Invent. math. (2022) 227:1093–1167
https://doi.org/10.1007/s00222-021-01077-7
A homological approach to pseudoisotopy theory. I
Manuel Krannich1
Received: 1 April 2020 / Accepted: 23 September 2021 / Published online: 31 January 2022
© The Author(s) 2022
Abstract We construct a zig–zag from the once delooped space of pseudoiso-
topies of a closed 2n-disc to the once looped algebraic K -theory space of the
integers and show that the maps involved are p-locally (2n − 4)-connected
for n > 3 and large primes p. The proof uses the computation of the sta-
ble homology of the moduli space of high-dimensional handlebodies due to
Botvinnik–Perlmutter and is independent of the classical approach to pseu-
doisotopy theory based on Igusa’s stability theorem and work of Waldhausen.
Combined with a result of Randal-Williams, one consequence of this identi-
fication is a calculation of the rational homotopy groups of BDiff∂(D2n+1) in
degrees up to 2n − 5.
Mathematics Subject Classification 57R52 · 19D50 · 57R65 · 55P47
Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
2 Surgery theory and spaces of block diffeomorphisms . . . . . . . . . . . . . . . . . 1111
3 High-dimensional handlebodies and their mapping classes . . . . . . . . . . . . . . 1119
4 Relative homotopy automorphisms of handlebodies . . . . . . . . . . . . . . . . . 1133
5 The proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148
Appendix A. Stable tangential bundle maps are stable bundle maps . . . . . . . . . . . 1158
Appendix B. A lemma on symplectic derivations . . . . . . . . . . . . . . . . . . . . 1164
B Manuel Krannich
krannich@dpmms.cam.ac.uk
1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
123
1094 M. Krannich
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165
The homotopy type of the group of smooth pseudoisotopies, or concordance
diffeomorphisms,
C(M) = {φ : M × [0, 1] ∼=−→ M × [0, 1] | φ|M×{0}∪∂M×[0,1] = id},
of a smooth compact d-dimensional manifold M in the smooth topology has
been an object of interest to geometric topologists for many years, not least
because of its intimate connection to algebraic K -theory already visible on the
level of path components. Building on Cerf’s proof that C(M) is connected if
M is simply connected and d ≥ 5 [11], Hatcher and Wagoner [20] computed
the group π0C(M) of isotopy classes of concordances in high dimensions by
relating it to the lower algebraic K -groups of the integral group ring Z[π1 M]
of the fundamental group of M .1 Beyond its components, the homotopy type
of the space of concordances C(M) and its relation to K -theory has so far been
studied in two steps: deep work of Igusa [22] shows that the stabilisation map
C(M) −→ C(M × [0, 1])
induced by crossing with an interval is min(d−43 ,
d−7
2 )-connected, so in this
range up to about a third of the dimension, one may consider the stable
concordance space colimkC(M ×[0, 1]k) instead, which in turn admits a com-
plete description in terms of Waldhausen’s generalised algebraic K -theory for
spaces by Waldhausen, Jahren, and Rognes’ foundational stable parametrised
h-cobordism theorem [48].
In this work, we focus on the case M = D2n of a closed disc of even
dimension and study 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 elab-
orate on this at a later point. Our main result relates the delooped concordance
space BC(D2n) to the once looped algebraic K -theory space of the integers
∞+1K(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 > 3, there exists a zig–zag
BC(D2n) −→ · ←− ∞+10 K(Z)
whose maps are p-locally min(2n − 4, 2p − 4 − n)-connected for primes p.
1 See also Igusa’s corrections in [21].
123
A homological approach to pseudoisotopy theory. I 1095
Remark The result we prove is slightly stronger than stated here (see The-
orem 5.1) and implies for instance that π2n−4BC(D2n) ⊗ Q surjects onto
K2n−3(Z) ⊗ Q as long as n > 3.
When combined with Borel’s work on the stable rational cohomology of
arithmetic groups [9], Theorem A provides an isomorphism
π∗BC(D2n) ⊗ Q ∼= K∗+1(Z) ⊗ Q
∼=
{Q if ∗ ≡ 0 (mod 4)
0 otherwise for 0 < ∗ < 2n − 4,
and an epimorphism π∗BC(D2n)⊗Q → K∗+1(Z)⊗Q in degree 2n−4, which
goes significantly beyond the range that was previously accessible by relying
on Igusa’s stability result and shows for instance that BC(D8) is nontrivial,
even rationally. Given that the K -groups K∗(Z) are known to contain p-torsion
for comparatively large primes with respect to the degree due to contributions
from Bernoulli numbers, Theorem A also exhibits many new torsion elements
in π∗BC(D2n), such as one of order 691 in π21BC(D2n) as long as n > 12
resulting from the fact that K22(Z) is cyclic of that order.
In the remainder of this introduction, we explain more direct applications
of Theorem A and conclude by indicating some ideas that go into its proof.
Diffeomorphisms and concordances of odd discs
Restricting a concordance to the moving part of its boundary induces a homo-
topy fibre sequence
Diff∂(Dd+1) −→ C(Dd) −→ Diff∂(Dd)
that compares the group of concordances C(Dd) of a d-disc to its group of
its diffeomorphisms Diff∂(Dd) fixing the boundary pointwise. By a result of
Randal-Williams [37, Thm 4.1] based on Morlet’s lemma of disjunction and
work of Berglund and Madsen [8] (a combination which incidentally inspired
parts of our strategy to prove Theorem A), the space BDiff∂(D2n) is rationally
(2n − 5)-connected, so the delooped maps
BDiff∂(D2n+1) −→ BC(D2n) and BC(D2n+1) −→ BDiff∂(D2n+1)
are rationally (2n − 5)-connected as well, resulting in the following corollary
of Theorem A.
Corollary B There exist isomorphisms
π∗BDiff∂(D2n+1) ⊗ Q ∼= π∗BC(D2n+1) ⊗ Q ∼= K∗+1(Z) ⊗ Q
123
1096 M. Krannich
in degrees ∗ < 2n − 5 and epimorphisms
π∗BDiff∂ (D2n+1) ⊗ Q → K∗+1(Z) ⊗ Q π∗BC(D2n+1) ⊗ Q → K∗+1(Z) ⊗ Q
in degree 2n − 5.
Remark
(i) The range in Corollary B is nearly optimal: by work of Watanabe [45] or
as consequence of Weiss’ results on topological Pontryagin classes [46],
the group π2n−2BDiff∂(D2n+1) ⊗ Q is known to be nontrivial for many
values of n for which K2n−1(Z) ⊗ Q vanishes by Borel’s work.
(ii) A combination of the strengthening of Theorem A mentioned earlier with
recent work of Kupers and Randal-Williams [27] improves the range of
Corollary B by one degree from 2n − 5 to 2n − 4.
(iii) In the range captured by Igusa’s stability result, i.e. up to approximately
degree 2n/3, Corollary B was previously known as a result of a classi-
cal computation due to Farrell and Hsiang [16] based on Waldhausen’s
approach to pseudoisotopy theory (of which the proof of Corollary B is
independent).
Homeomorphisms of Euclidean spaces
By an enhancement of a result due to Morlet (see e.g. [5, Thm 4.4]), there are
homotopy equivalences
BDiff∂(Dd)  d0Top(d)/O(d) and
BC(Dd)  dhofib
(
Top(d)/O(d) → Top(d + 1)/O(d + 1)
)
for d ≥ 5 that relate the groups of diffeomorphisms and concordances of a
d-disc to the homotopy fibre Top(d)/O(d) of the map BO(d) → BTop(d)
that classifies the inclusion of the orthogonal group O(d) into the topolog-
ical group Top(d) of homeomorphisms of Rd , and to its stabilisation map
Top(d)/O(d) → Top(d + 1)/O(d + 1) induced by taking products with R.
Theorem A and Corollary B can thus be reformulated in terms of these equiv-
alent spaces and result in particular in the following corollary, using the fact
that π∗Top(d)/O(d) is finite for ∗ < d + 2 and d = 4 [28, Essay V, 5.0].
Corollary C There exists an isomorphism
π∗Top(2n + 1)/O(2n + 1) ⊗ Q
∼=
{
0 for ∗ ≤ 2n + 1
K∗+2n+2(Z) ⊗ Q for 2n + 1 < ∗ < 4n − 4
123
A homological approach to pseudoisotopy theory. I 1097
and an epimorphism π∗Top(2n + 1)/O(2n + 1) ⊗ Q → K∗+2n+2(Z) ⊗ Q in
degree 4n − 4.
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 to achieve
a seemingly different task: relating the p-local homology of BC(D2n) to K -
theory in a range of degrees. This should, however, convey the main ideas;
the actual proof of Theorem A uses a similar strategy to construct a zig–
zag between BC(D2n) and ∞+10 K(Z) that consists of maps that are p-local
homology isomorphisms in a range and then argues that the maps are actually
p-locally highly connected. In the sketch that follows, we allow ourselves 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 D2n+1 as the 0th member of a whole family of manifolds—the high-
dimensional handlebodies
Vg := g Dn+1 × Sn
given as iterated boundary connected sums of Dn+1 × Sn . Comparing the
groups of diffeomorphisms Diff D2n (Vg) that pointwise fix a chosen disc D2n ⊂
∂Vg in the boundary to the corresponding block diffeomorphism groups yields
homotopy fibre sequences
˜Diff D2n (Vg)/Diff D2n (Vg) −→ BDiff D2n (Vg) −→ B˜Diff D2n (Vg),
one for each g. Varying g, these fibre sequences are connected by stabilisation
maps induced by extending (block) diffeomorphisms along the inclusion Vg ⊂
Vg+1 by the identity, and Morlet’s lemma of disjunction ensures that the map
between homotopy fibres
˜Diff D2n (Vg)/Diff D2n (Vg) −→ ˜Diff D2n (Vg+1)/Diff D2n (Vg+1)
is highly connected. It is not hard to see that the space BC(D2n) of interest is
equivalent to this fibre for g = 0, so to access BC(D2n) in a range, we may
as well study the homotopy fibre of the sequence obtained from the previous
one by taking homotopy colimits,
˜Diff D2n (V∞)/Diff D2n (V∞) −→ BDiff D2n (V∞) −→ B˜Diff D2n (V∞). (1)
123
1098 M. Krannich
By work of Botvinnik and Perlmutter [10], the homology of BDiff D2n (V∞) has
a surprisingly simple description in homotopy theoretical terms, so to compute
the homology of the fibre of (1) and hence that of BC(D2n) in a range, one
might try to compute the homology of B˜Diff D2n (V∞) and analyse the Serre
spectral sequence of (1). This is essentially what we do, and it involves several
steps of which some might be of independent interest:
(i) In Sect. 2, we use surgery theory to express the space of block diffeomor-
phisms of a general manifold triad satisfying a π -π -condition p-locally
for large primes in terms of its homotopy automorphisms covered by
certain bundle data. A similar result in the rational setting which inspired
ours but applies to another class of triads was obtained by Berglund and
Madsen [8] (see also Remark 2.3).
(ii) Sect. 3 serves to compute variants of the mapping class groupπ0Diff D2n (Vg)
up to extensions in terms of automorphisms groups of the integral homol-
ogy of Vg.
(iii) In Sect. 4, we calculate the p-local homotopy and homology groups of the
delooped space of homotopy automorphisms BhAutD2n (Vg, Wg,1) of Vg
that fix D2n and restrict to a homotopy automorphism of the complement
of the boundary as a module over the group π0hAutD2n (Vg, Wg,1) in
a range of degrees. This uses some pieces of the apparatus of rational
homotopy theory, as well as an ad-hoc p-local generalisation we provide
along the way.
(iv) The action on the nth homology group Hn(Vg; Z) ∼= Zg induces a map
B˜Diff D2n (V∞) −→ BGL∞(Z)+  ∞0 K(Z),
a variant of which we show in Sect. 5 with the help of all previous steps
to be a p-local homology isomorphism in a range of degrees.
Outlook
In [24], we will take a different approach and study concordance spaces C(M)
without restriction on the p-torsion. As a byproduct, the setup of [24] will also
make apparent that the zig–zag of Theorem A is compatible with the iterated
stabilisation map C(D2n) → C(D2n × [0, 1]2)  C(D2n+2) and agrees up to
equivalence with the zig–zag
BC(D2n) → ∞+10 WhDiff(∗) = ∞0 fib
(
S → K (S))
→ ∞0 fib
(
S → K (Z)) ← ∞+10 K (Z)
known from Waldhausen’s work [42] (see also Sect. 5.4).
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A homological approach to pseudoisotopy theory. I 1099
1 Preliminaries
We start off with a lemma on semi-simplicial actions and a short recollection
on nilpotent spaces for later reference, followed by foundational material on
various types automorphisms of manifolds with bundle data. Primarily, this
serves us to set up a convenient theory 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 the right is the semi-simplicial
space X•//M•whose space of p-simplices is defined as the bar-construction
B(X p, Mp, ∗), 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•//M• by BM•. The unique semi-simplicial map X• → ∗• induces a natural
map X•//M• → BM• which is well-known to geometrically realise to a quasi-
fibration with fibre the realisation of X• if M• is a group-like simplicial monoid
acting simplicially on a simplicial set X•. To explain a generalisation of this
fact for semi-simplicial M• and X•, we denote the geometric realisation 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 underlying semi-simplicial sets of M• and X• admit degeneracies, i.e. if
they agree (as semi-simplicial sets) with the underlying semi-simplicial sets
of simplicial sets, then the left arrow is an equivalence (see e.g. [15, Thm 7.2]),
so a contractible 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 an equivalence for all m ∈ |M•|. The following
lemma shows that this is sufficient to conclude that the natural map X•//M• →
BM• 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•//M• −→ BM•
induces a quasi-fibration on geometric realisations.
123
1100 M. Krannich
Remark 1.2
(i) In all situations we encounter, the condition that M• and X• admit degen-
eracies is ensured by them being Kan (every semi-simplicial Kan complex
admits degeneracies [23]) and the condition that M• acts on X• by equiv-
alences by M• being group-like, i.e. the monoid π0|M•| having inverses.
Note that Lemma 1.1 neither requires the degeneracies of M• to be com-
patible with the monoid 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-
simplicial set X• = G≤0• which agrees with G0 in degree 0 and is empty
otherwise, semi-simplicially acted upon by G• via right translations. In
this case, the realisation |X•//G•| is contractible, but |G≤0• |  G0 is rarely
equivalent to |BG•|  |G•|.
(iii) Under the assumption of Lemma 1.1, the long exact sequence induced by
the quasi-fibration yields a bijection π0|X•|/π0|M•| ∼= π0|X•//M•|
Proof of Lemma 1.1 The sequence in question is the geometric realisation of
a sequence
X• −→ B(X•, M•, ∗) −→ B M• (2)
of simplicial semi-simplicial sets, where the simplicial (bar-)direction is indi-
cated by the square  and the semi-simplicial direction by the bullet •; the
fibre X• is constant in the -direction. Since the realisation of a simplicial set
is canonically equivalent to the realisation of its underlying semi-simplicial set
[15, Lem. 1.7] and the realisation of a bi-semi-simplicial sets is independent of
which direction one realises first [15, p. 2106], we may rely on a result of Segal
[15, Thm 2.12] and the second part of its simplification from [15, Lem. 2.11]
to reduce the claim to showing that the commutative squares
|Bp(X•, M•, ∗)| |Bp−1(X•, M•, ∗)|
|Bp M•| |Bp−1 M•|
dp
dp
and
|B1(X•, M•, ∗)| |B0(X•, M•, ∗)|
|B1 M•| |B0 M•|
d0
d0
obtained from the simplicial structure of B(X•, M•, ∗) → B(M•) by realis-
ing the •-direction are homotopy cartesian for p ≥ 0. For the left square, this
follows directly from the definition of the bar-construction together with the
above mentioned fact that the realisation of the product of two semi-simplicial
123
A homological approach to pseudoisotopy theory. I 1101
sets that admit degeneracies is canonically equivalent to the product of their
realisations. Spelling out the definition of the bar-construction, the right hand
square is given as
|X• × M•| |X•|
|M•| ∗,
|(pr2)•|
|μ•|
involving the action μ• and the projection (pr2)• on the second coordinate,
so we are left to show that the shear map (|μ•|, |(pr2)•|) : |X• × M•| →
|X•| × |M•| is an equivalence. This map fits into a commutative triangle
|X• × M•| |X•| × |M•|
|M•|
(|μ•|,|(pr2)•|)
|(pr2)•| |(pr2)•|
whose map on diagonal homotopy fibres at m ∈ |M•| agrees up to equivalence
with the action map μ(−, m) : |X•| → |X•|. The latter is an equivalence by
assumption, so the shear map is an equivalence and the claim follows. unionsq
1.2 Nilpotent spaces
A space is nilpotent if it is path connected and its fundamental group is nilpotent
and acts nilpotently on all higher homotopy groups. Such spaces have an unam-
biguous 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 algebraic sense [31, Thm 6.1.2]. Localisations are defined in terms of a
universal property [31, Def. 5.2.3], which ensures that they are unique and
functorial up to homotopy. A map between nilpotent spaces is p-locally k-
connected for a prime p and k ≥ 1 if the induced map on p-localisations is
k-connected in the usual sense.
Lemma 1.3 For a map X → Y between nilpotent spaces, a prime p ≥ 2, and
k ≥ 1, the following statements are equivalent:
(i) The map X → Y is p-locally k-connected.
(ii) The induced map on p-localised homotopy groups (π∗X)(p) → (π∗Y )(p)
is an isomorphism for ∗ < k and surjective for ∗ = k.
(iii) The induced map on p-local homology groups H∗(X; Z(p)) → H∗
(Y ; Z(p)) is an isomorphism for ∗ < k and surjective for ∗ = k.
123
1102 M. Krannich
Proof The above mentioned fact that p-localisation of nilpotent spaces com-
mutes with taking homotopy groups shows that the first two items are
equivalent. The equivalence between the second two follows for k = 1 from
the well-known fact that a morphism G → H between nilpotent groups is
surjective if and only if it is surjective on abelianisations which one sees as
follows: writing 1(G) = G and n(G) = [n−1(G), G] for the lower central
series, there is a natural epimorphism ⊗nZGab → n(G)/n+1(G) induced
by taking left-normed commutators (see e.g. [44, Thm 3.1]), so one may prove
the claim by an induction on the nilpotence degree. For k ≥ 2, the equivalence
between the second two items is a consequence of the relative Hurewicz the-
orem for nilpotent spaces (see e.g. [19, Cor. 3.4]) applied to the p-localisation
of the map in question. unionsq
At several points in this work, we will make use of the fact that nilpo-
tent spaces behave well with respect to taking homotopy fibres (see e.g. [31,
Prop. 4.4.1] for a proof).
Lemma 1.4 Let π : E → B be a fibration, e ∈ E a point, and Ee ⊂ E
and Fe ⊂ π−1(π(e)) the respective path components containing e. If Ee is
nilpotent, then Fe is nilpotent as well.
1.3 Block spaces
As is customary, we denote the standard p-simplex byp ⊂ Rp+1 and identify
its facesσ : q → p with their images. A p-block space is a space X together
with a map π : X → p. For a q-face σ ⊂ p, the preimage Xσ := π−1(σ )
becomes a q-block space by pulling back π along σ . A block map between
block p-spaces is a map f : X → Y between underlying spaces such that
f (Xσ ) ⊂ Yσ for all faces σ ⊂ p and a block homotopy equivalence is a
block map f : X → Y such that the induced map fσ : Xσ → Yσ 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 = 0, . . . , p, the map
ci : [0, 1] × 
p−1 −→ p
(s, t0, . . . , tp−1) −→ ((1 − s)t0, . . . , (1 − s)ti−1, s, (1 − s)ti , . . . , (1 − s)tp−1)
induces for 0 < 
 ≤ 1 a diffeomorphism of manifolds with corners
ci,
 : [0, ε) × p−1 −→ ci ([0, 
) × p−1) =: pi,

