д , θ ∗ γ ; ℓ ) whose first two isomorphisms are the evident ones and whose third is induced by thechoice of stable framing ℓ , using that π Bun D n ( τ sV д , sfr ∗ γ ; ℓ ) is a torsor over the group π Maps D n ( V д , SO ) equipped with the pointwise multiplication. Remark . As the stabilisation map SO ( n + ) → SO is 2 n -connected, the induced mapMaps D n ( V д , SO ( n + )) → Maps D n ( V д , SO ) is n -connected, which implies that there is nodifference between equivalence classes of stable and unstable framings, so the discussionof this subsection applies equally well to unstable framings instead of stable ones.To relate the subgroups π Diff D n ( V д ) ℓ ⊂ π Diff D n ( V д ) and G ext д ,ℓ ⊂ G ext д to the se-quences established in Theorem 3.3, we define the GL ( H V д ) -submodule(1) N sfr д ⊂ N д as the intersection of N д with M д ⊕ h τ S n + i H Vд , where h τ S n + i is thesubgroup of π n SO ( n + ) generated by the tangent bundle τ S n + and(2) M sfr д ⊂ M д as the collection of (− ) n + -symmetric bilinear forms µ ∈ ( H V д ⊗ H V д ) ∨ that are even, i.e. µ ( x , x ) ∈ · Z for all x ∈ H V д , which is automatic if n is even. Standard arguments involving the long exact sequences in homotopy groups of the usualfibration SO ( d ) → SO ( d + ) → S d (cf. [Lev85, §1B)]) show that the sequence (22) restrictsto an exact sequence of GL ( H V д ) -modules of the form0 −→ H ∨ V д ⊗ ker (cid:0) Sπ n SO ( n ) → π n SO (cid:1) −→ N sfr д −→ M sfr д −→ . Proposition 3.7.
Let n ≥ .(i) The action of π Diff D n ( V д ) on the set π Bun D n ( τ sV д , sfr ∗ γ ; ℓ ) of equivalence classesof stable framings is transitive.(ii) For any stable framing ℓ : τ sV д → sfr ∗ γ , the sequences of Theorem 3.3 restrict to exactsequences of the form −→ N sfr д −→ π Diff D n ( V д ) ℓ −→ GL ( H V д ) −→ , −→ M sfr д −→ G ext д ,ℓ −→ GL ( H V д ) −→ , and −→ H ∨ V д ⊗ ker (cid:0) Sπ n SO ( n ) → π n SO (cid:1) −→ π Diff D n ( V д ) ℓ −→ G ext д ,ℓ −→ . Remark . The calculation of im ( ∂ n + ) in Remark 3.2 shows that the inclusion M sfr д ⊂ M д is an equality for n , , ,
7, so the same holds for G ext д ,ℓ ⊂ G ext д as a result of Proposition 3.7. Proof of Proposition 3.7.
The first sequence of Theorem 3.3 fits into a diagram0 N д π Diff D n ( V д ) GL ( H V д ) H ∨ V д ⊗ π n SOwhose vertical map is the crossed homomorphism (26), which is up to isomorphism givenby the action of π Diff D n ( V д ) on ℓ , so the first part of the statement is equivalent to its sur-jectivity. The diagonal arrow is the morphism which assigns an element ( µ , β ) ∈ N д to thecomposition of β : H V д → π n SO ( n + ) with the stabilisation map π n SO ( n + ) → π n SO.As τ S n + ∈ π n SO ( n + ) is stably trivial, it follows from the second defining propertyof N д that this composition is additive, so indeed defines an element of H ∨ V д ⊗ π n SO (cid:27) Hom ( H V д , π n SO ) . As the morphism π n SO ( n + ) → π n SO is surjective, the diagonal arrowin the above diagram is surjective as well, which reduces the first claim to the commuta-tivity of the triangle. The latter follows from the description of the composition (25) wegave in the proof of Theorem 3.3 in a straight-forward manner.The kernel of the vertical map agrees with the stabiliser π Diff D n ( V д ) ℓ , so the diagramalso shows that this stabiliser surjects onto GL ( H V д ) and that its kernel agrees with thekernel of the diagonal arrow. But this kernel is exactly the submodule N sfr д ⊂ N д , becausethe kernel of the stabilisation map π n SO ( n + ) → π n SO agrees with the kernel of π n SO ( n + ) → π n SO ( n + ) , i.e. the subgroup generated by τ S n + . This establishes the first sequenceof the second claim. For the second, note that the surjectivity of the morphism π Diff D n ( V д ) ℓ −→ GL ( H V д ) implies that also the morphism G ext д ,ℓ −→ GL ( H V д ) is surjective. Comparing the first two sequences of Theorem 3.3, we see that its kernelagrees with the image of N sfr д in M д under the projection N д → M д , i.e. with those bilinearforms µ ∈ M д that satisfy µ ( x , x ) ∈ im ( ∂ n + h τ S n + i) . The image of τ S n + ∈ π n SO ( n + ) under ∂ n + is the Euler characteristic of S n + (see [Lev85, §1B)]), so the kernel in questionagrees with M sfr д as claimed. To establish the last sequence, one first observes that π Diff D n ( V д ) ℓ −→ G ext д ,ℓ HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 23 is surjective by definition of the target, and then compares the first two sequences in theclaim to see that its kernel agrees with the kernel of N sfr д → M sfr д , which agrees with H ∨ V д ⊗ ker ( Sπ n SO ( n ) → π n SO ) as already noted in the discussion prior to this proof. (cid:3) Lemma 3.9.
The negative of the identity − id ∈ GL ( H W д ) is contained in the subgroup G ext д ,ℓ ⊂ GL ( H W д ) for all stable framings ℓ : τ sV д → sfr ∗ γ .Proof. We first restrict to the case д = D n + × S n ⊂ R n + × R n + . The diffeomorphism(27) D n + × S n −→ D n + × S n (( x , . . . , x n + ) , ( y , . . . , y n + )) 7−→ ((− x , x , . . . , x n + ) , (− y , y , . . . , y n + )) is orientation preserving, maps to − id ∈ GL ( H W ) , and has constant derivative, so it pre-serves the stable framing as required. It does not preserve a disc in the boundary, but byLemma 3.1, we can rectify this by an isotopy. To extend this argument to higher genera,we take the boundary connected sum of two copies of this diffeomorphism using the fixeddiscs in the boundary to obtain a diffeomorphism of V whose image in GL ( H W ) is − id,which we can again isotope so it preserves a disc as required. Continuing like this yieldsa sequence of isotopy classes in π Diff D n ( V д ) for all д ≥ д , butimplies the general case, the reason being that − id ∈ GL ( H W д ) is central and all subgroups G ext д ,ℓ ⊂ G ext д are conjugate, because the different stabilisers π Diff D n ( V д ) ℓ are as the actionis transitive by Proposition 3.7. This concludes the proof. (cid:3) The homotopy mapping class group.
Not only the smooth mapping class groups π Diff D n ( V д ) and π Diff ∂ ( W д , ) act on the homology H W д , , also their homotopical cousins π hAut D n ( V д , W д , ) and π hAut ∂ ( W д , ) do. Restricting these actions to the images π hAut (cid:27) D n ( V д , W д , ) ⊂ π hAut D n ( V д , W д , ) and π hAut (cid:27) ∂ ( W д , ) ⊂ π hAut ∂ ( W д , ) of the canonical maps (see Section 4 for the notation) π Diff D n ( V д , W д , ) −→ π hAut D n ( V д , W д , ) and π Diff ∂ ( W д , ) −→ π hAut ∂ ( W д , ) , they land in the above described subgroups G ext д and G д of GL ( H W д , ) , respectively. Lemma 3.10.
Let n ≥ . The morphisms induced by the action on H W д , = H n ( W д , ; Z ) π hAut (cid:27) ∂ ( W д , ) −→ G д and π hAut (cid:27) D n ( V д , W д , ) −→ G ext д are surjective. Moreover, their kernels are finite and p -torsion free as long as n < p − .Proof. In the course of this proof, we shall make frequent use of the fact that the homotopygroups π n + k (∨ h S n ) of a bouquet of n -spheres with n ≥ p -torsion free for k < p − h = π Diff ∂ ( W д , ) → G д and π Diff ( V д ) → G ext д by Theorem 3.3 and the discussion it precedes. Evidently, the kernel of the first morphismof the statement is contained in that of π hAut ∂ ( W д , ) → GL ( H W д , ) , which enjoys theclaimed finiteness and torsion property by an application of the fibre sequencehAut ∂ ( W д , ) −→ hAut ∗ ( W д , ) −→ Maps ∗ ( ∂ W д , , W д , ) , using the canonical isomorphism π hAut ∗ ( W д , ) (cid:27) GL ( H W д , ) and the fact that the group π Maps ∗ ( ∂ W д , , W д , ) (cid:27) π n (∨ д S n ) is finite and without p -torsion for n < p −
3. To de-duce the claim for the kernel of second morphism from this, we consult the fibre sequencehAut ∂ ( V д ) −→ hAut D n ( V д , W д , ) −→ hAut ∂ ( W д , ) to realise that it suffices to show that π hAut ∂ ( V д ) is finite and p -torsion free for n < p − ∂ ( V д ) −→ hAut ∗ ( V д ) −→ Maps ( W д , V д ) and the observation that its fibre inclusion is trivial on path components since π hAut ∂ ( V д ) acts trivially on homology and π hAut ∗ ( V д ) (cid:27) GL ( H V д ) , we see that π hAut ∂ ( V д ) is aquotient of the fundamental group π Maps ∗ ( W д , V д ) based at the inclusion ι . Finally, notethat the fibre sequenceMaps ∗ ( W д , V д ) −→ Maps ∗ ( W д , , V д ) −→ Maps ∗ ( ∂ W д , , V д ) induces an exact sequence π n + (∨ д S n ) ⊕ д −→ π ( Maps ∗ ( W д , V д ) ; ι ) −→ π n + (∨ д S n ) ⊕ д whose outer groups are finite and p -torsion free for n < p −
4, so the claim follows. (cid:3)
Stabilisation.