onto an open neighborhood pi,
 of the i th codimension 1 face 
p
i ⊂ p. A
block map f : p × M → p × N for spaces M and N is collared if there
exists an ε > 0 such that
123
A homological approach to pseudoisotopy theory. I 1103
(i) f (pi,
 × M) ⊂ pi,ε × N and
(ii) (c−1i,
 × idN ) ◦ f |pi,
×M ◦ (ci,
 × idM) = fpi × id[0,
)
are satisfied for i = 0, . . . , p.
1.4 Diffeomorphisms
For a compact smooth d-manifold W and two compact submanifolds M, N ⊂
W , we denote by ˜Diff M(W, N )• the semi-simplicial group of block diffeo-
morphisms whose p-simplices consists of all diffeomorphisms of p × W
that are collared block 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 and in the case M = ∂W ,
we write ˜Diff∂(W, N ) instead of ˜Diff M(W, N ). Making use of the collaring
condition, one shows that ˜Diff M(W, N )• satisfies the Kan property (see [18,
p. 58-59] for a proof in the case M = N = ∅; the general case follows in the
same way2). The semi-simplicial subgroup of diffeomorphisms
Diff M(W, N )• ⊂ ˜Diff M(W, N )•
is defined by requiring the diffeomorphisms of p × M to commute with the
projection to the simplex p instead of just preserving its faces. This semi-
simplicial subgroup agrees with the (collared and smooth) singular set of the
topological group Diff M(W, N ) of diffeomorphisms by which we mean the set
of 0-simplices Diff M(W, N )0 equipped with the smooth 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 block homotopy automor-
phisms ˜hAutM(W, N )• are the block homotopy automorphisms ofp×W that
fix p × M pointwise and restrict to homotopy automorphisms of p × N .
Unlike for ˜Diff M(W, N )•, it is straight-forward to see that ˜hAutM(W, N )•
2 In [6, App. A §3 a], the authors attempt to construct degeneracy maps for certain semi-
simplicial sets of block embeddings, which would in particular enhance ˜Diff∂ (W )• to a
simplicial group and thus imply that it satisfies the Kan property as every simplicial group
does. However, the argument in [6] is flawed: on two of the faces, the proposed degeneracy
maps violate the collaring condition described on p. 116 loc. cit. The more laborious argument
in [18] appears to be correct.
123
1104 M. Krannich
is Kan: a map from the semi-simplicial horn (pi )• is represented by a
homotopy equivalence ψ : pi × W → pj × W and a lift to a p-simplex
p×W → p ×W is given by (ϕi ×idW )◦(id[0,1]×ψ)◦(ϕ−1i ×idW ), where
ϕi : [0, 1] × pi → p is any homeomorphism that extends the inclusion on
{0}×pi ⊂ p and restricts to a homeomorphism from [0, 1]×∂pi ∪{1}×pi
onto the i th face pi ⊂ p. As for diffeomorphisms, insisting that the homo-
topy equivalences of p×M be over p defines a sub semi-simplicial monoid
of homotopy automorphisms
hAutM(W, N )• ⊂ ˜hAutM(W, N )•, (3)
which agrees with the singular set of the space hAutM(W, N ) obtained by
equipping the set of homotopy equivalences hAutM(W, N )0 with the compact
open topology, so also |hAutM(W, N )•| and hAutM(W, N ) are canonically
equivalent. An aspect which distinguishes the situation for homotopy auto-
morphisms from that for diffeomorphisms is that the inclusion (3) of Kan
complexes induces an equivalence on geometric realisation, which one can
see from the combinatorial description of their homotopy groups together
with the contractibility of hAut∂p(p).
1.6 Bundle maps, unstably
A bundle map between two vector bundles ξ → X and ν → Y over CW
complexes X and Y is a commutative square of the form
ξ ν
X Y
φ
φ¯
whose induced maps on vertical fibres are linear isomorphisms. Of course the
underlying map of spaces φ¯ can be recovered from φ, so we often omit it. Given
a subcomplex A ⊂ X and a bundle map 0 : ξ |A → ν defined on the restriction
of ξ to A, the semi-simplicial set of block bundle maps˜BunA(ξ, ν; 0)• has as
its p-simplices the bundle maps p × ξ → p × ν that agree with idp × 0
on p × ξ |A and whose underlying map p × X → p × Y is a block map.
As before, the semi-simplicial structure is induced by restriction to subspaces
σ × X for faces σ ⊂ p. Insisting that the underlying map between base
spaces be over p defines the sub semi-simplicial set
BunA(ξ, ν; 0)• ⊂ ˜BunA(ξ, ν; 0)•
123
A homological approach to pseudoisotopy theory. I 1105
of bundle maps, which agrees with the singular set of the space BunA(ξ, ν; 0)
obtained by equipping the set BunA(ξ, ν; 0)0 of bundle maps ξ → ν relative
to 0 with the compact-open topology. If ξ = ν and 0 = inc then the semi-
simplicial sets of (block) bundle maps˜BunA(ξ, ξ ; inc)• and BunA(ξ, ξ ; inc)•
are semi-simplicial monoids under composition, and they act by precompo-
sition on ˜BunA(ξ, ν; 0)• respectively BunA(ξ, ν; 0)• for any bundle ν and
bundle map 0 : ξ |A → ν. For a subcomplex C ⊂ X , we denote by
˜hAutA(ξ, C)• ⊂ ˜BunA(ξ, ξ ; inc)•
the submonoid of block bundle maps whose underlying selfmap of p × X
is a homotopy equivalence that restricts to an equivalence of p × C . The
submonoid
hAutA(ξ, C)• ⊂ BunA(ξ ; ξ, inc)•
is defined analogously.
1.6.1 Tangential bundle maps
Introducing yet another variant of bundle maps, we define the semi-simplicial
set of tangential block bundle maps ˜BunA(ξ, ν; 0)τ• as follows: writing τM
for the tangent bundle of a manifold M and ε for the trivial line bundle, the
p-simplices of ˜BunA(ξ, ν; 0)τ• are the bundle maps ϕ : τp × ξ → τp × ν
that agree with idτp × 0 on τp × ξ |A, satisfy ϕ(τpi × ξ) ⊂ τpi × ν for
0 ≤ i ≤ p, and make the diagram
τp |pi × ξ τp |pi × ν
(τpi
⊕ ε) × ξ (τpi ⊕ ε) × ν
ϕ
(4)
commute, where the bottom horizontal map is given by the restriction of ϕ
on τpi
× ξ and the identity on ε, and the vertical maps are induced by the
canonical trivialisation of τ[0,1] and the derivative of the map ci defined in
Sect. 1.3. Requiring the underlying map p × X → p × Y to be over p
defines the sub semi-simplicial set
BunA(ξ, ν; 0)τ• ⊂ ˜BunA(ξ, ν; 0)τ• (5)
123
1106 M. Krannich
of tangential bundle maps. As before, we have a chain of semi-simplicial
monoids
hAutA(ξ, C)τ• ⊂ ˜hAutA(ξ, C)τ• ⊂ ˜BunA(ξ, ξ ; inc)τ•
which are defined in the same way as for non-tangential (block) bundle maps.
Note that there is a canonical map
BunA(ξ, ν; 0)• −→ BunA(ξ, ν; 0)τ• (6)
given by extending a bundle map p × ξ → p × ν over p to τp × ξ →
τp × ν by the identity. This map is not an equivalence, but we shall see in the
next paragraph that it becomes one after stabilisation.
Remark 1.5 Note that the map (6) does not extend to a map˜BunA(ξ, ν; 0)• →
˜BunA(ξ, ν; 0)τ• in an obvious way. This is because, for a bundle map φ : p ×
ξ → p×ν covering a block map, the map τp×ξ → τp×ν that maps (x, y)
to (x, φ(x, y)) is not a bundle map unless φ commutes with the projection on
p. Here x ∈ p is the underlying point of x ∈ τp .
1.7 Bundle maps, stably
A stable vector bundle is a sequence of vector bundles ψ = {ψk → Bk}k≥l
for some l ≥ 0, where ψk is k-dimensional, together with structure maps
ψk ⊕ ε → ψk+1 for k ≥ l, covering maps of the form Bk → Bk+1. Given a d-
dimensional vector bundle ξ , its stabilisation is the stable vector bundle ξ s with
ξ sd+k = ξ ⊕ 
k and the identity as structure maps. For a d-dimensional vector
bundle ξ → X , a stable vector bundle ψ , and a bundle map 0 : ξ |A ⊕ εk →
ψd+k for some k, the semi-simplicial set of stable bundle maps is the colimit
BunA(ξ s, ψ; 0)• := colimm≥kBunA(ξ ⊕ εm, ψd+m; 0)•
over the stabilisation maps
BunA(ξ ⊕ εm, ψd+m; 0)• −→ BunA(ξ ⊕ εm+1, ψd+m+1; 0)•
given by adding a trivial line bundle followed by postcomposition with the
structure map ψd+m ⊕ε → ψd+m+1. Analogously, we define stable tangential
bundle maps as the colimit
BunA(ξ s, ψ; 0)τ• := colimm≥kBunA(ξ ⊕ εm, ψd+m; 0)τ• .
123
A homological approach to pseudoisotopy theory. I 1107
As in Section 1.6, there are semi-simplicial sub-monoids
hAutA(ξ s; C)• ⊂ BunA(ξ s, ξ s; inc)• and
hAutA(ξ s; C)τ• ⊂ BunA(ξ s, ξ s; inc)τ•,
and also block variants of these semi-simplicial sets, defined by adding appro-
priate tildes. As the extension map (6) is compatible with the stabilisation
maps, it gives rise to maps
BunA(ξ s, ψ; 0)• −→ BunA(ξ s, ψ; 0)τ• and
hAutA(ξ s; C)• −→ hAutA(ξ s; C)τ•
(7)
which we show in Lemma A.4 to be equivalences if X is a finite CW complex.
1.8 Tangential structures
A d-dimensional tangential structure is a fibration θ : Bd → BO(d) whose
target is the base of the universal d-dimensional vector bundle γd → BO(d).
The semi-simplicial set of θ -structures on a d-dimensional vector bundle ξ over
a CW complex X is the semi-simplicial set of bundle maps BunA(ξ, θ∗γd; 0)
from ξ to the pullback θ∗γd of the universal bundle along θ , relative to a
fixed bundle map 0 : ξ |A → θ∗d γd . We denote the homotopy quotient (in the
sense of Sect. 1.1) of the action of hAutA(ξ, C)• ⊂ BunA(ξ ; ξ, inc)• on the
semi-simplicial set of θ -structures by
BhAutθA(ξ, C; 0)• := BunA(ξ, θ∗γd; 0)•//hAutA(ξ, C)•.
Remark 1.6 Note that BhAutθA(ξ, C; 0)• is in many cases empty or discon-
nected, so despite the suggestive notation, it is in general not the classifying
space of any kind of group or monoid, (semi-)simplicial or topological.
1.8.1 Stable tangential structures
A stable tangential structure is a fibration of the form  : B → BO, which
induces a d-dimensional tangential structure d : Bd → BO(d) for any d ≥ 0
by pulling back  along the stabilisation map BO(d) → BO. Note, however,
that not all d-dimensional tangential structures arise this way, for instance the
tangential structure EO(d) → BO(d) encoding unstable framings does not. A
stable tangential structure defines a stable vector bundle ∗γ = {∗dγd}d≥0
whose structure maps are induced by the canonical bundle map γd ⊕ε → γd+1
covering the usual stabilisation map BO(d) → BO(d + 1). Given a stable
123
1108 M. Krannich
bundle map 0 : ξ s |A → ∗γ , we call BunA(ξ s, ∗γ ; 0) the semi-simplicial
set of stable θ -structures on ξ . As in the unstable case, we abbreviate
BhAutA(ξ
s, C; 0)• := BunA(ξ s, ∗γ ; 0)•//hAutA(ξ s, C)•. (8)
Tangential and or block variants of the previous definitions are defined by
using the respective variants of bundle maps and adding appropriate tildes and
or τ -superscripts.
1.8.2 Forgetting tangential structures
It follows from Lemma 1.1 that the sequence of semi-simplicial spaces
BunA(ξ, θ∗γd; 0)• −→ BhAutθA(ξ, C; 0)• −→ BhAutA(ξ, C)• (9)
realises to a quasi-fibration, because BunA(ξ, θ∗γd; 0)• and hAutA(ξ, C)•
admit degeneracies since they satisfy the Kan property (they agree with the
singular complex of a space, see Sect. 1.6) and the action is by equivalences as
hAutA(ξ, C)• is group-like. The same argument applies to the stable analogue
of this sequence involving (8) and, using Corollary A.2 (assuming that X
is a finite complex), also to its variants involving tangential and or block
bundle maps. By definition of the universal bundle, the semi-simplicial set
BunA(ξ, θ∗γd; 0)• is contractible in the case θ = id, so the second map in
(9) is an equivalence for this particular choice of θ . The analogous statement
holds in the stable case as well and, as a result of Lemmas A.3 and A.4, also
for the tangential and or block variants.
1.9 The derivative maps
Taking fibrewise derivatives of diffeomorphisms ϕ : p ×W → p ×W over
p induces a canonical semi-simplicial map
Diff M(W, N )• −→ hAutM(τW , N )•, (10)
which we call the derivative map. Furthermore, the notion of a tangential
bundle map is designed exactly so that there is a block derivative map
˜Diff M(W, N )• −→ ˜hAutM(τ sW , N )τ• . (11)
given by assigning a block diffeomorphism p ×W → p ×W its derivative
τp × τW → τp × τW , which indeed makes the square (4) commute as ϕ is
assumed to be collared in the sense of Section 1.3.
123
A homological approach to pseudoisotopy theory. I 1109
Remark 1.7
(i) A different model of the block derivative map (11) was considered by
Berglund and Madsen in their prominent study of the rational homotopy
type of spaces of block diffeomorphisms of manifolds with certain bound-
ary conditions [8] (see Section 4.3 loc. cit. and Remark 2.3 below).
(ii) Lemmas A.3 and A.4 provide an equivalence
˜hAutM(τ sW , N )
τ•  hAutM(τ sW , N )•,
so the delooped space B˜hAutM(τ sW , N )τ• classifies fibrations πW :
E → B with fibre W together with the following data:
(1) maps of fibrations πM → πW ← πN over the identity whose induced
maps on fibres is equivalent to the system of inclusions M ⊂ W ⊃ N ,
(2) a trivialisation of πM , and
(3) a stable vector bundle E → BO over the total space of πW whose
restriction to 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 reminiscent of the fact that a block bundle has an underlying fibration
and a stable vertical tangent bundle by [14]. However, somewhat curi-
ously, the block derivative map (11) obviously factors over the variant
˜hAutM(τW , N )τ• involving the unstable tangent bundle of W , giving rise
to an unstable block derivative map
B˜Diff M(W, N )• −→ B˜hAutM(τW , N )τ• .
The target of this map is neither equivalent to BhAutM(τ sW , N )• nor
BhAutM(τW , N )•, and it would be interesting to have a good descrip-
tion of what it classifies.
We denote the submonoids of the components hit by the derivative maps by
hAut∼=M(τW , N )• ⊂ hAutM(τW , N )• and
˜hAut
∼=
M(τ
s
W , N )
τ• ⊂ ˜hAutM(τ sW , N )τ•
(12)
and add the same∼=-superscript to (8) and its tangential block variant to indicate
when we take homotopy quotients by the submonoids (12) instead of the full
monoids. Defining
B˜Diff

M(W, N ; 0)• := ˜BunM(τ sW , ∗γ ; 0)τ•//˜Diff M(W, N )• (13)
123
1110 M. Krannich
for a stable tangential structure  and a -structure 0 on τ sW |M and
BDiffθM(W, N ; 0)• := BunM(τW , θ∗γd; 0)•//Diff M(W, N )•
in the unstable case, the two derivative maps fit into a commutative square
BDiffdM (W, N ; 0)• BhAutd ,
∼=
M (τW , N ; 0)•
B˜Diff

M(W, N ; 0)• B˜hAut
,∼=
M (τ
s
W , N ; 0)τ•
(14)
whose vertical maps are induced by the composition
BunM(τW , ∗dγd; 0)• → BunM(τ sW , ∗γ ; 0)•
→ BunM(τ sW , ∗γ ; 0)τ• ⊂ ˜BunM(τ sW , ∗γ ; 0)τ• .
All maps in this composition are equivalences, the first one by an exercise in
obstruction theory and the second map as well as the final inclusion as a result
of Lemmas A.3 and A.4. The composition of equivalences just discussed also
induces the vertical maps in the commutative diagram
BDiffdM (W, N ; 0)• BDiff M(W, N )•
B˜Diff

M(W, N ; 0)• B˜Diff M(W, N )•
(15)
whose horizontal maps are induced by forgetting tangential structures.
Lemma 1.8 The geometric realisation of (15) is homotopy cartesian.
Proof By construction the induced map on horizontal strict fibres agrees with
the composition of equivalences discussed above (15), so it suffices to show that
the horizontal maps of the square realise to quasi-fibrations. As Diff M(W, N )•
and˜Diff M(W, N )• are Kan and group-like (see Section 1.4), this follows from
Lemma 1.1 and Corollary A.2. unionsq
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A homological approach to pseudoisotopy theory. I 1111
2 Surgery theory and spaces of block diffeomorphisms
We use surgery theory to give a partial p-local description of the space
B˜Diff

∂0W (W, ∂1W ) of block diffeomorphisms with tangential structures in
terms of spaces of homotopy automorphisms with bundle data for manifold
triads (W ; ∂0W, ∂1W ) of dimension d ≥ 6 satisfying a π -π -condition.
Notation As a point of notation, we refer to the geometric realisation of any
of the semi-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 ; ∂0W, ∂1W ) consisting of
a compact smooth d-manifold W (possibly with corners) and (possibly empty)
submanifolds ∂0W ⊂ W and ∂1W ⊂ W such that
∂W = ∂0W ∪ ∂1W and ∂(∂0W ) = ∂0W ∩ ∂1W = ∂(∂1W ).
A diffeomorphism or (simple) homotopy equivalence (W ; ∂0W, ∂1W ) →
(W ′; ∂0W ′, ∂1W ′) between two manifold triads is a diffeomorphism or (sim-
ple) homotopy equivalence W → W ′ which restricts to a map of this kind
between ∂0W and ∂0W ′, between ∂1W and ∂1W ′, and between their intersec-
tions. At times, we omit ∂0W and ∂1W from the notation and abbreviate a triad
(W ; ∂0W, ∂1W ) simply by W . The smooth structure set S(W ) of a triad W (see
e.g. [43, Ch. 10]) is the collection of equivalence classes of simple homotopy
equivalences of triads N → W that restrict to a diffeomorphism ∂0 N → ∂0W ,
where two such equivalences N → W and N ′ → W are equivalent if there
exists a diffeomorphism of triads N → N ′ that makes the triangle of triads
N
W
N ′