To relate the automorphism spaces of the handlebody V д = ♮ д D n + × S n relative to the chosen disc D n ⊂ ∂ V д to the ones of V д + , it is convenient to modify V д by introducing codimension 2 corners at the boundary of the disc D n ⊂ V д in theboundary so that there is smooth boundary preserving embedding c : (− , ] × D n → V д whose restriction to { } × D n agrees with the chosen disc. Abusing common terminology,we call such an embedding a collar . Fixing another disc D n − ⊂ ∂ D n , we consider H ≔ ([ , ] × D n ) ♮ ( D n + × S n ) , where the boundary connected sum is performed away from the union D ≔ { , } × D n ∪ [ , ] × D n − , and think of V д + as being obtained by gluing V д to H along the two collared discs D n ⊂ V д and { } × D n ⊂ H , where we declare { } × D n ⊂ H ⊂ V д + to the new distinguisheddisc in the boundary, which comes with a preferred collar. Given a tangential structure θ : B → BO ( d ) , a choice of bundle map ℓ : ε ⊕ τ D n → θ ∗ γ n + induces canonical θ -structures on τ V д | D n and τ V д + | D n by using the fixed collars, and also one on τ H | D bymaking use of the canonical trivialisation of τ [ , ] . With respect to these θ -structures,which we generically denote by ℓ , there is an evident gluing map for tangential structuresBun D n ( τ V д , θ ∗ γ n + ; ℓ ) × Bun D ( τ H , θ ∗ γ n + ; ℓ ) −→ Bun D n ( τ V д + , θ ∗ n + γ ; ℓ ) that is equivariant with respect to the gluing morphism(28) Diff D n ( V д ) × Diff D ( H ) −→ Diff D n ( V д + ) for diffeomorphisms. Taking homotopy orbits, this induces a map of the formBDiff θD n ( V д ; ℓ ) × BDiff θD ( H ; ℓ ) −→ BDiff θD n ( V д + ; ℓ ) and hence a homotopy class of stabilisation maps (29) BDiff θD n ( V д ; ℓ ) → BDiff θD n ( V д + ; ℓ ) that in general depends on the choice of a component of BDiff θD ( H ; ℓ ) , but not in any ofthe cases we shall be interested in, because of the following lemma. Lemma 3.11. If B is n -connected, then BDiff θD ( H ; ℓ ) is nonempty and connected.Proof. Up to smoothing corners, the pairs ( H , D ) and ( V , D n ) are diffeomorphic, so therewe have an equivalence BDiff θD ( H ; ℓ ) ≃ BDiff θD n ( V ; ℓ ) , which shows that the claim isequivalent to the transitivity of the action of π Diff D n ( V ) on π Bun D n ( τ V , θ ∗ γ n + ; ℓ ) ,using that the latter set is nonempty as V is parallelizable. As B is n -connected, it is aconsequence of obstruction theory that every θ -structure on V д is induced by a framing,so it suffices to consider the case θ : EO ( n + ) → BO ( n + ) , which we have alreadysettled as the first part of Proposition 3.7 (see also Remark 3.6). (cid:3) HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 25
A similar discussion results in analogous stabilisation maps of the formB g Diff Ξ D n ( V д ; ℓ ) −→ B g Diff Ξ D n ( V д + ; ℓ ) for stable tangential structures Ξ : B → BO and bundle maps ℓ : τ sV д → Ξ ∗ γ , which arecompatible with the non-block variants defined above and are again unique up to homo-topy if B is n -connected. Moreover, using the splittings of the inclusions H W д , → H W д + , and H V д → H V д + suggested by the decomposition V д + = V д ∪ H , there are stabilisationmaps for the respective linear groups on H W д , and H V д given by extending automorphismsby the identity and these are related to the stabilisation maps described above via the ac-tion on the homology of V д and W д , (see Section 3.2), so there is commutative diagram ofcompatible stabilisation maps that has the formBDiff Ξ n + D n ( V д ; ℓ ) B g Diff Ξ D n ( V д ; ℓ ) BG ext д BGL ( H V д ) BDiff Ξ n + D n ( V д + ; ℓ ) B g Diff Ξ D n ( V д + ; ℓ ) BG ext д + BGL ( H V д + ) .
4. Relative homotopy automorphisms of handlebodies
Theorem 2.2 illustrates that the space of block diffeomorphisms g Diff D n ( V д ) is closely re-lated to the space (cid:157) hAut D n ( V д , W д , ) of relative block homotopy automorphisms or, equiva-lently, to its non-block variant hAut D n ( V д , W д , ) (see Section 1.5). To access the homologyof the classifying space of this space of homotopy automorphisms, one might try to studythe Serre spectral sequence of the fibration sequenceBhAut id D n ( V д , W д , ) −→ BhAut D n ( V д , W д , ) −→ B π hAut D n ( V д , W д , ) for which one ought to know at least the homology of the fibre as a module over the group π hAut D n ( V д , W д , ) . This is what this section aims to compute— p -locally and in a rangeof degrees—by first calculating the p -local homotopy groups in a range using some toolsfrom rational homotopy theory combined with an ad-hoc extension to the p -local settingtailored to our situation, and then pass from homotopy to homology groups.4.1. Conventions on gradings.
Essentially all objects in this section carry a Z -grading,and we shall keep track of it throughout. For instance, we consider the (reduced) homologyof a space X always with its natural grading, even if it is supported in a single degree.We denote the k -fold suspension of a graded R -module A over some commutative ring R by s k A , the free graded commutative R -algebra on A by S ∗ A , the graded R -module whosedegree k piece consists of R -module morphisms raising the degree by k by Hom ( A , B ) forgraded modules A and B , and the dual of A by A ∨ ≔ Hom ( A , R [ ]) where R [ ] is the basering concentrated in degree zero. The graded tensor product A ⊗ B is defined in the usualway. The degreewise rationalisation or p -localisation of a graded Z -module A is denotedby A Q or A ( p ) respectively and we view it as a graded Q - respectively Z ( p ) -module.4.2. Lie algebras and their derivations.
We consider differential graded (short dg) Liealgebras over a commutative ring R . However, most of the dg Lie algebras which we shallencounter actually have trivial differential. Examples include the free Lie algebra L ( V ) ona graded R -module A or the onefold shift of the homotopy groups π ∗ + X of a based space X with its canonical Lie algebra structure over Z given by the Whitehead bracket. Wewrite L + for the positive truncation of a dg Lie algebra L , i.e. the sub dg Lie algebra whichcoincides with L in all degrees ≥
2, agrees with the kernel of the differential d : L → L in degree 1, and is trivial in nonpositive degrees. The guiding principle of this section isthat the homotopy type of mapping spaces is closely related to certain chain complexes of f -derivations by which we mean the following: for a morphism f : ( L , d L ) → ( L ′ , d ′ ) of dg Lie algebras, an f -derivation of degree k is a linear map θ : L → L ′ that raises the degreeby k and satisfies the identity θ ([ x , y ]) = [ θ ( x ) , f ( y )] + (− ) k | x | [ f ( x ) , θ ( y ] . These derivations form the degree k piece of the chain complex Der f ( L , L ′ ) of f -derivations over R whose differential is defined as d ( θ ) = d L ′ θ − (− ) | θ | θd L , so it vanishes if both L and L ′ have trivial differential. Given a cycle ω ∈ L , we denote the subcomplex of f -derivations that vanish on ω by Der fω ( L , L ′ ) ⊂ Der f ( L , L ′ ) . In the case L = L ′ and f = id,we abbreviate the complex of id-derivations Der id ( L , L ) by Der ( L ) .4.3. Rational homotopy theory, Quillen style.
Recall Quillen’s functor λ , which as-signs a simply connected based space X a dg Lie algebra λ ( X ) over the rationals, one ofwhose many properties is that it captures the rationalised homotopy Lie algebra of X viaa natural isomorphism H ∗ ( λ ( X )) (cid:27) π ∗ + ( X ) Q of graded Lie algebras. A Lie model of X is a rational dg Lie algebra L Q X quasi-isomorphic to λ ( X ) . Such a model is called free ifthe underlying graded Lie algebra of L Q X is isomorphic to a free graded Lie algebra L ( V ) on a graded Q -vector space V and minimal if it is free and has decomposable differen-tial, i.e. d ( L Q X ) ⊂ [ L Q X , L Q X ] . Any simply connected based space has a minimal Lie model L Q X , unique up to (non-canonical) isomorphism, and a based map between such spaces f : X → Y gives rise to a map f : L Q X → L Q Y between their minimal models.4.4. Derivations and mapping spaces.
As mentioned earlier, the homotopy theory ofmapping spaces is tightly connected to derivations of dg Lie algebras. In the rationalsetting, this is made precise for instance by a result of Lupton–Smith [LS07, Thm 3.1].The version of their result we shall need is marginally stronger than stated in [LS07], butfollows from the given proof in a straight-forward way (see also [BM20, Thm 3.6]).