∼=

homotopy commute relative to ∂0 N . The structure set S(W ) is canonically
based; the identity serves as basepoint. The main tool to access S(W ) is the
surgery exact sequence
. . . S(W × Dk) N (W × Dk) L(W × Dk) S(W × Dk−1)
. . . L(W × D1) S(W ) N (W ) L(W )
(16)
123
1112 M. Krannich
which relates the structure sets S(W × Dk) of the triads
W × Dk = (W × Dk; ∂0W × Dk ∪ W × ∂ Dk; ∂1W × Dk)
to the sets of normal invariants N (W × Dk) and the L-groups L(W × Dk).
Assuming d ≥ 6, this sequence is an exact sequence of abelian groups until
L(W × D1) where it continues as an exact sequence of based sets (see e.g. [43,
Ch. 10]). The similarity with the long exact sequence of homotopy groups
induced by a fibration is no coincidence: Quinn’s surgery fibration [34,35] is
a homotopy fibration of based spaces
S˜(W ) −→ N(W ) −→ L(W ) (17)
that induces (16) on homotopy groups (see also [43, Ch. 17A], or [32] for a
detailed account in the topological category). We refrain from describing (16)
or (17) in detail; all we 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 theorem results in a
preferred homotopy equivalence
˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W )  S˜(W )id (18)
between the homotopy fibre of the map B˜Diff∂0W (W ) → B˜hAut
∼=
∂0W (W, ∂1W )
induced by inclusion and the basepoint component S˜(W )id ⊂ S˜(W ) of the
block structure space (cf. [43, Ch. 17A] or [7, p. 33-34]). Here
˜hAut
∼=
∂0W (W, ∂1W ) ⊂ ˜hAut∂0W (W, ∂1W ) (19)
are the components in the image of the map˜Diff∂0W (W ) →˜hAut∂0W (W, ∂1W ).
Note that a diffeomorphism of W that fixes ∂0W pointwise automatically
preserves ∂1W setwise, since ∂1W is the complement of the interior of
∂0W ⊂ ∂W . On homotopy groups, the equivalence (18) can be described as
follows: using the combinatorial description of the relative homotopy groups
of a semi-simplicial Kan pair, a class in
πk(˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W ); id)
∼= πk(˜hAut
∼=
∂0W (W, ∂1W ),˜Diff∂0W (W ); id)
123
A homological approach to pseudoisotopy theory. I 1113
is represented by a simple homotopy equivalence of triads W × Dk → W ×
Dk which is the identity on ∂0W × Dk and restricts to a diffeomorphism on
W ×∂ Dk , so it defines a class in the structure set S(W × Dk) ∼= πk(S˜(W ); id).
2.1.2 The space of normal invariants
The space of normal invariants N(W ) admits a preferred homotopy equivalence
to the pointed mapping space Maps∗(W/∂0W, G/O)based at the constant map,
where G/O is the homotopy fibre of the canonical map BO → BG witnessing
the fact that a stable vector bundle has an underlying stable spherical fibration
(see e.g. [34] or [43, 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
space Maps∗(W/∂0W, G/O). On homotopy groups, the composition
S˜(W ) → N(W )  Maps∗(W/∂0W, G/O) → Maps∗(W/∂0W, BO)
has the following geometric description (see e.g. [43, p. 113-114]): given a
class in the structure set πk(S˜(W ); ∗) ∼= S(W × Dk) represented by a simple
homotopy equivalence ϕ : N → W × Dk , choose a homotopy inverse ϕ˜ : W ×
Dk → N of triads that agrees with (ϕ|∂0 N )−1 on ∂0(W × Dk). Writing νs and
τ s for the stable normal respectively tangent bundle the stable vector bundle
(ϕ˜∗νsN ) ⊕ τ sW×Dk on W × Dk comes with a trivialisation on the subspace
∂0(W × Dk) = ∂0W × Dk ∪ W × ∂ Dk
by making use of the diffeomorphism ϕ˜|∂0(W×Dk), and hence gives a class
[(ϕ˜∗νsN ) ⊕ τ sW×Dk ] ∈ πk(Maps∗(W/∂0W, BO); ∗).
2.1.3 The L-theory space
The L-theory space L(W ) is an infinite loop space as well (see e.g. [32,
Prop. 2.2.2]), and its homotopy groups are canonically isomorphic to Wall’s
quadratic L-groups (see e.g. [32, Prop. 2.2.4]). We shall not need to know
much about these groups, except that πk(L(W ); ∗) ∼= L(W × Dk) vanishes if
W satisfies the π -π -condition, i.e. if the inclusion ∂1W ⊂ W induces an equiv-
alence on fundamental groupoids. This is a consequence of the exact sequence
of L-groups of a triad (or more generally, n-ad) described for instance in [43,
Thm 3.1]. In other words, under these assumptions, the L-theory space L(W ) is
weakly contractible, so (17) induces a preferred equivalence S˜(W )  N(W )—
an instance of the so-called π -π -theorem.
123
1114 M. Krannich
2.2 The block derivative map of a triad
With these basics of space-level surgery theory in mind, we now turn towards
studying connectivity properties of the block derivative map resulting from
(11) and (12)
B˜Diff∂0W (W ) −→ B˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ
and its enhancement involving tangential structures (i.e. the realisation of the
bottom row of (14)), beginning with a technical but useful lemma on the
homotopy fibre featuring in (18). We refer Sect. 1.2 for a recollection on
nilpotent spaces.
Lemma 2.1 For a manifold triad (W ; ∂0W, ∂1W ) of dimension d ≥ 6, the
homotopy fibre
˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W )
= hofib(B˜Diff∂0W (W ) → B˜hAut
∼=
∂0W (W, ∂1W ))
is a nilpotent space.
Proof By definition of the subspace ˜hAut
∼=
∂0W (W, ∂1W ) ⊂ ˜hAut∂0W (W, ∂1W )
in Sect. 2.1.1, the map B˜Diff∂0W (W ) → BhAut∼=∂0W (W, ∂1W ) is surjective
on fundamental groups, so its homotopy fibre is connected. To see that it
is nilpotent, we use the equivalence (18) to the identity component S˜(W )id
of the block structure space, which is itself equivalent to a component of
the homotopy fibre of the surgery obstruction map N(W ) → L(W ) of (17).
Given this description of the space in consideration, the claim follows 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 Sect. 2.1.2). unionsq
To state the main result of this section, some notation is in order. For a
stable tangential structure  : B → BO and a -structure  : τ sW → ∗γ (see
Sect. 1.8.1), we write 0 for the restriction of  to τ sW |∂0W and denote by
B˜Diff

∂0W (W ; 0) and B˜hAut
,∼=
∂0W (τ
s
W , ∂1W ; 0)τ
the components of the spaces (see Sects.1.8.1 and 1.9 for the notation)
B˜Diff

∂0W (W ; 0) = ˜Bun∂0W (τ sW , ∗γ ; 0)τ //˜Diff∂0W (W )
123
A homological approach to pseudoisotopy theory. I 1115
and
B˜hAut
,∼=
∂0W (τ
s
W , ∂1W ; 0)τ = ˜Bun∂0W (τ sW , ∗γ ; 0)τ //˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ
that correspond to  under the bijection
π0B˜hAut
,∼=
∂0W (τ
s
W , ∂1W ; 0)τ ∼= π0B˜Diff

∂0W (W ; 0)
∼= π0˜Bun∂0W (τ sW , ∗γ ; 0)τ /π0˜Diff∂0W (W )
resulting from Remark 1.2 1.2 and the discussion in Sects.1.8.1 and 1.9, using
that the map induced by the block derivative
π0˜Diff∂0W (W ) −→ π0˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ
is surjective by the definition of the target in (12). Recall from Sects.2.1.3
that a manifold triad W = (W ; ∂0W, ∂1W ) satisfies the π -π -condition if the
inclusion ∂1W ⊂ W induces an equivalence on fundamental groupoids.
Theorem 2.2 Let d ≥ 6 and W be a d-dimensional triad satisfying the π -π -
condition. For a stable tangential structure  and a -structure  on τ sW , the
homotopy fibre of the map
B˜Diff

∂0W (W ; 0) −→ B˜hAut
,∼=
∂0W (τ
s
W , ∂1W ; 0)τ
is nilpotent and has finite homotopy groups. Moreover, this fibre is p-locally
(2p−4−k)-connected for primes p, where k is the relative handle dimension
of the inclusion ∂0W ⊂ W .
Remark 2.3 Theorem 2.2 is inspired by a similar result of Berglund and Mad-
sen [8, Thm 1.1], which applies to a different class of triads, namely those
satisfying ∂0W = ∂W ∼= Sd−1. Another point in which their result differs
from ours is that it is purely rational, and does in fact not seem to admit a p-
local refinement analogous to Theorem 2.2. This is because 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.
123
1116 M. Krannich
Proof of Theorem 2.2 Using Corollary A.2, an application of Lemma 1.1 to
the horizontal arrows of the canonical square
B˜Diff

∂0W (W ; 0)τ B˜Diff∂0W (W )
B˜hAut
,∼=
∂0W (τ
s
W , ∂1W ; 0)τ B˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ ,
(20)
identifies the horizontal homotopy fibres with the union of components of
the space˜Bun∂0W (τ sW , 
∗γ ; 0)τ given by the π0˜Diff∂0W (W )-orbit of the -
structure , so (20) is homotopy cartesian, which reduces the proof to the case
 = id in which both rows of the square are equivalences (see Sect. 1.8.1).
To settle the case  = id, we consult the map of fibre sequences induced by
the block derivative map
˜hAut
∼=
∂0W (W,∂1W )
˜Diff∂0W (W )
B˜Diff∂0W (W ) B˜hAut
∼=
∂0W (W, ∂1W )
˜hAut
∼=
∂0W (W,∂1W )
˜hAut
∼=
∂0W (τ
s
W ,∂1W )τ
B˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ B˜hAut
∼=
∂0W (W, ∂1W )
(21)
in order to see that the homotopy fibre in question agrees with the homotopy
fibre of the left vertical map and is therefore nilpotent by Lemmas 1.4 and 2.1.
The bottom right horizontal map of (21) forms the rightmost column of a
commutative diagram
BhAut∼=∂0W (τ
s
W , ∂1W ) BhAut
∼=
∂0W (τ
s
W , ∂1W )
τ B˜hAut
∼=
∂0W (τ
s
W , ∂1W )
τ
BhAut∼=∂0W (W, ∂1W ) BhAut
∼=
∂0W (W, ∂1W ) B˜hAut
∼=
∂0W (W, ∂1W )
 

(22)
whose right horizontal maps are induced by inclusion and are equivalences;
the upper by Lemma A.3 and the lower by the discussion in Sect. 1.5. The
left upper horizontal map is the equivalence from (7) and the leftmost vertical
arrow fits into a commutative square induced by inclusion
BhAut∼=∂0W (τ
s
W , ∂1W ) BhAut∂0W (τ
s
W , ∂1W )
BhAut∼=∂0W (W, ∂1W ) BhAut∂0W (W, ∂1W )τ sW ,
(23)
123
A homological approach to pseudoisotopy theory. I 1117
where hAut∂0W (W, ∂1W )τ sW are the components in the image of the map
hAut∂0W (τ sW , ∂1W ) −→ hAut∂0W (W, ∂1W ) (24)
that forgets bundle data. Before taking geometric realisation, the map (24) is
easily seen to be a Kan fibration, so the homotopy fibre of the right vertical map
in (23) is equivalent to the classifying space of the gauge group of τ sW relative to
∂0W , which is in turn canonically equivalent to the space Maps∂0W (W, BO)τ sW
of maps homotopic to a choice of classifying map for τ sW relative to ∂0W .
The horizontal arrows in (23) are 1-coconnected by construction, so the same
holds for the induced map on vertical homotopy fibres. Combining this with
Lemma 4.8 (i), we see that the left vertical map in (21) agrees, up to canonical
equivalence and postcomposition with a 1-coconnected map, with a map
˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W ) −→ Maps∂0W (W, BO)τ sW ,
so it suffices to show that this map between nilpotent spaces is p-locally
(2p − 3 − k)-connected and that its homotopy fibre has finite homotopy
groups. On homotopy groups, this map has the following description: a class in
πk(˜hAut
∼=
∂0W (W, ∂1W ),˜Diff∂0W (W ); id) is represented by a homotopy equiv-
alence of triads ϕ : W × Dk → W × Dk that is the identity on Dk × ∂0W
and restricts to a diffeomorphism on ∂ Dk × W that is the identity on ∗ × M
for a base point ∗ ∈ ∂ Dk . The pullback ϕ∗τ sW×Dk is a stable vector bundle
over W × Dk that agrees with τ sW×Dk over Dk × ∂0W ∪ {∗} × W and comes
with a canonical identification ϕ∗τ sW×Dk |∂ Dk×M ∼= τ sW×Dk |∂ Dk×M given by
the derivative of ϕ|∂ Dk×M , so it defines a class in the kth homotopy group of
Maps∂0W (W, BO)τ sW based at a choice of classifying map for τ
s
W . Comparing
this description with those of the equivalence
˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W )  S˜(W )id
and the map S˜(W )id → Maps∗(W/∂0W, BO)0 on homotopy groups explained
in Sections 2.1.1 and 2.1.2, one sees that the diagram of nilpotent spaces
˜hAut
∼=
∂0W (W, ∂1W )/˜Diff∂0W (W ) Maps∂0W (W, BO)τ sW
S˜(W )id
Maps∗(W/∂0W, G/O)0 Maps∗(W/∂0W, BO)0,

ι◦((−)·νsW )

123
1118 M. Krannich
commutes upon taking homotopy groups. Here the upper horizontal map is the
one we just discussed, the right vertical equivalence is induced by multiplica-
tion with the stable normal bundle νsW using the infinite loop space structure
induced from that of BO followed by the involution on Maps∂0W (W, BO)0
induced by the canonical involution on BO, the left 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/∂0W, BO)0 is simple, so its homotopy groups at different base
points are canonically identified. The bottom arrow is a map of infinite loop
spaces and its homotopy fibre equivalent to a collection of components of the
infinite loop space Maps∗(W/∂0W, G), so to finish the proof, we are left to
show that the homotopy groups of this infinite loop space (including in degree
0) are finite and vanish p-locally in degrees∗ < (2p−3−k). This follows from
an application of obstruction theory, since W/∂0W has no cohomology above
degree k by the assumption on the relative handle dimension of ∂0W ⊂ W and
the homotopy groups of G are Z/2 in degree 0 and agree the stable homotopy
groups of spheres in positive degrees, which are finite and p-torsion free in
degrees ∗ < 2p − 3 by a result of Serre [38, p. 498, Prop. 5]. unionsq
The stable tangential structure we shall be primarily interested in is the one
encoding stable framings, which we denote by sfr : EO → BO. In this case, the
p-local approximation of the space of block diffeomorphisms with tangential
structures provided by Theorem 2.2 can be further simplified in terms of the
union of components
˜hAut
∼=
∂0W (W, ∂1W ) ⊂ ˜hAut
∼=
∂0W (W, ∂1W )
given by the image of the canonical map that forgets tangential structures
B˜hAut
sfr,∼=
∂0W (τ
s
W , ∂1W ; 0)τ −→ B˜hAut
∼=
∂0W (W, ∂1W )
on fundamental groups based at a fixed stable framing  of τW (see (12) and
(19) for the notation). Loosely speaking, these are the components of homotopy
equivalences of triads that are homotopic to a diffeomorphism preserving the
component of the stable framing .
Corollary 2.4 Let d ≥ 6 and W be a d-dimensional manifold triad satisfying
the π -π -condition. For a stable framing  of W , the homotopy fibre of
B˜Diff
sfr
∂0W (W ; 0) −→ B˜hAut
∼=
∂0W (W, ∂1W )
123
A homological approach to pseudoisotopy theory. I 1119
is nilpotent and has finite homotopy groups. Moreover, this fibre is p-locally
(2p−4−k)-connected for primes p, where k is the relative handle dimension
of the inclusion ∂0W ⊂ W .
Proof Once we show that the map
B˜hAut
sfr,∼=
∂0W (τ
s
W , ∂1W ; 0)τ −→ B˜hAut
∼=
∂0W (W, ∂1W )
is an equivalence, the statement is a consequence of Theorem 2.2. By con-
struction, the homotopy fibre of this map is connected. Using (22) and (23),
one sees that it is thus sufficient to show that the map
BhAutsfr∂0W (τ
s
W , ∂1W ; 0) −→ BhAut∂0W (W, ∂1W )
is an equivalence. Taking vertical homotopy fibres in the map of fibre sequences
Bun∂0W (τ sW , sfr
∗γ ; 0) BhAutsfr∂0W (τ sW , ∂1W ; 0) BhAut∂0W (τ sW , ∂1W )
∗ BhAut∂0W (W, ∂1W ) BhAut∂0W (W, ∂1W ),
where Bun∂0W (τ sW , sfr
∗γ ; 0) ⊂ Bun∂0W (τ sW , sfr∗γ ; 0) are the components
of the -orbit of the π0hAut∂0W (τ sW , ∂1W )-action, we see that it suffices to
show that the induced action of the loop space of the homotopy fibre of
the right vertical map on Bun∂0W (τ sW , sfr
∗γ ; 0) is a torsor in the homo-
topical sense, i.e. that the map given by acting on  is an equivalence. Since
hAut∂0W (τ sW , ∂1W )• → hAut∂0W (W, ∂1W )• is a Kan fibration, this loop space
is canonically equivalent to the space of bundle self-maps of τ sW that cover the
identity and agree with the identity on τ sW |∂0W . Moreover, by construction
of the top fibration, the induced action on Bun∂0W (τ sW , sfr
∗γ ; 0) is given
by precomposition. Suitably modeled, this action is simply transitive, so the
assertion follows. unionsq
3 High-dimensional handlebodies and their mapping classes
This section serves to compute variants of the mapping class group of a high-
dimensional handlebody up to extensions in terms of automorphisms of the
integral homology.
123
1120 M. Krannich
3.1 Automorphisms of handlebodies
The proof of Theorem A relies on considering a more general family of man-
ifold than discs, the boundary connected sums
Vg := g(Dn+1 × Sn),
and their boundaries as well as the manifolds obtained by cutting out a fixed
embedded disc D2n ⊂ ∂Vg, denoted by
Wg := ∂Vg ∼= g(Sn × Sn) and Wg,1 := Wg\int(D2n).
This includes the case g = 0, that is V0 = D2n+1, W0 = S2n , and W0,1 =
D2n . Using the notation introduced in Sections 1.4 and 1.5, the restriction of
diffeomorphisms and relative homotopy automorphisms of Vg to its boundary
induces a commutative diagram
Diff∂(Vg) Diff D2n (Vg) Diff∂(Wg,1)
hAut∂(Vg) hAutD2n (Vg, Wg,1) hAut∂(Wg,1)
(25)
where the right horizontal maps are fibrations and the left maps the inclusions
of the fibres over the identity. These fibrations need not be surjective; we denote
their images by
Diffext∂ (Wg,1) ⊂ Diff∂(Wg,1) and hAutext∂ (Wg,1) ⊂ hAut∂(Wg,1).
Furthermore, in agreement with (19) we write
hAut∼=∂ (Vg), hAut
∼=
D2n (Vg, Wg,1), and hAut
∼=
∂ (Wg,1)
for the components hit by the vertical maps. Block variants of all of the above
automorphism spaces are defined in the same way.
3.2 The mapping class group
As a first step in our analysis of the mapping class group π0Diff D2n (Vg), we
observe that we may equally study orientation-preserving diffeomorphisms of
Vg that do not necessarily fix the embedded disc in the boundary D2n ⊂ ∂Vg,
or diffeomorphisms that preserve a disc D2n+1 ⊂ int(Vg) in the interior set-
or pointwise.
123
A homological approach to pseudoisotopy theory. I 1121
Lemma 3.1 For n ≥ 2 and discs D2n ⊂ ∂Vg and D2n+1 ⊂ int(Vg), the
compositions induced by inclusion
π0Diff D2n+1(Vg) −→ π0Diff+(Vg, D2n+1) −→ π0Diff+(Vg) and
π0Diff D2n (Vg) −→ π0Diff+(Vg, D2n) −→ π0Diff+(Vg)
consist of isomorphisms.
Proof Up to isotopy, preserving a disc in the interior setwise is equivalent to
preserving a point, so the group π0Diff+(Vg, D2n+1) agrees with the group
of path components of the subgroup Diff+(Vg, ∗) ⊂ Diff+(Vg) of diffeomor-
phisms that fix the centre∗ ∈ D2n+1 ⊂ int(Vg). As Vg is (n−1)-connected, the
long exact sequence of the fibration Diff+(Vg) → int(Vg) given by evaluating
diffeomorphisms at ∗ implies that π0Diff+(Vg, ∗) agrees with π0Diff+(Vg),
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
Diff D2n+1(Vg) −→ Diff+(Vg, ∗) d−→ SO(2n + 1) (26)
and hence an exact sequence
π1SO(2n + 1) t−→ π0Diff D2n+1(Vg) −→ π0Diff+(Vg, ∗) −→ 0.
On the subgroup SO(n) ⊂ SO(2n + 1), the derivative map d has a sec-
tion since Vg admits a smooth SO(n)-action with fixed point ∗ ∈ Vg whose
tangential representation agrees with the restriction of the standard represen-
tation to the subgroup SO(n): take the g-fold equivariant boundary connected
sums of Dn+1 × Sn with SO(n) acting by rotating Sn along an axis. As
SO(n) ⊂ SO(2n + 1) is (n − 1)-connected, this section ensures that the
long exact sequence on homotopy groups of (26) splits in degrees ∗ ≤ n − 1
into short exact sequences and shows in particular that t is trivial for n ≥ 2,
so π0Diff D2n+1(Vg) ∼= π0Diff+(Vg) holds as claimed. Replacing (26) by the
fibration sequence Diff D2n (Vg) → Diff+(Vg, ∗) → SO(2n) induced by tak-
ing the derivative at the centre ∗ ∈ D2n ⊂ ∂Vg of the disc in the boundary,
the proof of the claim regarding the maps in the second composition of the
statement proceeds analogous to the first part of the proof. unionsq
3.2.1 The homology action
To obtain further information on the mapping class group π0Diff D2n (Vg),
we consider its action on the nth integral homology of Vg and its boundary
123
1122 M. Krannich
∂Vg = Wg, which we abbreviate by
HWg,1 := Hn(Wg,1; Z) and HVg := Hn(Vg; Z).
This action preserves further structure, such as the intersection pairing
λ : HWg,1 ⊗ HWg,1 −→ Z,
which equips HWg,1 with a nondegenerate (−1)n-symmetric form by Poincaré
duality. In addition, any automorphism of HWg,1 induced by an orientation-
preserving diffeomorphism of Wg,1 has to preserve the function
α : HWg,1 −→ πnBSO(n)
given by representing a homology class by an embedded n-sphere and taking
its normal bundle. Wall [39, Thm 2] has shown that λ and α satisfy the relations
(i) λ(x, x) = ∂nα(x)
(ii) α(x + y) = α(x) + α(y) + λ(x, y) · τSn
as long as n ≥ 3, where
∂n : πnBSO(n) −→ πn−1Sn−1 ∼= Z
is induced by the fibration Sn−1 → BSO(n − 1) → BSO(n) and τSn ∈
πnBSO(n) is the class representing the tangent bundle of the n-sphere. In
sum, we arrive at a morphism
π0Diff(Wg,1) −→ Gg := Aut(HWg,1, λ, α)
to the subgroup Gg ⊂ GL(HWg ) of automorphisms preserving λ and α, which
is surjective by [40, Lem. 10]. However, we are interested in the mapping
class group π0Diff D2n (Vg) and not every automorphism in Gg is realised by
a diffeomorphism of Wg,1 that extends to one of Vg; it would at least have to
preserve the Lagrangian subspace
Kg := ker
(
HWg,1 → HVg
)
,
so there is a canonical map π0Diff D2n (Vg) → Gextg to the subgroup
Gextg := { ∈ Gg | (Kg) ⊂ Kg}
of automorphisms that preserve this Lagrangian, given by acting on the homol-
ogy of the boundary. Using the canonical isomorphism HWg,1/Kg ∼= HVg , the
123
A homological approach to pseudoisotopy theory. I 1123
subgroup Gextg maps further to GL(HVg ). The resulting composition
π0Diff D2n (Vg) −→ Gextg −→ GL(HVg ) (27)
agrees with the action on the homology of Vg and one may ask for the (co)kernel
of the three maps involved. Extending work of Wall [40,41], we express the
answer in Theorem 3.3 below in terms of an exact sequence
0 −→ H∨Vg ⊗ SπnSO(n) −→ Ng −→ Mg −→ 0, (28)
of GL(HVg )-modules where
(1) SπnSO(n) ⊂ πnSO(n + 1) is the image of the stabilisation map
πnSO(n) → πnSO(n + 1),
(2) Mg ⊂ (HVg ⊗ HVg )∨ is the submodule of bilinear forms μ ∈ (HVg ⊗
HVg )∨ that are
• (−1)n+1-symmetric and satisfy
• μ(x, x) ∈ im(∂n+1 : πnSO(n + 1) → Z) for x ∈ HVg , and
(3) Ng ⊂ Mg ⊕ (πnSO(n + 1))HVg is the submodule of pairs (μ, β) of a
bilinear form μ ∈ Mg and a function β : HVg → πnSO(n + 1) that fulfil
the conditions
• μ(x, x) = ∂n+1β(x) for x ∈ HVg and• β(x + y) = β(x) + β(y) + μ(x, y) · τSn+1 for x, y ∈ HVg ,
all equipped with the evident GL(HVg )-action through HVg . Here we denoted
the integral dual of a G-module M by M∨ := Hom(M, Z).
Remark 3.2 The image of the map ∂n+1 : πnSO(n + 1) → πn Sn ∼= Z is
generated by the order of the tangent bundle τSn ∈ πn−1SO(n), so we have
(see [29, §1B)])
im(∂n+1) =
⎧⎪⎨
⎪⎩
0 for n even
Z for n = 1, 3, 7
2 · Z otherwise,
(29)
which exhibits the condition μ(x, x) ∈ im(∂n+1) in (2) as vacuous unless
n = 1, 3, 7 is odd. Moreover, this shows that, after choosing a basis HVg ∼= Zg,
the module Mg can be described equivalently in terms of (−1)n+1-symmetric
integral (g × g)-matrices, with even diagonal entries if n = 1, 3, 7 is odd.
Theorem 3.3 Let n ≥ 3.
(1) The action of π0Diff D2n (Vg) on HVg gives rise to an extension
0 −→ Ng −→ π0Diff D2n (Vg) −→ GL(HVg ) −→ 0.
123
1124 M. Krannich
(2) The morphism Gextg → GL(HVg ) fits into an extension of the form
0 −→ Mg −→ Gextg −→ GL(HVg ) −→ 0.
(3) The action of π0Diff D2n (Vg) on HWg,1 induces an extension
0 −→ H∨Vg ⊗ SπnSO(n) −→ π0Diff D2n (Vg) −→ Gextg −→ 0.
Moreover, the induced outer actions of these extensions is as specified above
and the second extension admits a preferred splitting.
Remark 3.4 For a complete description of π0Diff D2n (Vg), one still needs to
determine the extension problems of the first or third part of the theorem, which
we do not pursue at this point. Similar extensions by Kreck [26] describing
the closely related mapping class group π0Diff∂(Wg,1) for n ≥ 3 have been
resolved in [25] for 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
instead of isotopy, but these notions agree in our situation by [11].
(2) Justified by Lemma 3.1, we do not distinguish between seemingly dif-
ferent variants of π0Diff D2n (Vg) fixing various discs point- or setwise.
(3) We identify Kg canonically with the dual HVg ∨ as a Gextg -module via the
isomorphism induced by the form λ, and dually HVg with K ∨g .
Wall [40, Lem. 10] showed that the action π0Diff D2n (Vg) → GL(HVg ) is sur-
jective and identified its kernel with those isotopy classes of diffeomorphisms
ϕ that are homotopic to the identity. Moreover, in [41, p. 298], he defined a
complete obstruction (μϕ, βϕ) ∈ Ng for such a homotopically trivial diffeo-
morphism to be isotopic to the identity. By [40, Lem. 12–13], these obstructions
are additive and exhaust Ng, so the resulting function
ker
(
π0Diff(Vg) → GL(HVg )
) −→ Ng (30)
is an isomorphism of groups, which establishes the first of the three claims.
To demonstrate the second, note that, as automorphisms in Gextg preserve the
form λ, the composition
Gextg −→ GL(HVg ) ((−)
−1)∨−→ GL(Kg)
agrees with the restriction to Kg. In particular, this shows that the kernel of
the first map in this composition acts trivially on Kg, so there is a canonical
123
A homological approach to pseudoisotopy theory. I 1125
monomorphism
ker(Gextg → GL(HVg ))