Theorem 4.1 (Lupton–Smith) . Let f : X → Y be a map between simply connected finitebased CW-complexes, with minimal Lie model f : L Q X → L Q Y . There is an isomorphism π ∗ ( Maps ∗ ( X , Y ) ; f ) Q (cid:27) −→ H ∗ ( Der f ( L Q X , L Q Y )) for ∗ ≥ , which is natural in both X and Y . For X a co-H-space, this also holds for ∗ = . A p -local generalisation. From the point of view of Quillen’s approach to rationalhomotopy theory, the spaces we shall be applying Theorem 4.1 to are of the simplestnature possible: they are equivalent to boquets of equidimensional spheres. The minimalmodel of such a space X ≃ ∨ д S n for n ≥ L Q X = L ( s − H Q X ) (cid:27) π ∗ + X Q on H Q X ≔ e H ∗ ( X ; Q ) , equipped with the trivial differential. Given a map between spaces of this kind, the inducedmap on minimals models is simply given by the induced map on rational homotopy groups.It is a consequence of the Hilton–Milnor theorem that, p -locally in small degrees withrespect to p , the homotopy Lie algebra π ∗ + X is free even before rationalisation. To makethis precise, we abbreviate the integral and p -local analogue of (30) by L X ≔ L ( s − H X ) and L ( p ) X ≔ L ( s − H ( p ) X ) where H X ≔ e H ∗ ( X ; Z ) and H ( p ) X ≔ e H ∗ ( X ; Z ( p ) ) . The inverse of the Hurewicz map H n ( X ; Z ( p ) ) (cid:27) π n X ( p ) induces a map L ( p ) X → π ∗ + X ( p ) ofgraded Lie algebras over Z ( p ) , which turns out to be an isomorphism in a range of degrees. Lemma 4.2.
For a based space X ≃ ∨ д S n with n ≥ and an odd prime p , the morphism L ( p ) X −→ π ∗ + X ( p ) is an isomorphism on torsion free quotients. Moreover, the right hand side is torsion free indegrees ∗ < p − + n , so the map is an isomorphism in this range. HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 27
Proof.
The rationalisation of this morphism agrees with (30), so to prove the first part ofthe statement, it suffices to show that all classes in π ∗ + X ( p ) of infinite order are in the im-age. By the Hilton–Milnor theorem, every class in π k X ( p ) can be written as a composition ( y ◦ x ) of some x ∈ π k S m ( p ) with m ≥ n and a class y ∈ π m X ( p ) in the image of the map inquestion. The group π k S m ( p ) is finite unless k = m , where it is generated by the identity, or m = n and k = n −
1, where it is is generated by [ id S n , id S n ] , since this element hasHopf invariant 2 and we assumed p to be odd. As [ id S n , id S n ] ◦ x = [ x , x ] ∈ π n − X ( p ) and the image of L ( p ) X → π ∗ + X ( p ) is closed under taking brackets, this implies the firstpart of the claim. The second part follows from Serre’s result that π k S m is p -torsion freefor k − m < p − (cid:3) As a result of Lemma 4.2, every map f : X → Y between bouquets of equidimensionalspheres induces a morphism f ∗ : L ( p ) X → L ( p ) Y by taking torsion free quotients of p -localhomotopy groups, so the following extension of Theorem 4.1 might not come as a surprise. Proposition 4.3.
For a map f : X → Y between based spaces X ≃ ∨ д S n and Y ≃ ∨ h S m with n , m ≥ , the morphism of Theorem 4.1 fits into a commutative diagram for ∗ > π ∗ ( Maps ∗ ( X , Y ) ; f ) ( p ) Der f ( L ( p ) X , L ( p ) Y ) π ∗ ( Maps ∗ ( X , Y ) ; f ) Q Der f ( L Q X , L Q Y ) (−)⊗ Q (−)⊗ Q (cid:27) which is natural in X and Y and whose upper arrow is an isomorphism for ∗ < p − −( n − m ) .Remark . Dwyer’s tame homotopy theory [Dwy79] provides a p -local generalisation ofQuillen’s rational homotopy theory for primes p that are just large enough with respectto the degree to prevent stable k -invariants from appearing. It is not unlikely that Theo-rem 4.1 could be generalised to this setting, but our layman extension Proposition 4.3 forbouquets of spheres suffices for the applications we have in mind. Proof of Proposition 4.3.
We begin with a twofold simplification of the statement. Firstly,the claimed naturality is automatic, since the vertical maps are evidently natural, the bot-tom map is natural by Theorem 4.1, and the right vertical one is injective, so it sufficesto construct a top arrow with the desired properties for X = ∨ д S n . Secondly, there is acommutative diagram Der f ( L ( p ) X , L ( p ) Y ) Hom ( s − H ( p ) X , L ( p ) Y ) Der f ( L Q X , L Q Y ) Hom ( s − H Q X , L Q Y ) , (cid:27) (−)⊗ Q (−)⊗ Q (cid:27) induced by restricting derivations to generators, which shows that it is enough to producea dashed arrow making the diagram(31) π ∗ ( Maps ∗ ( X , Y ) ; f ) ( p ) Hom ( s − H ( p ) X , L ( p ) Y ) π ∗ ( Maps ∗ ( X , Y ) ; f ) Q Der f ( L Q X , L Q Y ) Hom ( s − H Q X , L Q Y ) , (−)⊗ Q (−)⊗ Q (cid:27) (cid:27) commute. To do so, we consider the composition(32) π ∗ ( Maps ∗ ( X , Y ) ; f ) (− f ) ∗ −−−−→ (cid:27) π ∗ ( Maps ∗ ( X , Y ) ; ∗) (cid:27) Hom ( s − H X , π ∗ + Y ) whose first isomorphism is given by acting with the inverse of f , using the loop spacestructure on Maps ∗ ( X , Y ) , and whose second isomorphism is induced by mapping a class in π k ( Maps ∗ ( X , Y ) ; ∗) represented by a pointed map д : S k → Maps ∗ ( X , Y ) to the compositionH n ( X ) (cid:27) H n + k ( S k ∧ X ) (cid:27) π n + k ( S k ∧ X ) д ∗ −→ π n + k ( Y ) , involving the suspension isomorphism, the inverse of the Hurewicz map, and the adjointof д . Postcomposing (32) with the map of Lemma 4.2 given by taking p -localisation andtorsion free quotients results in a dashed map with the claimed connectivity, so we are leftto show that this choice does make the diagram (31) commute, i.e. that the rationalisationof (32) agrees with the bottom map of (31). The adjoint of a map h : S k → Maps ∗ ( X , Y ) representing a class in π ∗ ( Maps ∗ ( X , Y ) ; f ) forms the top arrow of the diagram S k × X YS k + ∧ XX ∨ S k ∧ X , h ≃ f ∨(− f ) ∗ ( h ) whose middle diagonal arrow is induced by h via the canonical homeomorphism ( S k × X )/( S k ∨ ∗) (cid:27) S k + ∧ X and whose vertical equivalence is given as the composition S k + ∧ X id Sk + ∧∇ −−−−−−→ S k + ∧ X ∨ S k + ∧ X c −→ X ∨ S k ∧ X using the co-H-space structure ∇ of X and the evident collapse map c . The map (− f ) ∗ ( h ) is the adjoint of a representative of the image of h under the first map in (32), so thediagram commutes up to changing h within its class in π k ( Maps ∗ ( X , Y ) ; f ) . We thus obtaina rational model for the top arrow as the composition (cid:16) L (cid:16) s − H Q X , s − + k H Q X , s − H Q S k (cid:17) , d (cid:17) −→ L Q S k ∨ S k ∧ X π ∗ + ( f ∨(− f ) ∗ ( д ))⊗ Q −−−−−−−−−−−−−−−→ L Q Y , where the source is the Lie model of S k × X described in [LS07, Cor. 2.2], i.e. its differential d is trivial except on elements x ∈ s − + k H Q X on which it is given by the bracket (− ) k − [ z , x ′ ] ,where z ∈ s − H Q S k is the generator and x ′ ∈ s − H Q X the element corresponding to x underthe canonical identification s − H Q X (cid:27) s − + k H Q X of ungraded vector spaces. The first mapin the composition takes the quotient by the dg Lie subalgebra generated by s − H Q S k andthe second map is defined as indicated. Using this particular choice of rational model inthe definition of the isomorphism of Theorem 4.1 in [LS07, p. 176–177], the image of theclass defined by h under the bottom horizontal composition in (31) is precisely its imageunder (32) after rationalisation, so the claim follows. (cid:3) p -local homotopy groups of BhAut id D n ( V д , W д , ) . The theory set up in the previ-ous paragraphs will allow us to compute the p -local homotopy groups of the classifyingspace BhAut id D n ( V д , W д , ) of the identity component of the topological monoid of relativehomotopy automorphisms as defined in Section 3.1 as a module over π BhAut D n ( V д , W д , ) (cid:27) π hAut D n ( V д , W д , ) . We adopt the notation set up in the previous subsection for the three spaces involved ∂ W д , ⊂ W д , ⊂ V д that are equivalent to bouquets of spheres. In addition, we abbreviate K ( p ) д ≔ ker (cid:16) H ( p ) W д , → H ( p ) V д (cid:17) and L ( p ) K д ≔ ker (cid:16) L ( p ) W д , → L ( p ) V д (cid:17) (cid:27) L ( s − K ( p ) д ) HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 29 and omit the ( p ) -superscripts to denote their integral analogues. As a final piece of nota-tion, we let ω ∈ L W д , be the class representing the inclusion S n − = ∂ W д , ⊂ W д , of theboundary of W д , = ♯ д ( S n × S n )\ int ( D n ) , i.e. the attaching map ω = Í дi = [ e i , f i ] ∈ π n − W д , , where e i , f i ∈ π n W д , correspond to the first respectively second S n -summand in the i thsummand of W д , (cid:27) ♯ д S n × S n \ int ( D n ) ≃ ∨ д ( S n ∨ S n ) . As the inclusion ∂ W д , ⊂ V д is nullhomotopic, this element is contained in the kernel L K д ⊂ L W д , , but although it is decomposable in L W д , , it is not decomposable in L K д . Theorem 4.5.
For n ≥ , there is an isomorphism of graded π hAut D n ( V д , W д , ) -modules π ∗ + BhAut id D n ( V д , W д , ) ( p ) (cid:27) ker (cid:0) Der + ω ( L ( p ) W д , ) φ −→ Der , + ω ( L ( p ) K д , L ( p ) V д ) (cid:1) in degrees ∗ < p − − n , where the morphism φ is induced by the inclusion K д ⊂ H W д , andthe projection H W д , → H V д . In particular, Theorem 4.5 implies that the π hAut D n ( V д , W д , ) -action on the p -localhomotopy groups in consideration factors through the morphism π hAut D n ( V д , W д , ) l → { ϕ ∈ GL ( H W д , ) | ϕ ( K д ) ⊂ K д } induced by the action on the homology of W д , . During the proof of Theorem 4.5, which isdivided into a sequence of auxiliary lemmas, we frequently use Proposition 4.3 to implicitlyidentify p -local homotopy groups of path components of pointed mapping spaces betweenbouquets of spheres with derivations of Lie algebras in a range of degrees. Moreover, wegenerically write ι for any combination of the inclusions ∂ W д , ⊂ W д , ⊂ W д ⊂ V д and denote by Maps f ∗ ( X , Y ) for a map of based spaces f : X → Y the corresponding com-ponent of the mapping space, pointed by f . Lemma 4.6.
Let n ≥ . In the range ∗ < p − − n , the morphism induced by inclusion π ∗ Maps id ∂ ( W д , , W д , ) ( p ) −→ π ∗ Maps id ∗ ( W д , , W д , ) ( p ) (cid:27) Der + ( L ( p ) W д , ) is injective and has image Der + ω ( L ( p ) W д , ) ⊂ Der + ( L ( p ) W д , ) .Remark. Rationally, this lemma is due Berglund and Madsen [BM20, Prop. 5.6].
Proof of Lemma 4.6.
The inclusion ∂ W д , ⊂ W д , yields a fibration Maps ∗ ( W д , , W д , ) → Maps ∗ ( ∂ W д , , W д , ) whose fibre at ι is Maps ∂ ( W д , , W д , ) . The induced maps on homotopygroups fits in the range 0 < ∗ < p − − n into a diagram(33) π ∗ Maps id ∗ ( W д , , W д , ) ( p ) π ∗ Maps ι ∗ ( ∂ W д , , W д , ) ( p ) Der ( L ( p ) W д , ) Der ι ( L ( p ) ∂ W д , , L ( p ) W д , ) s −( n − ) H ( p ) W д , ⊗ L ( p ) W д , s −( n − ) [ L ( p ) W д , , L ( p ) W д , ] (cid:27) (cid:27)(cid:27)(cid:27) [− , −] whose upper square is provided by Proposition 4.3 and whose lower square is given asfollows: the right vertical map evaluates at the fundamental class [ ∂ W д , ] ∈ s − H ( p ) ∂ W д , , which provides an isomorphism as indicated for degree reasons, and the left vertical mapis given as the composition s −( n − ) H ( p ) W д , ⊗ L ( p ) W д , (cid:27) −→ ( s − H ( p ) W д , ) ∨ ⊗ L ( p ) W д , (cid:27) Hom ( s − H ( p ) W д , , L ( p ) W д , ) (cid:27) ←− Der ( L ( p ) W д , ) , where the first isomorphism is induced by the intersection form on H ( p ) W д , , the second isthe canonical one, and the third is given by restricting to generators. A calculation showsthat the resulting square commutes (see [BM20, Prop. 3.9] for a proof in the rational case,which also applies p -locally) and since the lower horizontal map is clearly surjective, themiddle horizontal arrow is as well, and hence so is the top one. A consultation of the longexact sequence in homotopy groups induced by the fibration we began with shows thatit only remains to identify the kernel of the middle horizontal arrow of the diagram withDer ω ( L ( p ) W д , ) . But this is clear because the compositionDer ( L ( p ) W д , ) −→ s −( n − ) [ L ( p ) W д , , L ( p ) W д , ] coincides with the evaluation at the class ω that represents the inclusion ∂ W д , ⊂ W д , . (cid:3) Lemma 4.7.
Let n ≥ . In the range ∗ < p − − n , the morphism induced by inclusion π ∗ Maps ι ∗ ( W д , V д ) ( p ) −→ π ∗ Maps ι ∗ ( W д , , V д ) ( p ) (cid:27) Der ι , + ( L ( p ) W д , , L ( p ) V д ) is injective and has image Der ι , + ω ( L ( p ) W д , , L ( p ) V д ) ⊂ Der ι , + ( L ( p ) W д , , L ( p ) V д ) .Proof. Using the fibration sequence induced by applying Maps ∗ (− , V д ) to the cofibrationsequence ∂ W д , → W д , → W д , the claim follows analogous to the proof of Lemma 4.6. (cid:3) Lemma 4.8.
Let n ≥ . The boundary map π ∗ Maps ι ∗ ( W д , V д ) −→ π ∗− Maps id ∂ ( V д , V д ) of the fibration Maps ∗ ( V д , V д ) → Maps ∗ ( W д , V д ) induced by the inclusion W д ⊂ V д is surjec-tive and induces in degrees ∗ < p − − n an isomorphism π ∗− Maps id ∂ ( V д , V д ) ( p ) (cid:27) Der , + ω ( L ( p ) K д , L ( p ) V д ) . Proof.
The morphism π ∗ Maps id ∗ ( V д , V д ) → π ∗ Maps ι ∗ ( W д , V д ) is injective because its compo-sition with the morphism π ∗ Maps ι ∗ ( W д , V д ) −→ π ∗ Maps ι ∗ ( W д , , V д ) induced by restrictionalong the inclusion W д , ⊂ W д is a retract as the inclusion ∨ д ( S n ∨ S n ) ≃ W д , ⊂ V д ≃ ∨ д S n is a homotopy retraction. The long exact sequence in homotopy groups induced by thefibration of the claim thus splits into short exact sequences, which are by Lemma 4.7 after p -locatlisation and in the range ∗ < p − − n of the form0 Der + ( L ( p ) V д ) Der ι , + ω ( L ( p ) W д , , L ( p ) V д ) π ∗− Maps id ∂ ( V д , V д ) ( p ) . Combining this with the isomorphismcoker (cid:0)
Der + ( L ( p ) V д ) → Der ι , + ω ( L ( p ) W д , , L ( p ) V д ) (cid:1) (cid:27) −→ Der , + ω ( L ( p ) K д , L ( p ) V д ) induced by the exact sequence 0 → K д → H W д , → H V д →
0, we conclude the claim. (cid:3)
Equipped with the previous three lemmas, we are ready to prove Theorem 4.5.
HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 31
Proof of Theorem 4.5.
Consider the map of horizontal fibration sequences(34) Maps ∂ ( V д , V д ) Maps D n (( V д , W д , ) , ( V д , W д , )) Maps ∂ ( W д , , W д , ) Maps ∂ ( V д , V д ) Maps ∗ ( V д , V д ) Maps ∗ ( W д , V д ) , where the middle vertical arrow is the evident one and the right vertical arrow is given byextending a selfmap of W д , relative to the boundary over the complement of W д , ⊂ W д by the identity, followed by postcomposition with the inclusion W д ⊂ V д . By naturality,the boundary map of the upper fibre sequence factors as π ∗ Maps id ∂ ( W д , , W д , ) −→ π ∗ Maps ι ∗ ( W д , V д ) −→ π ∗− Maps id ∂ ( V д , V д ) and is hence by Lemmas 4.6–4.8 in the range ∗ < p − − n and after p -localisation naturallyisomorphic to the morphism φ of the claim. Now consider the diagram(35) 0 Der ω ( L ( p ) W д , ) s −( n − ) H ( p ) W д , ⊗ L ( p ) W д , s −( n − ) [ L ( p ) W д , , L ( p ) W д , ]
00 Der ω ( L ( p ) K д , L ( p ) V д ) s −( n − ) H ( p ) V д ⊗ L ( p ) V д s −( n − ) [ L ( p ) V д , L ( p ) V д ] , φ [− , −][− , −] which commutes and has exact rows, where the upper row results from (33) and the loweris given analogously by using the canonical isomorphism K ∨ д (cid:27) H V д induced by the inter-section form on H W д , . The middle and right vertical maps are induced by the canonicalprojection H W д , → H V д . A choice of splitting of this projection induces compatible split-ting of the middle and right vertical maps and hence a splitting of φ , so φ is surjective.Combined with the long exact sequence in homotopy groups of the upper fibre sequenceof (34), we obtain a sequence of isomorphism π ∗ + BhAut id ( V д , W д , ) ( p ) (cid:27) π ∗ Maps id D n (( V д , W д , ) , ( V д , W д , )) ( p ) (cid:27) ker ( φ ) , valid in degrees ∗ < p − − n , and are left to show that this identification is equi-variant with respect to the canonical π hAut D n ( V д , W д , ) -action on both sides. With re-spect to the first isomorphism, the action on π ∗ + BhAut id ( V д , W д , ) agrees with the ac-tion on π ∗ Maps id D n (( V д , W д , ) , ( V д , W д , )) by conjugation, so it suffices to show that thesecond of the isomorphisms is equivariant. The latter is the restriction of the isomor-phism π ∗ Maps id ∂ ( W д , , W д , ) ( p ) (cid:27) Der ω ( L ( p ) W д , ) provided by Lemma 4.6, so it is enough toconvince ourselves of the equivariance of this isomorphism, which is, by another con-sultation of Lemma 4.6, implied by the π hAut ∗ ( W д , ) -equivariance of the isomorphism π ∗ hAut ∂ ( W д , ) ( p ) (cid:27) Der + ( L ( p ) W д , ) resulting from the naturality of Proposition 4.3. (cid:3) In a range of degrees, the particular shape of the p -local homotopy groups of the spaceBhAut id D n ( V д , W д , ) ensured by Proposition 4.3 allows us to pass from homotopy to homol-ogy groups, which is what we are actually interested in. Corollary 4.9.