↪−→ Hom(HVg , Kg) ∼= (HVg ⊗ HVg )∨
defined by sending φ ∈ Gextg to the linear map (φ) : HVg → Kg induced
by the difference (φ − id) : HWg,1 → Kg. This leaves us with identifying
the image of  with Mg for which it is helpful to note that a morphism
f ∈ Hom(HVg , Kg) lies in the subspace
Mg ⊂ (HVg ⊗ HVg )∨ ∼= Hom(HVg , Kg)
if and only if f ∨ = (−1)n+1 f and λ( f (x), x˜) ∈ im(∂n+1) holds for all
x ∈ HVg . Here x˜ ∈ HWg,1 is a choice of preimage of x under the projection
HWg,1 → HVg , but the value λ( f (x), x˜) is independent of this choice x˜ since
Kg is Lagrangian. For elements of the form (φ), the first property (φ)∨ =
(−1)n+1(φ) follows from the fact that φ preserves the form λ. To see the
second, we note that, since Kg is Lagrangian, the function α is additive on Kg,
so it vanishes on it since the images of the second Sn-factors of the connected
sum Wg,1 ∼= g(Sn × Sn)\Dn in Vg = g(Dn+1 × Sn) induce a basis of Kg
and have trivial normal bundle. Using the fact that φ ∈ Gextg preserves α and
property (ii) of α, we compute
α(x˜) = α((φ)(x) + x˜) = α((φ)(x)) + α(x˜) + λ((φ)(x), x˜) · τSn
= α(x˜) + λ((φ)(x), x˜) · τSn
and conclude that λ
(
(φ)(x), x˜
) ·τSn vanishes, so λ((φ)(x), x˜) ∈ im(∂n+1)
holds as claimed. This proves that the image of  is contained in Mg, and to
show that it agrees with it, we consider the commutative diagram
0 Ng π0Diff(Vg) GL(HVg ) 0
0 ker(Gextg → GL(HVg )) Gextg GL(HVg ) 0
Mg

whose vertical arrow Ng → ker(Gextg → GL(HVg )) is induced by the iso-
morphism (30). By [41, Lem. 24], the vertical composition in the diagram
agrees with the projection Ng → Mg in (28), so it is surjective. Consequently,
 is surjective as well and hence an isomorphism, which proves the sec-
ond claim of the statement. To show the third, we observe that, given that
123
1126 M. Krannich
 is an isomorphism and Ng → Mg is surjective, the diagram implies that
π0Diff(Vg) → Gextg is surjective and moreover that its kernel agrees with the
kernel of Ng → Mg, which is H∨Vg ⊗ SπnSO(n) as claimed.
We now identify the actions as asserted. For the second extension, one
can argue as follows: an automorphism ϕ ∈ GL(HVg ) acts on ker(Gextg →
GL(HVg )) by conjugating with a choice of lift ϕ˜ ∈ Gextg , so the isomorphism 
is equivariant simply because of the identity (ϕ˜(φ−id)ϕ˜−1) = (ϕ˜φϕ˜−1−id) in
Hom(HVg , Kg) for all φ ∈ Hom(HVg , Kg). For the first extension, we use that
the left vertical composition in the diagram above agrees with the projection
Ng → Mg, so it suffices to show that the composition
ker(π0Diff D2n (Vg) → GL(HVg ))
∼=−→ Ng −→ (πnSO(n + 1))HVg (31)
of (30) with the projection is equivariant. From Wall’s definition [40, p. 267]
of the invariant βϕ of an element ϕ in the kernel, we see that this composi-
tion can be described as follows: representing a homology class [e] ∈ HVg
by an embedded sphere e : Sn → Vg, we can 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) ∼= εn+1 which induces an element
βϕ([e]) ∈ πnSO(n +1). This uses a trivialisation of ν(e), but the resulting ele-
ment βϕ([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 from that of the first two by chasing through the
diagram obtained by extending the diagram above by taking vertical kernels.
To see that the second extension splits, note that the second Sn-factors in the
connected sum decomposition of Wg,1 = g Sn × Sn\int(D2n) ⊂ g Dn+1 ×
Sn = Vg induce a splitting of the canonical map HWg,1 → HVg and thus
an isomorphism of the form Kg ⊕ HVg ∼= HWg,1 . Using this splitting of the
homology, we can define a morphism GL(HVg ) → Gextg by assigning φ ∈
GL(HVg ) the automorphism
HWg,1 ∼= Kg ⊕ HVg
(φ−1)∨⊕φ−−−−−−→ Kg ⊕ HVg ∼= HWg,1,
which clearly preserves Kg. Moreover, its induced automorphism of HWg,1
agrees with φ by construction, so we obtain a splitting as desired. unionsq
Remark 3.5 With respect to the basis HWg,1 ∼= Z2g suggested by the connected
sum decomposition Wg,1 ∼= g(Sn × Sn)\int(D2n), the subgroup Gextg ⊂
GL(HWg,1) ∼= GL2g(Z) agrees with the group of block matrices of the form
(
A M
0 (A−1)T
)
123
A homological approach to pseudoisotopy theory. I 1127
with M ∈ Mg, using the matrix description of Mg explained in Remark 3.2.
From this point of view, the splitting GLg(Z) → Gextg described in the proof
of Theorem 3.3 is the obvious one that sends a matrix A ∈ GLg(Z) to the
block diagonal matrix with M = 0.
3.3 Stable framings
Choosing the canonical map sfr : EO → BO as a stable tangential structure in
the sense of Section 1.8.1, the space of sfr-structures
BunD2n (τ sVg , sfr
∗γ ; 0)
as defined in that section is the space of stable framings of Vg relative to a fixed
stable framing 0 : τ sVg |D2n → sfr∗γ . We denote the stabiliser of the canonical
action of π0Diff D2n (Vg) on the set of components π0BunD2n (τ sVg , sfr
∗γ ; 0)
and its image in Gextg by
π0Diff D2n (Vg) ⊂ π0Diff D2n (Vg) and Gextg, ⊂ Gextg .
Note that π0Diff D2n (Vg) agrees with the image of the canonical map
BDiffsfrD2n (Vg) → BDiff D2n (Vg) on fundamental groups based at the point
induced by the stable framing , or equivalently, with the kernel of the crossed
homomorphism
π0Diff D2n (Vg) −→ H∨Vg ⊗ πnSO (32)
given by acting on . This uses the identification
H∨Vg ⊗ πnSO ∼= Hom(HVg , πnSO)
∼= π0MapsD2n (Vg, SO) ∼= π0BunD2n (τ sVg , θ∗γ ; 0)
whose first two isomorphisms are the evident ones and whose third is induced
by the choice of stable framing , using that π0BunD2n (τ sVg , sfr
∗γ ; 0) is a
torsor over π0MapsD2n (Vg, SO) with pointwise multiplication.
Remark 3.6 As the stabilisation map SO(2n + 1) → SO is 2n-connected, the
induced map MapsD2n (Vg, SO(2n+1)) → MapsD2n (Vg, SO) is n-connected,
which implies that there is no difference between equivalence classes of stable
and unstable framings, so the discussion of this subsection applies equally well
to unstable framings instead of stable ones.
To relate the subgroups π0Diff D2n (Vg) ⊂ π0Diff D2n (Vg) and Gextg, ⊂ Gextg
to the sequences of Theorem 3.3, we define the GL(HVg )-submodule
123
1128 M. Krannich
(1) N sfrg ⊂ Ng as the intersection of Ng with Mg ⊕〈τSn+1〉HVg , where 〈τSn+1〉
is the subgroup of πnSO(n + 1) generated by the tangent bundle τSn+1
(2) Msfrg ⊂ Mg as the collection of (−1)n+1-symmetric bilinear forms μ ∈
(HVg ⊗ HVg )∨ that are even, i.e.μ(x, x) ∈ 2 · Z for all x ∈ HVg , which
is automatic if n is even.
Standard arguments involving the long exact sequences in homotopy groups
of the usual fibration SO(d) → SO(d + 1) → Sd (cf. [29, §1B)]) show that
the sequence (28) restricts to an exact sequence of GL(HVg )-modules
0 −→ H∨Vg ⊗ ker
(
SπnSO(n) → πnSO
) −→ N sfrg −→ Msfrg −→ 0.
Proposition 3.7 Let n ≥ 3.
(i) The action of π0Diff D2n (Vg) on the set π0BunD2n (τ sVg , sfr∗γ ; 0) of
equivalence classes of stable framings is transitive.
(ii) For any stable framing  : τ sVg → sfr∗γ , the sequences of Theorem 3.3
restrict to exact sequences of the form
0 −→ N sfrg −→ π0Diff D2n (Vg) −→ GL(HVg ) −→ 0,
0 −→ Msfrg −→ Gextg, −→ GL(HVg ) −→ 0, and
0 −→ H∨Vg ⊗ ker
(
SπnSO(n) → πnSO
) −→ π0Diff D2n (Vg) −→ Gextg, −→ 0.
Remark 3.8 The calculation of im(∂n+1) in Remark 3.2 shows that the inclu-
sion Msfrg ⊂ Mg is an equality for n = 1, 3, 7, so the same holds for
Gextg, ⊂ Gextg as a result of Proposition 3.7.
Proof of Proposition 3.7 The first sequence of Theorem 3.3 fits into a diagram
0 Ng π0Diff D2n (Vg) GL(HVg ) 0
H∨Vg ⊗ πnSO
τ
where τ is the crossed homomorphism (32), which is up to isomorphism
given by the action of π0Diff D2n (Vg) on , so the first part of the statement is
equivalent to the surjectivity of τ . The diagonal arrow is the morphism which
assigns an element (μ, β) ∈ Ng the composition of β : HVg → πnSO(n + 1)
with the stabilisation map πnSO(n + 1) → πnSO. As τSn+1 ∈ πnSO(n + 1)
is stably trivial, it follows from the second defining property of Ng that this
composition is additive, so indeed defines an element of HVg ∨ ⊗ πnSO ∼=
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A homological approach to pseudoisotopy theory. I 1129
Hom(HVg , πnSO). As a next step, observe that the diagonal map is surjective,
since the projection
Ng ⊂ Mg ⊕ (πnSO(n + 1))HVg → (πnSO(n + 1))HVg
has a section over the subspace of linear maps HVg ∨ ⊗ πnSO(n + 1) ⊂
(πnSO(n + 1))HVg by setting the Mg-coordinate to zero and because the sta-
bilisation map πnSO(n + 1) → πnSO is surjective. This reduces the first
claim of the statement to the commutativity of the triangle, which follows
from the geometric description of the composition (31) we gave in the proof
of Theorem 3.3 in a straight-forward manner.
The kernel of the vertical map agrees with the stabiliser π0Diff D2n (Vg), so
the surjectivity of the diagonal arrow in the diagram also shows that this sta-
biliser surjects onto GL(HVg ) and that the kernel of the restriction ker(τ ) →
GL(HVg ) agrees with the kernel of the diagonal arrow. But this kernel is
exactly the submodule N sfrg ⊂ Ng, because the kernel of the stabilisation map
πnSO(n+1) → πnSO agrees with the kernel of πnSO(n+1) → πnSO(n+2),
i.e. the subgroup generated by τSn+1 . This establishes the first sequence of the
second claim. For the second, note that the surjectivity of the morphism
π0Diff D2n (Vg) −→ GL(HVg )
implies that also the morphism
Gextg, −→ GL(HVg )
is surjective. Comparing the first two sequences of Theorem 3.3, we see that
its kernel agrees with the image of N sfrg in Mg under the projection Ng → Mg,
i.e. with those bilinear forms μ ∈ Mg that satisfy μ(x, x) ∈ im(∂n+1〈τSn+1〉).
The image of τSn+1 ∈ πnSO(n + 1) under ∂n+1 is the Euler characteristic of
Sn+1 (see [29, §1B)]), so the kernel in question agrees with Msfrg as claimed.
To establish the last sequence, one first observes that
π0Diff D2n (Vg) −→ Gextg,
is surjective by definition of the target, and then compares the first two
sequences in the claim to see that its kernel agrees with the kernel of
N sfrg → Msfrg , which agrees with H∨Vg ⊗ ker(SπnSO(n) → πnSO) as already
noted in the discussion prior to this proof. unionsq
Lemma 3.9 The negative of the identity −id ∈ GL(HWg ) is contained in the
subgroup Gextg, ⊂ GL(HWg ) for all stable framings  : τ sVg → sfr∗γ .
123
1130 M. Krannich
Proof We first restrict to the case g = 1 and a particular choice of stable
framing, namely that induced by the standard embedding Dn+1×Sn ⊂ Rn+1×
Rn+1. The diffeomorphism
Dn+1 × Sn −→ Dn+1 × Sn
((x1, . . . , xn+1), (y1, . . . , yn+1)) −→ ((−x1, x2, . . . , xn+1), (−y1, y2, . . . , yn+1)) (33)
is orientation preserving, maps to −id ∈ GL(HW1), and has constant deriva-
tive, so it preserves the stable framing as required. It does not preserve a disc in
the boundary, but by Lemma 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 fixed discs in the boundary to obtain
a diffeomorphism of V2 whose image in GL(HW2) is −id, which we can again
isotope so it preserves a disc as required. Continuing like this yields a sequence
of isotopy classes in π0Diff D2n (Vg) for all g ≥ 0 that satisfy the requirements
of the claim. This establishes the statement for one specific stable framing for
each g, but implies the general case, the reason being that −id ∈ GL(HWg )
is central and all subgroups Gextg, ⊂ Gextg are conjugate, because the different
stabilisers π0Diff D2n (Vg) are, as the action is transitive by Proposition 3.7.
This concludes the proof. unionsq
3.4 The homotopy mapping class group
Not only the smooth mapping class groups π0Diff D2n (Vg) and π0Diff∂(Wg,1)
act on the homology HWg,1 , also their homotopical cousins π0hAutD2n
(Vg, Wg,1) and π0hAut∂(Wg,1) do. Restricting these actions to the images
π0hAut
∼=
D2n (Vg, Wg,1) ⊂ π0hAutD2n (Vg, Wg,1) and
π0hAut
∼=
∂ (Wg,1) ⊂ π0hAut∂(Wg,1)
of the canonical maps
π0Diff D2n (Vg, Wg,1) → π0hAutD2n (Vg, Wg,1) and
π0Diff∂(Wg,1) → π0hAut∂(Wg,1),
they land in the subgroups Gextg and Gg of GL(HWg,1), respectively.
Lemma 3.10 Let n ≥ 3. The morphisms induced by the action on HWg,1
π0hAut
∼=
∂ (Wg,1) −→ Gg and π0hAut∼=D2n (Vg, Wg,1) −→ Gextg
are surjective. Moreover, their kernels are finite and p-torsion free as long as
n < 2p − 4.
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A homological approach to pseudoisotopy theory. I 1131
Proof In the course of this proof, we shall make frequent use of the fact that the
homotopy groups πn+k(∨h Sn) with n ≥ 3 are p-torsion free for k < 2p − 3
as a result of the Hilton–Milnor theorem (see (37) below) and the case h = 1
due to Serre [38, p. 498, Prop. 5].
To begin with the actual proof, note that the two morphisms of the statement
are certainly surjective, since this holds already for π0Diff∂(Wg,1) → Gg and
π0Diff(Vg) → Gextg by Theorem 3.3 and the discussion it precedes. Evi-
dently, the kernel of the first morphism of the statement is contained in that
of π0hAut∂(Wg,1) → GL(HWg,1), which enjoys the claimed finiteness and
torsion property by an application of the fibre sequence induced by restriction
along the boundary inclusion
hAut∂(Wg,1) −→ hAut∗(Wg,1) −→ Maps∗(∂Wg,1, Wg,1),
using the isomorphism π0hAut∗(Wg,1) ∼= GL(HWg,1) induced by the homol-
ogy action and the fact that the group π1Maps∗(∂Wg,1, Wg,1) ∼= π2n(∨2g Sn)
is finite and without p-torsion for n < 2p − 3. To deduce the claim for
the kernel of second morphism from this, we consult the fibre sequence
hAut∂(Vg) → hAutD2n (Vg, Wg,1) → hAut∂(Wg,1) to realise that it suf-
fices to show that π0hAut∂(Vg) is finite and p-torsion free for n < 2p − 4.
Using the fibre sequence hAut∂(Vg) → hAut∗(Vg) → Maps(Wg, Vg) and
the observation that its fibre inclusion is trivial on path components since
π0hAut∂(Vg) acts trivially on homology and π0hAut∗(Vg) ∼= GL(HVg ), we
see thatπ0hAut∂(Vg) is a quotient of the fundamental groupπ1Maps∗(Wg, Vg)
based at the inclusion ι. Finally, note that the fibre sequence Maps∗(Wg, Vg) →
Maps∗(Wg,1, Vg) → Maps∗(∂Wg,1, Vg) induces an exact sequence
π2n+1(∨g Sn)⊕g −→ π1(Maps∗(Wg, Vg); ι) −→ πn+1(∨g Sn)⊕2g
whose outer groups are finite and p-torsion free for n < 2p − 4, so we can
conclude the claim. unionsq
3.5 Stabilisation
To relate the automorphism spaces of the handlebody Vg = g Dn+1 × Sn
relative to the disc D2n ⊂ ∂Vg fixed in Section 3.1 to the ones of Vg+1,
it is convenient to modify Vg by introducing codimension 2 corners at the
boundary of the disc D2n ⊂ Vg in the boundary so that there is smooth
boundary preserving embedding c : (−1, 0]× D2n → Vg whose restriction to
{0} × D2n agrees with the chosen disc. Abusing common terminology, we call
123
1132 M. Krannich
such an embedding a collar. Fixing another disc D2n−1 ⊂ ∂ D2n , we consider
H := ([0, 1] × D2n)(Dn+1 × Sn),
where the boundary connected sum is performed away from the union
D := {0, 1} × D2n ∪ [0, 1] × D2n−1 ⊂ [0, 1] × D2n,
and think of Vg+1 as being obtained by gluing Vg to H along the two collared
discs D2n ⊂ Vg and {0}×D2n ⊂ H , where we declare {1}×D2n ⊂ H ⊂ Vg+1
to the new distinguished disc in the boundary, which comes with a preferred
collar. Given a tangential structure θ : B → BO(d), a choice of bundle map
0 : ε ⊕ τD2n → θ∗γ2n+1 induces canonical θ -structures on τVg |D2n and
τVg+1 |D2n by using the fixed collars, and also one on τH |D by making use of the
canonical trivialisation of τ[0,1]. With respect to these θ -structures, generically
denoted by 0, there is an evident gluing map for tangential structures
BunD2n (τVg , θ
∗γ2n+1; 0) × BunD(τH , θ∗γ2n+1; 0) −→ BunD2n (τVg+1 , θ∗2n+1γ ; 0)
that is equivariant with respect to the gluing morphism
Diff D2n (Vg) × Diff D(H) −→ Diff D2n (Vg+1) (34)
for diffeomorphisms. Taking homotopy orbits, this induces a map of the form
BDiffθD2n (Vg; 0) × BDiffθD(H ; 0) −→ BDiffθD2n (Vg+1; 0)
and hence a homotopy class of stabilisation maps
BDiffθD2n (Vg; 0) → BDiffθD2n (Vg+1; 0) (35)
that in general depends on the choice of a component of BDiffθD(H ; 0), but
not in any of the cases we shall be interested in, because of the following.
Lemma 3.11 If B is n-connected, then the space BDiffθD(H ; 0) is nonempty
and connected.
Proof Up to smoothing corners, the pairs (H, D) and (V1, D2n) are diffeomor-
phic, so there we have an equivalence BDiffθD(H ; 0)  BDiffθD2n (V1; 0),
which shows that the claim is equivalent to the transitivity of the action of
π0Diff D2n (V1) on π0BunD2n (τV1, θ∗γ2n+1; 0), using that the latter set is
nonempty as V1 is parallelizable. As B is n-connected, it is a consequence
of obstruction theory that every θ -structure on Vg is induced by a framing, so
it suffices to consider the case θ : EO(2n + 1) → BO(2n + 1), which we have
already settled as the first part of Proposition 3.7 (see also Remark 3.6). unionsq
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A homological approach to pseudoisotopy theory. I 1133
A similar discussion results in analogous stabilisation maps of the form
B˜Diff