For n ≥ , there is an isomorphism of graded π hAut D n ( V д , W д , ) -modules e H ∗ ( BhAut id D n ( V д , W д , ) ; Z ( p ) ) (cid:27) s − n (cid:0) ker ( S H ( p ) W д , ι ∗ −→ S H ( p ) V д ) (cid:1) in degrees ∗ < min ( n − , p − − n ) , where ι ∗ is induced by the inclusion W д , ⊂ V д .Proof. We may assume p >
3, since otherwise the claim has no content. (Anti-)symmetrisationinduces a canonical isomorphism [ s − H ( p ) W д , , s − H ( p ) W д , ] (cid:27) (( s − H ( p ) W д , ) ⊗ ⊗ ( Z −( p ) ) ⊗ n ) Σ and an analogous one for [ s − H ( p ) V д , s − H ( p ) V д ] , where the symmetric group acts on tensorpowers by permutation, (−) Σ denotes taking invariants, and Z −( p ) is the sign representationconcentrated in degree 0. Using this, a straight-forward calculation based on the two exactrows of (35) shows that there are natural isomorphismsDer + ω ( L ( p ) W д , ) (cid:27) s − n − (( H ( p ) W д , ) ⊗ ⊗ ( Z −( p ) ) ⊗ n ) Σ and Der , + ω ( L ( p ) K д , L ( p ) V д ) (cid:27) s − n − (( H ( p ) V д ) ⊗ ⊗ ( Z −( p ) ) ⊗ n ) Σ in degrees ∗ < n −
2. Taking quotients, these invariants are canonically isomorphic to s − n − S H ( p ) W д , respectively s − n − S H ( p ) V д , so together with Theorem 4.5, we arrive at anisomorphism of π hAut D n ( V д , W д , ) -modules π ∗ BhAut id D n ( V д , W д , ) ( p ) (cid:27) s − n ker (cid:0) S H ( p ) W д , → S H ( p ) V д (cid:1) in degrees ∗ < k ≔ min ( n − , p − − n ) . The right hand side of this isomorphism isconcentrated in degree n , so by truncating, we obtain a p -locally k -connected map(36) BhAut id D n ( V д , W д , ) −→ K ( A , n ) to the Eilenberg–MacLane space on the underlying ungraded module A ofker ( S H ( p ) W д , → S H ( p ) V д ) and thus leaves us with showing that K ( A , n ) has trivial Z ( p ) -homology in the range n < ∗ < k . As A is free as a Z ( p ) -module, it suffices to show that H ∗ ( K ( Z ( p ) , n ) ; Z ( p ) ) (cid:27) H ∗ ( K ( Z , n ) ; Z ( p ) ) vanishes in this range, which is certainly true rationally, so we may provethat H ∗ ( K ( Z , n ) ; F p ) vanishes for n + < ∗ < k + ∗ + n ( K ( Z , n ) ; F p ) −→ colim n H ∗ + n ( K ( Z , n ) ; F p ) = HF ∗ p ( HZ ) is an isomorphism in degrees ∗ < n , this follows from showing that the spectrum coho-mology HF ∗ p ( HZ ) vanishes in degrees ∗ < min ( n , p − − n ) . But HF ∗ p ( HZ ) is a quotient ofthe mod p Steenrod algebra HF ∗ p ( HF p ) by an ideal containing the Bockstein, so HF ∗ p ( HZ ) vanishes in degrees ∗ < p − (cid:3)
5. The proof of Theorem A
This final section is devoted to the proof of the following refinement of Theorem A.
Theorem 5.1.
For n > , there is a nilpotent space X and a zig-zag BC ( D n ) ϕ −→ X ψ ←− Ω ∞ + K ( Z ) such that p -locally, ϕ is min ( n − , p − − n ) -connected and ψ is min ( n , p − ) -connected.Remark . An extension of the proof of Theorem 5.1 presented in this section yields toan explicit description of the space X as an infinite loop space and moreover to a slightlystronger version of the theorem (see Remark 5.10).As a first step towards proving Theorem 5.1, we replace BC ( D n ) by an equivalentspace that is more convenient to compare to the various automorphism spaces of high-dimensional handlebodies V д = ♮ д D n + × S n studied in the previous sections. Lemma 5.3.
For d ≥ , there exists an equivalence BC ( D d ) ≃ g Diff D d ( D d + )/ Diff D d ( D d + ) HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 33
Proof.
A choice of identification D d × I (cid:27) D d + by smoothing corners induces an equiv-alence BC ( D d ) ≃ BDiff D d ( D d + ) , so the claim is equivalent to showing that the space ofblock diffeomorphisms g Diff D d ( D d + ) is contractible. This follows from Cerf’s result thatevery concordance of a manifold of dimension at least 5 is isotopic to the identity [Cer70],together with the chain of isomorphisms π k ( g Diff D d ( D d + ) ; id ) (cid:27) π Diff D d + k ( D d + k + ) (cid:27) π C ( D d + k ) whose first isomorphism is most easily seen from the combinatorial description of thehomotopy groups of the Kan complex g Diff D d ( D d + ) • . (cid:3) The alternative point of view on BC ( D n ) as g Diff D n ( D n + )/ Diff D n ( D n + ) = g Diff D n ( V )/ Diff D n ( V ) is advantageous as it makes a stabilisation map of the form(37) BC ( D n ) ≃ g Diff D n ( V )/ Diff D n ( V ) −→ g Diff D n ( V д )/ Diff D n ( V д ) , apparent, which is induced by iterating the stabilisation maps for BDiff D n ( V д ) and itsblock analogue explained in Section 3.5. It is a consequence of Morlet’s lemma of disjunc-tion that this map is highly connected: Lemma 5.4.
The stabilisation map (37) is ( n − ) -connected.Proof. Taking vertical homotopy fibres in the diagram (see Section 3.1 for the notation)BDiff ∂ ( V д ) BDiff D n ( V д ) BDiff ext ∂ ( W д , ) BDiff ∂ ( D n + ) BDiff D n ( D n + ) BDiff ext ∂ ( D n ) B g Diff ∂ ( V д ) B g Diff D n ( V д ) B g Diff ext ∂ ( W д , ) B g Diff ∂ ( D n + ) B g Diff D n ( D n + ) B g Diff ext ∂ ( D n ) of fibre sequences whose diagonal arrows are given by the iterated stabilisation mapsresults in a map of fibre sequences of the form g Diff ∂ ( V д ) Diff ∂ ( V д ) g Diff D n ( V д ) Diff D n ( V д ) g Diff ext ∂ ( W д , ) Diff ext ∂ ( W д , ) g Diff ∂ ( W д , ) Diff ∂ ( W д , ) g Diff ∂ ( D n + ) Diff ∂ ( D n + ) g Diff D n ( D n + ) Diff D n ( D n + ) g Diff ext ∂ ( D n ) Diff ext ∂ ( D n ) g Diff ∂ ( D n ) Diff ∂ ( D n ) , ≃≃ whose inner diagonal map is the map in question and whose rightmost equivalences followfrom another application of Cerf’s result mentioned in the previous proof. As the inclu-sions D n + ⊂ V д and D n ⊂ W д , have relative handle dimension n and the manifolds V д and W д , are ( n − ) -connected, the two outer diagonal maps are ( n − ) -connected bya form of Morlet’s lemma of disjunction [BLR75, p. 28, Lem. a], so the claim follows fromthe induced ladder of long exact sequences in homotopy groups. (cid:3) Combining the previous two lemmas results in a ( n − ) -connected map(38) BC ( D n ) −→ g Diff D n ( V ∞ )/ Diff D n ( V ∞ ) ≔ hocolim д g Diff D n ( V д )/ Diff D n ( V д ) to the homotopy colimit over the stabilisation maps, so in order to prove Theorem 5.1,it remains to establish a zig-zag with the claimed connectivity properties between this homotopy colimit and the unit component of the once looped algebraic K -theory space ofthe integers Ω ∞ + K ( Z ) , which we model as the plus construction Ω ∞ + K ( Z ) ≃ BGL ∞ ( Z ) + of the homotopy colimit BGL ∞ ( Z ) ≔ hocolim д BGL д ( Z ) over the stabilisation maps in-duced by the usual block inclusions GL д ( Z ) ⊂ GL д + ( Z ) . The zig-zag we construct arisesas part of a zig-zag of horizontal homotopy fibre sequences of the form(39) g Diff D n ( V ∞ )/ Diff D n ( V ∞ ) BDiff sfr D n ( V ∞ ; ℓ ) ℓ B g Diff sfr D n ( V ∞ ; ℓ ) ℓ X BDiff sfr ( V ∞ ; ℓ ) + ℓ BGL ∞ ( Z ) + Ω BGL ∞ ( Z ) + ∗ BGL ∞ ( Z ) + h i , (cid:13) (cid:13) (cid:13) which we shall explain now. Denoting the tangential structure encoding stable framingsby sfr : EO → BO, the upper right corner is defined as the homotopy colimitB g Diff sfr D n ( V ∞ ; ℓ ) ℓ ≔ hocolim д B g Diff sfr D n ( V ∞ ; ℓ ) ℓ along the stabilisation maps explained in Section 3.5, and the space BDiff sfr D n ( V ∞ ; ℓ ) ℓ isdefined analogously, using the unstable tangential structure induced by sfr : EO → BO,which we denote by the same symbol to simplify the notation (see Section 1.8). The upperright horizontal map is the homotopy colimit of the comparison mapBDiff sfr D n ( V д ; ℓ ) ℓ −→ B g Diff sfr D n ( V д ; ℓ ) ℓ whose homotopy fibre at the base point is canonically equivalent to g Diff D n ( V ∞ )/ Diff D n ( V ∞ ) in view of the homotopy cartesian square (12) and the fact that homotopy fibres commutewith sequential homotopy colimits. Using a functorial model of the plus construction, theupper right square is induced by the homotopy colimit of the compositionBDiff sfr ( V д ; ℓ ) ℓ −→ B g Diff sfr ( V д ; ℓ ) ℓ −→ BGL ( H V д ) (cid:27) BGL д ( Z ) along the stabilisation maps (see Section 3.5), where the second map is induced by theaction on the middle homology of V д . The space X is defined as the homotopy fibre ofthe map BDiff sfr ( V ∞ ; ℓ ) + ℓ → BGL ∞ ( Z ) + , and it receives a map from the top left cornerinduced by the commutativity of the upper right square. The bottom row is induced bythe inclusion of the basepoint in the universal cover of BGL ∞ ( Z ) + whose homotopy fibreagrees with the base point component of the loop space of BGL ∞ ( Z ) + . This explains thediagram (39), aside from the map of fibre sequences from the bottom to the middle row,which is induced by the 1-connected cover BGL ∞ ( Z ) + h i → BGL ∞ ( Z ) + and the basepointinclusion of BDiff sfr ( V ∞ ; ℓ ) + ℓ .In the two following subsections, we analyse the vertical maps 1 (cid:13) and 2 (cid:13) in order toprepare the proof of Theorem 5.1.5.1. The stable homology of
BDiff D n ( V д ) and the map (cid:13) . Botvinnik and Perlmutter[BP17] have computed the stable homology of BDiff θD n ( V д ; ℓ ) ℓ in homotopy theoreticalterms for all tangential structures θ : B → BSO ( n + ) whose space B is n -connected. Forus, their main result [BP17, Cor. 6.8.1, Prop. 6.14] is most conveniently expressed as anidentification of the group completion of the disjoint union M θ ≔ Ý д ≥ BDiff θD n ( V д ; ℓ ) ℓ , Unless said otherwise, we take all plus constructions with respect to the unique maximally perfect subgroup.
HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 35 which becomes a homotopy commutative topological monoid under boundary connectedsum when choosing an appropriate point-set model (see [BP17, Prop. 6.11, 6.14]).
Theorem 5.5 (Botvinnik–Perlmutter) . For n > , a tangential structure θ : B → BSO ( n + ) for which B is n -connected, and a θ -structure ℓ : τ V д → θ ∗ γ n + , it holds that Ω B M θ ≃ Ω ∞ Σ ∞ + B . As M θ is homotopy commutative, Randal-Williams’ elucidation of the group comple-tion theorem [RW13, Cor. 1.2] provides an equivalence of the form(40) BDiff θD n ( V ∞ ; ℓ ) + ℓ ≃ Ω B M θ , leading to the following consequence of Theorem 5.5 when specialised to the tangentialtangential structure encoding stable framings. Corollary 5.6.
The space
BDiff sfr D n ( V ∞ ; ℓ ) + ℓ is nilpotent and p -locally min ( n , p − ) -connected as long as n > .Proof. Connected H -spaces are nilpotent, so (40) settles the nilpotency claim. Regardingthe connectivity part of the statement, note that the unstable ( n + ) -dimensional tan-gential structure induced by the stable tangential structure sfr : EO → BO is equivalentto the inclusion O / O ( n + ) → BO ( n + ) of the homotopy fibre of the stabilisation mapBO ( n + ) → BO, so the discussion preceding this corollary shows that the space in ques-tion is equivalent to Ω ∞ Σ ∞ + O / O ( n + ) . This space has the claimed connectivity property,because O / O ( n + ) is 2 n -connected, so the homotopy groups of Ω ∞ Σ ∞ + O / O ( n + ) agreein positive degrees less than 2 n + p -torsion in degrees less than 2 p − (cid:3) In other words, for n >
3, the base point inclusion 3 (cid:13) of the space BDiff sfr D n ( V ∞ ; ℓ ) + ℓ is p -locally min ( n , p − ) -connected.5.2. The homology action and the map (cid:13) . As a consequence of Proposition 3.7, themap on fundamental group induced by 2 (cid:13) has infinite kernel, so the map 2 (cid:13) is as far frombeing highly connected as possible, even p -locally. Nevertheless, it turns out that it doesinduce an isomorphism on p -local homology groups in a range of degrees, which we shallprove by studying the effect on homology of the two maps in the factorisation(41) B g Diff sfr D n ( V д ; ℓ ) ℓ −→ BG ext д ,ℓ −→ BGL ( H V д ) of the canonical map B g Diff sfr D n ( V д ; ℓ ) ℓ → BGL ( H V д ) one at a time (see Section 3.3). Lemma 5.7.
For n ≥ and a prime p , the induced map H ∗ (cid:0) B g Diff sfr D n ( V д ; ℓ ) ℓ ; Z ( p ) (cid:1) −→ H ∗ (cid:0) BG ext д ,ℓ ; Z ( p ) (cid:1) is an isomorphism for ∗ < min ( n − , p − − n ) and a surjection in that degree.Proof. The claim is vacuous if p is even, so we assume otherwise. Consider the factorisa-tion of the map in questionB g Diff sfr D n ( V д ; ℓ ) ℓ −→ B (cid:157) hAut (cid:27) D n ( V д , W д , ) ℓ −→ B π (cid:157) hAut (cid:27) D n ( V д , W д , ) ℓ −→ BG ext д ,ℓ , where the first map is that of Corollary 2.4 applied to the triad ( V д ; D n , W д , ) , the secondmap is induced by taking path components, and the third is given by acting on the homol-ogy of the boundary of V д (see Section 3.3). By the corollary just mentioned, the first mapinduces an isomorphism in homology with Z ( p ) -coefficients in degrees ∗ < p − − n anda surjection in that degree since V д is obtained from D n by attaching n -handles and thetriad ( V д ; D n , W д , ) satisfies the π - π -condition as W д , and V д are simply connected for n ≥
2. To study the remaining maps, we may replace the space of block homotopy auto-morphisms by its equivalent non-block analogue BhAut (cid:27) D n ( V д , W д , ) ℓ (see Section 1.5), sothe E -page of the p -local Serre spectral sequence of the second map has the form(42) E p , q (cid:27) H p (cid:16) π hAut (cid:27) D n ( V д , W д , ) ℓ ; H q (cid:0) BhAut id D n ( V д , W д , ) ; Z ( p ) (cid:1) (cid:17) . To compute its entries for 0 < q < p − − n , we employ the Serre spectral sequence of0 −→ L д −→ π hAut (cid:27) D n ( V д , W д , ) ℓ −→ G ext д ,ℓ −→ , with coefficients in H q ( BhAut id D n ( V д , W д , ) ; Z ( p ) ) , where L д is the kernel as indicated. The E -page of this spectral sequence has the form E s , t (cid:27) H s (cid:16) G ext д ,ℓ ; H t (cid:0) L д ; H q ( BhAut id D n ( V д , W д , ) ; Z ( p ) ) (cid:1) (cid:17) and since L д is finite and p -torsion free by Lemma 3.10 as we have assumed 1 < p − − n and hence n < p −
4, this E -page is concentrated at the bottom row, so the groups (42)simplify in the range q < p − − n to E p , q (cid:27) H p (cid:16) G ext д ,ℓ ; H q (cid:0) BhAut id D n ( V д , W д , ) ; Z ( p ) (cid:1) L д (cid:17) , where (−) L д stands for taking coinvariants. By Corollary 4.9, there is an isomorphism(43) e H ∗ ( BhAut id D n ( V д , W д , ) ; Z ( p ) ) (cid:27) s − n ( ker ( S H ( p ) W д , → S H ( p ) V д )) of graded π hAut (cid:27) D n ( V д , W д , ) ℓ -modules in degrees ∗ < min ( n − , p − − n ) , so inparticular the kernel L д acts trivially in this range. Moreover, by Lemma 3.9, the group G ext д ,ℓ contains the negative of the identity, which is central, acts on (43) by −
1, and hence allowsus to apply the “centre kills”-trick to conclude that the G ext д ,ℓ -homology with coefficientsin (43) is 2-torsion and therefore trivial as p was assumed to be odd. This shows that thegroups E p , q vanish for 0 < q < min ( n − , p − − n ) , so the mapH ∗ ( B (cid:157) hAut (cid:27) D n ( V д , W д , ) ℓ ; Z ( p ) ) −→ H ∗ ( BG ext д ,ℓ ; Z ( p ) ) is an isomorphism for ∗ < min ( n − , p − − n ) and surjective in that degree. Combinedwith the first part of the proof, this implies the assertion. (cid:3) In contrast to the first map in (41), the second map might not induce a p -local homol-ogy isomorphism in a range of degrees, but we shall see below that its homotopy colimitBG ext ∞ ,ℓ → BGL ( H V ∞ ) with respect to the stabilisation maps explained in Section 3.5 does. Lemma 5.8.