D2n (Vg; 0) −→ B˜Diff

D2n (Vg+1; 0)
for stable tangential structures : B → BO and bundle maps 0 : τ sVg → ∗γ ,
which are compatible with the non-block variants defined above and are again
unique up to homotopy if B is n-connected. Moreover, using the splittings
of the inclusions HWg,1 → HWg+1,1 and HVg → HVg+1 suggested by the
decomposition Vg+1 = Vg ∪ H , there are stabilisation maps for the respective
linear groups on HWg,1 and HVg given by extending automorphisms by the
identity and these are related to the stabilisation maps described above via
the action on the homology of Vg and Wg,1 (see Section 3.2), so there is a
commutative diagram of compatible stabilisation maps that has the form
BDiff2n+1D2n (Vg; 0) B˜Diff

D2n (Vg; 0) BGextg BGL(HVg )
BDiff2n+1D2n (Vg+1; 0) B˜Diff

D2n (Vg+1; 0) BGextg+1 BGL(HVg+1 ).
4 Relative homotopy automorphisms of handlebodies
Theorem 2.2 illustrates that the space of block diffeomorphisms ˜Diff D2n (Vg)
is closely related to the space ˜hAutD2n (Vg, Wg,1) of relative block homotopy
automorphisms or, equivalently, to its non-block variant hAutD2n (Vg, Wg,1)
(see Section 1.5). To access the homology of the classifying space of this
space of homotopy automorphisms, one might try to study the Serre spectral
sequence of the fibration sequence induced by taking components
BhAutidD2n (Vg, Wg,1) −→ BhAutD2n (Vg, Wg,1) −→ Bπ0hAutD2n (Vg, Wg,1)
for which one ought to know at least the homology of the fibre as a module over
the group π0hAutD2n (Vg, Wg,1). This is what this section aims to compute—
p-locally and in a range of degrees—by first calculating the p-local homotopy
groups in a range using some tools from rational homotopy theory combined
with an ad-hoc extension to the p-local setting tailored to our situation, and
then pass from homotopy to homology groups.
123
1134 M. Krannich
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) homology of 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 com-
mutative ring R by sk A, the graded R-module whose degree k piece consists
of R-module morphisms raising the degree by k by Hom(A, B) for graded
modules A and B, the graded R-dual of A by A∨ := Hom(A, R[0]) where
R[0] is the base ring concentrated in degree zero, and the subspace of ele-
ments of strictly positive degrees by A+ ⊂ A. For an ungraded R-module M
we write M[k] for the graded R-module which is trivial in all degrees but k
where it agrees with M . The graded tensor product A ⊗ B is defined in the
usual way. Note that (sk A) ⊗ B = sk(A ⊗ B) = A ⊗ (sk B). The degreewise
rationalisation or p-localisation of a graded Z-module A is denoted by AQ 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) Lie algebras over a commutative
ring R. However, most of the dg Lie algebras which we shall encounter actu-
ally have trivial differential. Examples include the free graded Lie algebra
L(V ) on a graded R-module A or the onefold shift of the homotopy groups
π∗+1 X of a based space X with its canonical Lie algebra structure over Z
given by the Whitehead bracket (except for a 2-torsion subtlety that will not
play a role for us). Given a dg Lie algebra L , we write [L , L] ⊂ L for the
graded subspace generated by brackets. An important principle in this section
is that 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 , dL) → (L ′, d ′) of dg Lie algebras, an f -derivation of degree k is a
linear map θ : L → L ′ that raises the degree by k and satisfies
θ([x, y]) = [θ(x), f (y)] + (−1)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) = dL ′θ −
(−1)|θ |θdL , 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 Derid(L , L) by Der(L).
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A homological approach to pseudoisotopy theory. I 1135
4.3 Rational homotopy theory, Quillen style
Recall from [33] Quillen’s functor λ, which assigns a simply connected based
space X a dg Lie algebra λ(X) over the rationals, one of whose many prop-
erties is that it captures the rationalised homotopy Lie algebra of X via a
natural isomorphism H∗(λ(X)) ∼= π∗+1(X)Q of graded Lie algebras, where
H∗(λ(X)) = ker(dλ(X))/im(dλ(X)) is the homology Lie algebra of λ(X). A
Lie model of X is a rational dg Lie algebra L XQ quasi-isomorphic to λ(X). Such
a model is called free if the underlying graded Lie algebra of L XQ 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 differential, i.e. d(L XQ) ⊂ [L XQ, L XQ]. Any
simply connected based space has a minimal Lie model L XQ, unique up to (non-
canonical) isomorphism, and a based map between such spaces f : X → Y
gives rise to a map f : L XQ → LYQ between their minimal models.
4.4 Derivations and mapping spaces
As mentioned earlier, the homotopy theory of mapping spaces is tightly con-
nected to derivations of dg Lie algebras. In the rational setting, this is made
precise for instance by a result of Lupton–Smith [30, Thm 3.1]. The version of
their result we shall need is marginally stronger than stated in [30], but follows
from the given proof in a straight-forward way (see also [8, Thm 3.6]).
Theorem 4.1 (Lupton–Smith) Let f : X → Y be a map between simply con-
nected finite based CW-complexes, with minimal Lie model f : L XQ → LYQ.
There is an isomorphism
π∗(Maps∗(X, Y ); f )Q
∼=−→ H∗(Der f (L XQ, LYQ))
for ∗ ≥ 2, which is natural in both X and Y . For X a co-H-space, this also
holds for ∗ = 1.
4.5 A p-local generalisation
From the point of view of Quillen’s approach to rational homotopy theory,
the spaces we shall be applying Theorem 4.1 to are of the simplest nature
possible: they are homotopy equivalent to boquets of equidimensional spheres.
The minimal model of such a space X  ∨g Sn for n ≥ 2 agrees with the free
graded Lie algebra
L XQ := L(s−1 H XQ ) ∼= π∗+1 XQ on H XQ := H˜∗(X; Q), (36)
123
1136 M. Krannich
equipped with the trivial differential. Given a map between spaces of this kind,
the induced map 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 with respect to p, the homotopy Lie algebra
π∗+1 X is free even before rationalisation. To make this precise, we abbreviate
the integral and p-local analogue of (36) by
L X := L(s−1 H X ) and L X(p) := L(s−1 H X(p)) where
H X := H˜∗(X; Z) and H X(p) := H˜∗(X; Z(p)).
The inverse of the Hurewicz map Hn(X; Z(p)) ∼= πn X(p) induces a map
L X(p) → π∗+1 X(p) of graded Lie algebras over Z(p), which turns out to be
an isomorphism in a range of degrees.
Lemma 4.2 For an odd prime p and a based space X that is homotopy equiv-
alent to ∨g Sn with n ≥ 2, the morphism
L X(p) −→ π∗+1 X(p)
is an isomorphism on torsion free quotients. Moreover, the right hand side is
torsion free in degrees ∗ < 2p − 4 + n, so the map is an isomorphism in this
range.
Proof As a preparation to the proof, note that by specialising the Hilton–Milnor
theorem to ∨g Sn , we have an isomorphism
πi+1(∨g Sn) ∼= ⊕ω∈Lg πi+1(Sl(ω)(n−1)+1) (37)
where Lg denotes a Hall basis for the free ungraded Lie algebra in
g ordered generators and l(ω) is the word-length of ω. Here the map
πi+1(Sl(ω)(n−1)+1) → πi+1(∨g Sn) corresponding to ω ∈ Lg is given by
mapping a class x ∈ πi+1(Sl(ω)(n−1)+1) to the composition (ιω ◦ x), where
ιω ∈ πl(ω)(n−1)+1(∨g Sn) is the class obtained by taking Whitehead products
of the canonical classes ιi ∈ πn(∨g Sn) for 1 ≤ i ≤ g represented by the
inclusions of the summands as guided by the Lie word ω ∈ Lg. A proof can
be extracted from [47]: combine XI.6.6 and the subsequent discussion with
VII.2.6 and X.7.10.
To prove the asserted claim, we use that source and domain of the morphism
in the statement are both degreewise finitely generated and that the rationalisa-
tion of this morphism agrees with (36), so to prove the first part of the claim, it
suffices to show that all classes in π∗+1 X(p) of infinite order are in the image.
From (37), we see every class in πk X(p) is a composition (y ◦ x) of some
x ∈ πk Sm(p) with m ≥ n and a class y ∈ πm X(p) in the image of the map in
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A homological approach to pseudoisotopy theory. I 1137
question. The group πk Sm(p) is finite unless k = m, where it is generated by the
identity, or m = 2l and k = 4l − 1, where it is is generated by [idS2l , idS2l ],
since this element has Hopf invariant 2 and we assumed p to be odd. As
y ◦ [idS2l , idS2l ] = [y, y] in π4l−1 X(p) and the image of L X(p) → π∗+1 X(p) is
closed under taking brackets, this implies the first part of the claim. The second
part follows from Serre’s result [38, p. 498, Prop. 5] that πk Sm is p-torsion free
for k − m < 2p − 3 together with another application of (37). unionsq
As a result of Lemma 4.2, every map f : X → Y between bouquets of
equidimensional spheres induces a morphism f∗ : L X(p) → LY(p) by taking
torsion free quotients of p-local homotopy groups, so the following extension
of Theorem 4.1 might not come as a surprise.
Proposition 4.3 For an odd prime p and a map f : X → Y between based
spaces X  ∨g Sn and Y  ∨h Sm with n, m ≥ 2 , the map of Theorem 4.1
fits into a commutative square
π∗(Maps∗(X, Y ); f )(p) Der f (L X(p), LY(p))+
π∗(Maps∗(X, Y ); f )Q Der f (L XQ, LYQ)+
(−)⊗Q (−)⊗Q
∼=
for ∗ > 0,
which is natural in X and Y and whose upper arrow is an isomorphism for
∗ < 2p − 3 − (n − m).
Remark 4.4 Dwyer’s tame homotopy theory [13] provides a p-local general-
isation of Quillen’s rational homotopy theory for primes p that are just large
enough with respect to the degree to prevent stable k-invariants from appear-
ing. It is not unlikely that Theorem 4.1 could be generalised to this setting, but
our layman extension Proposition 4.3 for bouquets of spheres suffices for the
applications we have in mind.
Proof of Proposition 4.3 We begin with a twofold simplification of the state-
ment. Firstly, the claimed naturality is automatic, since the vertical maps are
evidently natural, the bottom map is natural by Theorem 4.1, and the right
vertical map is injective, so it suffices to construct a top arrow with the desired
properties for X = ∨g Sn . Secondly, there is a commutative diagram
Der f (L X(p), L
Y
(p)) Hom(s
−1 H X(p), L
Y
(p))
Der f (L XQ, L
Y
Q) Hom(s
−1 H XQ , LYQ),
∼=
(−)⊗Q (−)⊗Q
∼=
123
1138 M. Krannich
induced by restricting derivations to generators, which shows that it is enough
to produce a dashed arrow making the diagram
π∗(Maps∗(X, Y ); f )(p) Hom(s−1 H X(p), LY(p))+
π∗(Maps∗(X, Y ); f )Q Der f (L XQ, LYQ)+ Hom(s−1 H XQ , LYQ)+,
(−)⊗Q (−)⊗Q
∼= ∼=
(38)
commute. To do so, we consider the composition
π∗(Maps∗(X, Y ); f )
(− f )∗−−−→∼= π∗(Maps∗(X, Y ); ∗)
∼= Hom(s−1 H X , π∗+1Y )+ (39)
whose first isomorphism is given by acting with the inverse of f , using the loop
space structure on Maps∗(X, Y ), and whose second isomorphism is induced by
mapping a class inπk(Maps∗(X, Y ); ∗) represented by a pointed map g : Sk →
Maps∗(X, Y ) to the composition
Hn(X) ∼= Hn+k(Sk ∧ X) ∼= πn+k(Sk ∧ X) g∗−→ πn+k(Y ),
involving the suspension isomorphism, the inverse of the Hurewicz map, and
the adjoint of g. Postcomposing (39) with the map given by p-localising and
taking torsion free quotients results by Lemma 4.2 in a dashed map with the
claimed connectivity property, so we are left to show that this choice does make
the diagram (38) commute, i.e. that the rationalisation of (39) agrees with the
bottom map of (38). The adjoint of a map h : Sk → Maps∗(X, Y ) representing
a class in π∗(Maps∗(X, Y ); f ) forms the top arrow of the diagram
Sk × X Y
Sk+ ∧ X
X ∨ Sk ∧ X,
h