For odd primes p , the induced map H ∗ (cid:0) BG ext ∞ ,ℓ ; Z ( p ) (cid:1) −→ H ∗ (cid:0) BGL ( H V ∞ ) ; Z ( p ) (cid:1) is an isomorphism.Proof. By an application of the Serre spectral sequence of the extension0 −→ M sfr д −→ G ext д ,ℓ −→ GL ( H V д ) −→ д H ∗ (cid:0) GL ( H V д ) ; e H ∗ ( M sfr д ; Z ( p ) ) (cid:1) induced by the stabilisation maps vanishes. Recall from Section 3.3 that the GL ( H V д ) -module M sfr д ⊂ ( H V д ⊗ H V д ) ∨ consists of all even (− ) n + -symmetric bilinear forms µ ∈( H V д ⊗ H V д ) ∨ , so as p is assumed to be odd, its p -localisation M sfr д ⊗ Z ( p ) is isomorphic tothe dual of the symmetric square Sym ( H V д ) ∨ ⊗ Z ( p ) if n is odd and the dual of the exteriorsquare Λ ( H V д ) ∨ ⊗ Z ( p ) if n is even. In particular, by antisymmetrisation, the GL ( H V д ) -module M sfr д ⊗ Z ( p ) is a direct summand of ( H V д ⊗ H V д ) ∨ ⊗ Z ( p ) . Choosing a basis H V д (cid:27) Z д compatible with the stabilisation maps, this shows that it suffices to show that the stable HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 37 GL д ( Z ) -homology with coefficients in H k (( Z д ⊗ Z д ) ∨ ; Z ) (cid:27) Λ k ( Z д ⊗ Z д ) ∨ vanishes for k >
0. Pulling back this module along the automorphism of GL д ( Z ) given by takingtranspose inverse, we see that we may replace this module by its dual Λ k ( Z д ⊗ Z д ) whoseGL д ( Z ) -homology for large д with respect to k does indeed vanish by an application of aresult due to Betley [Bet89, Thm 3.1]. (cid:3) Corollary 5.9.
Let n ≥ and p a prime. The map on homology induced by the map (cid:13) , H ∗ ( B g Diff D n ( V ∞ ; ℓ ) ℓ ; Z ( p ) ) −→ H ∗ ( BGL ∞ ( Z ) + ; Z ( p ) ) , is an isomorphism for ∗ < min ( n − , p − − n ) and surjective in that degree.Proof. This is free of content if p is even and follows for p odd from a combination ofLemmas 5.7 and 5.8, using that the canonical map BGL ∞ ( Z ) → BGL ∞ ( Z ) + is acyclic. (cid:3) Proof of Theorem 5.1.
As in the previous proofs, the claim is vacuous if p is even,so we assume otherwise. Our goal is to demonstrate that the precomposition of the leftcolumn in (39) with the ( n − ) -connected map (38) provides a zig-zag as promised byTheorem 5.1. As a result of the discussion in Section 5.1, the space X is the homotopyfibre of a map out of an infinite loop space which is moreover surjective on fundamentalgroups as a result of Proposition 3.7, so it follows from Lemma 1.4 that X is nilpotent.That the map 3 (cid:13) is p -locally min ( n , p − ) -connected is a consequence of 1 (cid:13) having thisproperty by Corollary 5.6 and the 1-connected cover BGL ∞ ( Z ) + h i → BGL ∞ ( Z ) + being a p -local equivalence as π BGL ∞ ( Z ) + (cid:27) Z / p is odd. This leaves us with showing thatBC ( D n ) −→ g Diff D n ( V ∞ )/ g Diff D n ( V ∞ ) −→ X is p -locally min ( n − , p − − n ) -connected, which we can test on p -local homologygroups by Lemma 1.1 as source and target are nilpotent. The first map in this compositionis ( n − ) -connected by Lemma 5.4, so we may focus on the second and show that it inducesan isomorphism in p -local homology in the required range. Since plus constructions do notaffect homology groups and 2 (cid:13) is an isomorphism in homology with Z ( p ) -coefficients indegrees less than min ( n − , p − − n ) and an epimorphism in that degree by Corollary 5.9,the claim follows from an application of Zeeman’s comparison theorem (see e.g. [HR76,Thm. 3.2]) to the map of Serre spectral sequences induced by the first two rows of (39),provided we ensure that the action of the fundamental groups of the bases of both fibresequences on the Z ( p ) -homology of the respective fibres is trivial in this range. For thesecond row, this follows from the p -local high connectivity of the map 3 (cid:13) established in thefirst part of the proof, and for the first row, it is a consequence of Lemma 5.4 together withthe observation that any element in π Diff D n ( V д ) can be represented by a diffeomorphismthat fixes D n + = V ⊂ V д , so commutes with any diffeomorphism in the image of theiterated stabilisation map C ( D n ) ≃ Diff D n ( D n + ) → Diff D n ( V д ) . Remark . It is not necessary for the proof of Theorem 5.1, but with some more care,one can show that the map BDiff sfr ( V ∞ ; ℓ ) + → BGL ∞ ( Z ) + ≃ Ω ∞ K ( Z ) in (39) agrees withrespect to the equivalence BDiff sfr ( V ∞ ; ℓ ) + ≃ Ω ∞ Σ ∞ + SO / SO ( n + ) explained in Section 5.1with the induced map on infinite loop spaces of the 0-connected cover of the compositionof maps of spectra Σ ∞ + SO / SO ( n + ) −→ S −→ K ( Z ) , where the first map is the canonical projection and the second is the unit. This showsthat the space named X is equivalent to the infinite loop space of the homotopy fibre ofthis composition and moreover leads to a cleaner formulation of Theorem 5.1: there is a p -locally min ( n − , p − − n ) -connected mapBC ( D n ) −→ Ω ∞ hofib ( S → K ( Z )) , which can furthermore be shown to be compatible with the iterated stabilisation mapBC ( D n ) → BC ( D n × I ) ≃ BC ( D n + ) . We refrain from going through this additional work at this point, the reason being that it will be a direct consequence of what we havealready shown in this section once [Kra20] is available. Appendix A. Stable tangential bundle maps are stable bundle maps
Fix a d -dimensional vector bundle ξ over a finite CW-complex X , a stable vector bundle { ψ k → B k } k ≥ , subcomplexes A , C ⊂ X , and a bundle map ℓ : ξ | A ⊕ ε k → ψ d + k coveringa map ¯ ℓ : X → B d + k for some k ≥ . In addition to the notation introduced for varioustypes of bundle maps in Section 1, we abbreviate g Map A ( X , B ; ¯ ℓ ) • ≔ colim m ≥ k g Map A ( X , B m ; ¯ ℓ ) • , where g Map A ( X , B m ; ¯ ℓ ) • is the semi-simplicial set whose p -simplices are block maps ∆ p × X → ∆ p × B m which agree with id ∆ p × ¯ ℓ on ∆ p × A . The colimit is taken over the mapsinduced by post-composition with the maps B m → B m + underlying the structure mapsof ψ . The sub semi-simplicial set of maps ∆ p × X → ∆ p × B m over ∆ p is denoted byMaps A ( X , B ; ¯ ℓ ) • ⊂ g Map A ( X , B ; ¯ ℓ ) • . Lemma A.1.
The semi-simplicial maps induced by forgetting bundle maps g Bun A ( ξ s , ψ ; ℓ ) τ • −→ g Map A ( X , B ; ¯ ℓ ) • and (cid:157) hA ut A ( ξ s ; C ) τ • −→ (cid:157) hAut A ( X ; C ) • are Kan fibrations.Proof. For ψ = ξ s and ℓ = inc the inclusion, the second map agrees with the pullback ofthe first map along the inclusion (cid:157) hAut A ( X ; C ) • ⊂ g Map A ( X , X ; inc ) • , so it suffices to showthat the first map is a Kan fibration, and we shall do so in the case where A is empty; theargument for general A is similar. The upper horizontal map in a lifting problem ( Λ pj ) • g Bun ( ξ s , ψ ) τ • ( ∆ p ) • g Map ( X , B ) • , induces bundle maps ϕ i : τ ∆ pi × ξ ⊕ ε k −→ τ ∆ pi × ψ d + k for i , j that agree on their faces and cover the restriction of the map ¯ ϕ : ∆ p × X → ∆ p × B d + k induced by the bottom horizontal arrow to ∆ pi × X . By replacing ξ with itsstabilisation ξ ⊕ ε k , we may assume k = . There is an extension of ϕ i to τ ∆ p | ∆ pi given bythe composition τ ∆ p | ∆ pi × ξ (cid:27) −→ ε ⊕ τ ∆ pi × ξ id ε ⊕ ϕ i −−−−−→ ε ⊕ τ ∆ pi × ψ d (cid:27) −→ τ ∆ p | ∆ pi × ψ d whose outer isomorphisms are induced by the differential of the diffeomorphism c i , ϵ ofSection 1.3. The condition (4) in the definition of tangential bundle maps is made preciselysuch that these extensions agree on their intersections, so they assemble to a bundle map ϕ Λ pj : τ ∆ p | Λ pj × ξ −→ τ ∆ p | Λ pj × ψ d that covers the restriction of ¯ ϕ to Λ pj × X , so we are left to argue that this bundle map isthe restriction of a p -simplex in Bun ( ξ s , ψ ) τ • covering ¯ ϕ . As the inclusion Λ pj × X ⊂ ∆ p × X is a trivial cofibration, obstruction theory provides an extension of ϕ Λ pj to a bundle map ϕ ∆ p : τ ∆ p × ξ −→ τ ∆ p × ψ d that covers ¯ ϕ , but this extension might violate condition (4) on the j th face, i.e. the map ϕ j : ε ⊕ τ ∆ pj × ξ −→ ε ⊕ τ ∆ pj × ψ d HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 39 obtained by restricting ϕ ∆ p to ∆ pj × X and using the isomorphism τ ∆ p | ∆ pj (cid:27) ε ⊕ τ ∆ pi inducedby the derivative of c j , ϵ need not have the form id ε ⊕ ϕ j for a bundle map ϕ j : τ ∆ pj × ξ −→ τ ∆ pj × ψ d . Nevertheless, its restriction to ∂ ∆ pj × X does have this form and we will argue that afteradding a trivial bundle of sufficiently large dimension, say n , we can alter ϕ j by a homotopyof bundle maps relative to ∂ ∆ pi × X that covers ¯ ϕ | ∆ pi so that it does have the required form.This would show the claim because we can use this homotopy to change ϕ ∆ p ⊕ id ε n to a p -simplex in Bun ( ξ ⊕ ε n , ψ d ⊕ ε n ) τ • that covers ¯ ϕ and extends ϕ Λ pj ⊕ id ε n and thus providesa lift as required. To this end, we denote by Iso ( ν , η ) → Y for vector bundles ν and η overa space Y the fibre bundle whose sections correspond to bundle morphisms ν → η overthe identity, so the fibre over y ∈ Y is the space of isomorphisms ν y → η y between thefibres of ν and η . Abbreviating ν ≔ τ ∆ pj × ξ and η ≔ ¯ ϕ ∗ ( τ ∆ pj × ψ d ) , we consider the diagram ∂ ∆ pj × X Iso ( ν ⊕ ε n , η ⊕ ε n ) Iso ( ε ⊕ ν ⊕ ε n , ε ⊕ η ⊕ ε n ) ∆ pj × X ∆ pj × X ∆ pj × X id ε ⊕− in which the solid diagonal arrow is induced by ϕ j ⊕ id ε n and the upper left horizontalone by its restriction to ∂ ∆ pj × X , which makes the subdiagram of solid arrows commutestrictly. From this point of view, the task we set us is equivalent to constructing a dashedarrow for some n that makes the upper leftmost triangle commutes strictly and the oneformed by the two diagonal arrows up to homotopy relative ∂ ∆ pj × X . Using that X is afinite CW complex and that the connectivity of the map on vertical homotopy fibres ofthe right square increases in n as it agrees with the inclusion O ( p + d + n ) ⊂ O ( p + d + n + ) up to equivalence, this follows from an application of obstruction theory. (cid:3) Remark.
We learnt the “stabilisation trick” of the previous proof from Appendix D of[BM20], which contains results similar to those of this appendix.
Corollary A.2.
The following semi-simplicial sets satisfy the Kan property.(i) g Bun A ( ξ s , ψ ; ℓ ) τ • (ii) Bun A ( ξ s , ψ ; ℓ ) τ • (iii) (cid:157) hAut A ( ξ s ; C ) τ • (iv) hAut A ( ξ s ; C ) τ • Proof.
It is straight-forward to see that g Map A ( X , B ; ¯ ℓ ) • is Kan, so the first part followsfrom Lemma A.1 and the fact that the domain of a Kan fibration over a Kan complex is Kan.The same reasoning applies to the second semi-simplicial set, using that Bun A ( ξ s , ψ ; ℓ ) τ • → Maps A ( X , B ; ¯ ℓ ) • is a Kan fibration because it is the pullback of the first map of Lemma A.1along the inclusion Maps A ( X , B ; ¯ ℓ ) • ⊂ g Map A ( X , B ; ¯ ℓ ) • . Also the semi-simplicial sets hA ut A ( X ; C ) • and (cid:157) hAut A ( X ; C ) • are easily seen to be Kan, so the remaining claims can beproved in the same way. (cid:3) Lemma A.3.
The inclusions
Bun A ( ξ s , ψ ; ℓ ) τ • ⊂ g Bun A ( ξ s , ψ ; ℓ ) τ • and hAut A ( ξ s ; C ) τ • ⊂ (cid:157) hAut A ( ξ s ; C ) τ • are equivalences.Proof. These inclusions are pullbacks of the inclusions(44) Maps A ( X , B ; ¯ ℓ ) • ⊂ g Map A ( X , B ; ¯ ℓ ) • and hA ut A ( X ; C ) • ⊂ (cid:157) hAut A ( X ; C ) • along the two Kan fibrations discussed in Lemma A.1, so the claim follows from showingthat the inclusions (44) are equivalences. As already mentioned in the previous proof, it is straight-forward to show that these semi-simplicial sets are Kan. Using the combina-torial description of their homotopy groups, the claim follows from the contractibility ofMaps ∂ ∆ p ( ∆ p , ∆ p and hAut ∂ ∆ p ( ∆ p ) (cf. Section 1.5). (cid:3) Lemma A.4.
The canonical extension maps described in Section 1.7
Bun A ( ξ s , ψ ; ℓ ) • −→ Bun A ( ξ s , ψ ; ℓ ) τ • and hAut A ( ξ s ; C ) • −→ hAut A ( ξ s ; C ) τ • are equivalences.Proof. The proof for the two maps are essentially identical; we restrict our attention tothe first map. Its source and target have the Kan property (the source because it is thesingular complex of a space, the target by Corollary A.2), so we may test the claim onsemi-simplicial homotopy groups. The two semi-simplicial sets involved have the same0-simplices, so the map is clearly surjective on path components. To show that it is sur-jective on homotopy groups in positive degrees, we fix a semi-simplicial base point inBun A ( ξ s , ψ ; ℓ ) • by choosing a 0-simplex ℓ : ξ ⊕ ε l → ψ d + l and taking products with thetangent bundles τ ∆ p . Let ϕ : ξ ⊕ ε k × τ ∆ p −→ ψ d + k × τ ∆ p be a p -simplex that represents a class in π p ( Bun A ( ξ s , ψ ; ℓ ) τ • ; ℓ ) and choose 0 < ϵ < / ¯ ϕ : X × ∆ p → B d + k satisfies the collaring condition (4). Byreplacing ξ with ξ ⊕ ϵ k , we may assume k = . Fixing a trivialisation F : τ ∆ p (cid:27) R p × ∆ p , ourcandidate for a preimage in π p ( Bun A ( ξ s , ψ ; ℓ ) • ; ℓ ) is the class defined by the composition ξ ⊕ ε p × ∆ p id ξ × F − −→ ξ × τ ∆ p ϕ −→ ψ d × τ ∆ p id ψd × F −→ ψ d ⊕ ε p × ∆ p −→ ψ d + p × ∆ p , where the last arrow is induced by the structure map of ψ . To justify this, it suffices toshow the existence of a ( p + ) -simplex in Bun A ( ξ ⊕ ε p , ψ d + p ; ℓ ) τ • which on the p th faceagrees with the p -fold stabilisation of ϕ , on the ( p + ) st face with the image of the previouscomposition, denoted by ϕ F , and on the remaining faces with the basepoint. To this end,pick a path γ : [ , ] → GL ( R p × R p ) from the identity to the permutation of the twocoordinates that is constant in a neighborhood of [ , ϵ ] ∪ [ − ϵ , ] , and define a homotopy H t of bundle automorphisms of ε p ⊕ τ ∆ p covering the identity by ε p ⊕ τ ∆ p id R p × F −→ R p × R p × ∆ p γ t × ∆ p −→ R p × R p × ∆ p id R p × F − −→ ε p ⊕ τ ∆ p . Using H , we define a homotopy e H of bundle maps ξ ⊕ ε p × τ ∆ p → ψ d + p × τ ∆ p from thestabilisation of ϕ to ϕ F whose underlying homotopy of maps of spaces is constantly ¯ ϕ by ξ ⊕ ε p × τ ∆ p id ξ × H t −→ ξ ⊕ ε p × τ ∆ p ϕ −→ ψ d ⊕ ε p × τ ∆ p id ψd × H − t −→ ψ d ⊕ ε p × τ ∆ p −→ ψ d + p × τ ∆ p , where the last map is induced by the structure map of ψ . Using the canonical trivialisationof τ [ , ] , this homotopy gives rise to a bundle map e H : ξ ⊕ ε p × τ ∆ p ×[ , ] → ψ d + p × τ ∆ p ×[ , ] which one checks to descend uniquely to a dashed arrow making the diagram ξ ⊕ ε p × τ ∆ p ×[ , ] ψ d + p × τ ∆ p ×[ , ] ξ ⊕ ε p × τ ∆ p + ψ d + p × τ ∆ p + e H id ξ ⊕ εp × d c id ξ ⊕ εp × d c ¯ H commute, where c is the map c : ∆ p × [ , ] −→ ∆ p + (( x , . . . , x p ) , s ) 7−→ ( x , . . . , x p − , s · x p , ( − s ) · x p ) whose derivative is surjective (though not fibrewise). The resulting bundle map ¯ H hasthe correct behaviour on all faces, so almost provides a ( p + ) -simplex in Bun A ( ξ ⊕ HOMOLOGICAL APPROACH TO PSEUDOISOTOPY THEORY. I 41 ε p , ψ d + p ; ℓ ) τ • as wished. The only problem is that it does not satisfy the collaring con-dition (4) for i = p , but this can be rectified as follows: choosing a block diffeomorphism α : ∆ p + → ∆ p + which agrees with the identity on ∆ p + p + , δ for some δ > (see Section 1.3for the notation) and makes the diagram ∆ p + [ , ϵ ) × ∆ p \ ∆ pp , δ ′ ∆ p + αc p c commute for some δ ′ > ϵ , the composition ξ ⊕ ε p × τ ∆ p + id ξ ⊕ εp × d α −→ ξ ⊕ ε p × τ ∆ p + ¯ H −→ ψ d + p × τ ∆ p + id ξ ⊕ εp × d α − −→ ξ ⊕ ε p × τ ∆ p + defines a ( p + ) -simplex as required. This finishes the proof of surjectivity of the map onhomotopy groups and injectivity follows from a relative version of the argument. (cid:3) References [Bet89] S. Betley,
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E-mail address : [email protected]@dpmms.cam.ac.uk