f ∨(− f )∗(h)
(40)
whose middle diagonal arrow is induced by h via the canonical homeomor-
phism (Sk × X)/(Sk ∨ ∗) ∼= Sk+ ∧ X and whose vertical equivalence is given
as the composition
Sk+ ∧ X
idSk+
∧∇
−−−−→ Sk+ ∧ X ∨ Sk+ ∧ X c−→ X ∨ Sk ∧ X
123
A homological approach to pseudoisotopy theory. I 1139
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 (39), so (40) commutes up to changing h within its class in
πk(Maps∗(X, Y ); f ). We thus obtain a rational model for the top arrow in
(40) as the composition
(
L
(
s−1 H XQ ⊕ s−1+k H XQ ⊕ s−1 H S
k
Q
)
, d
)
−→ L Sk∨Sk∧XQ
π∗+1( f ∨(− f )∗(g))⊗Q−−−−−−−−−−−−−→ LYQ,
(41)
where the source is the Lie model of Sk × X described in [30, Cor. 2.2], i.e. its
differential d is trivial on s−1 H XQ ⊕ s−1 H S
k
Q and is on s
−1+k H XQ given as
s−1+k H XQ ∼= s−1 H XQ
(−1)k−1[z,−]−−−−−−−→ L
(
s−1 H XQ ⊕ s−1+k H XQ ⊕ s−1 H S
k
Q
)
where the first isomorphism is the canonical identification as ungraded vector
spaces induced by the identity and z ∈ s−1 H SkQ denotes the standard generator.
The first map in the composition (41) takes the quotient by the dg Lie ideal
generated by the subspace s−1 H SkQ and the second map is defined as indicated.
Using this particular choice of rational model in the definition of the isomor-
phism of Theorem 4.1 in [30, p. 176–177], the image of the class defined by h
under the bottom horizontal composition in (38) is precisely its image under
(39) after rationalisation, so the claim follows. unionsq
4.6 p-local homotopy groups of BhAutidD2n(Vg, Wg,1)
The theory set up in the previous paragraphs will allow us to compute the
p-local homotopy groups of the classifying space BhAutidD2n (Vg, Wg,1) of the
identity component of the topological monoid of relative homotopy automor-
phisms as defined in Sect. 3.1 as a module over the group
π1BhAutD2n (Vg, Wg,1) ∼= π0hAutD2n (Vg, Wg,1).
More generally, we will compute the p-local homotopy groups of the spaces
participating in the fibration sequence (see Section 3.1 for the notation)
BhAut∂(Vg) −→ BhAutD2n (Vg, Wg,1) −→ BhAutext∂ (Wg,1) (42)
induced by restriction, together with the induced action of π0hAutD2n
(Vg, Wg,1). To state the answer (and give the proof), we adopt the notation
of in the previous subsection for the three manifolds Wg,1, Vg, and ∂Wg,1
123
1140 M. Krannich
involved, which are homotopy equivalent to bouquets of spheres. We do how-
ever omit the g-superscripts to increase readability, so write
H W(p) = H˜∗(Wg,1; Z(p)), LW(p) = L(s−1 H W(p)),
H V(p) = H˜∗(Vg; Z(p)), and LV(p) = L(s−1 H V(p))
and omit the (p)-subscripts to denote the integral analogues. Moreover, we
generically write ι for any combination of the inclusions
∂Wg,1 ⊂ Wg,1 ⊂ Wg ⊂ Vg.
Finally, we let ω ∈ LWg,1 be the class that represents the inclusion S2n−1 =
∂Wg,1 ⊂ Wg,1 of the boundary of Wg,1 = g(Sn × Sn)\int(D2n), i.e. the
attaching map
ω = ∑gi=1[ei , fi ] ∈ π2n−1Wg,1, (43)
where ei , fi ∈ πnWg,1 correspond to the first respectively second Sn-summand
in the i th summand of
Wg,1 ∼= g Sn × Sn\int(D2n)  ∨g(Sn ∨ Sn).
Note that the inclusion ∂Wg,1 ⊂ Vg is trivial, since it factors over Wg =
Wg,1 ∪∂Wg,1 D2n .
Remark 4.5 Note that H W and H V stand for the graded Z-module given by
the reduced homology of Wg,1 and Vg, which shall not be confused with the
ungraded middle dimensional integral homology groups of these spaces that
featured in Section 3.2.1 as HWg,1 and HVg .
Theorem 4.6 Let n ≥ 2 and p an odd prime.
(i) The inclusion π0Maps∂(Vg, Vg) ⊂ π0hAut∂(Vg) is an equality. This
group is abelian.
(ii) In degrees 0 < ∗ < 2p − 3 − n, the boundary map of the fibration (42)
fits into a commutative diagram of graded Z(p)-modules with exact rows
0 π∗+1BhAutext∂ (Wg,1)(p) s−(2n−1) H
W
(p) ⊗ LW(p) s−(2n−2)[LW(p), LW(p)] 0
0 π∗BhAut∂ (Vg)(p) s−(2n−1) H V(p) ⊗ LV(p) s−(2n−2)[LV(p), LV(p)] 0.
∂ ι∗⊗ι∗
[−,−]
ι∗
[−,−]
which is π0hAutD2n (Vg, Wg,1)-equivariant with respect to the action on
the leftmost column induced by (42) and by the action through H W and
H V on the other columns.
(iii) Rationally, the conclusion of (ii) holds in all positive degrees.
123
A homological approach to pseudoisotopy theory. I 1141
A splitting of the canonical projection ι∗ : HWg,1 → HVg induces compatible
splittings of the rightmost two columns of the diagram in Theorem 4.6 (ii), so
the boundary map ∂ of the fibration (42) is p-locally split surjective in a range
and we conclude the following.
Corollary 4.7 Let n ≥ 2 and p an odd prime. In degrees 0 < ∗ < 2p−4−n,
the graded Z(p)[π0hAutD2n (Vg, Wg,1)]-module π∗+1BhAutD2n (Vg, Wg,1)(p)
is isomorphic to the common kernel of the maps
s−(2n−1)H W(p) ⊗ LW(p)
[−,−]−→ s−(2n−2)[LW(p), LW(p)] and
s−(2n−1)H W(p) ⊗ LW(p)
ι∗⊗ι∗−→ s−(2n−1)H V(p) ⊗ LV(p).
Rationally, this holds in all positive degrees.
In particular, Theorem 4.6 and Corollary 4.7 imply that the
π0hAutD2n (Vg, Wg,1)-action on the p-local higher homotopy groups of the
spaces participating in (42) factors in a range of degrees through the mor-
phism (recall Kg = ker(Hn(Wg,1) → Hn(Vg)) from Section 3.2.1)
π0hAutD2n (Vg, Wg,1) −→ {φ ∈ GL(HWg,1) | φ(Kg) ⊂ Kg}
induced by the action on the homology of Wg,1. During the proof of Theo-
rem 4.6 and the preceding Lemma 4.8, we frequently use Proposition 4.3 to
implicitly identify p-local homotopy groups of path components of pointed
mapping spaces between bouquets of equidimensional spheres with deriva-
tions of free graded Lie algebras in a range of degrees. We denote by
Maps f∗ (X, Y ) for a map of based spaces f : X → Y the corresponding compo-
nent of the mapping space, pointed by f . Reminding the reader of our notation
for spaces of derivations in Section 4.2, we begin with the following lemma,
whose first part is rationally due to Berglund and Madsen [8, Prop. 5.6].
Lemma 4.8 Let n ≥ 2 and p an odd prime.
(i) In degrees ∗ < 2p − 3 − n, the morphism induced by relaxing the
boundary condition
π∗Mapsid∂ (Wg,1, Wg,1)(p) −→ π∗Mapsid∗ (Wg,1, Wg,1)(p) ∼= Der(LW(p))+
is injective and has image Derω(LW(p))+ ⊂ Der(LW(p))+.(ii) In the range ∗ < 2p − 3 − n, the morphism induced by Wg,1 ⊂ Wg
π∗Mapsι∗(Wg, Vg)(p) −→ π∗Mapsι∗(Wg,1, Vg)(p) ∼= Derι(LW(p), LV(p))+
is injective and has image Derιω(LW(p), LV(p))+ ⊂ Derι(LW(p), LV(p))+.
123
1142 M. Krannich
Proof Restriction along the inclusion ∂Wg,1 ⊂ Wg,1 yields a fibration
Maps∗(Wg,1, Wg,1) −→ Maps∗(∂Wg,1, Wg,1) (44)
whose fibre at ι is Maps∂(Wg,1, Wg,1). The induced maps on homotopy groups
fits in the range 0 < ∗ < 2p − 2 − n into a diagram of the form
π∗Mapsid∗ (Wg,1, Wg,1)(p) π∗Mapsι∗(∂Wg,1, Wg,1)(p)
Der(LW(p))
+ Derι(L∂Wg,1(p) , L
W
(p))
+
(
s−(2n−1)LW(p) ⊗ H W(p)
)+ (
s−(2n−2)[LW(p), LW(p)]
)+
.
∼= ∼=
∼= ∼=
[−,−]
(45)
whose top square is provided by Proposition 4.3, so commutes. The bottom
square is given as follows: the bottom right vertical map is the evaluation at
the fundamental class
[∂Wg,1] ∈ s−1 H ∂Wg,1(p) ∼= Q[2n − 2],
which factors as a composition of isomorphisms
Derι(L∂Wg,1(p) , L
W
(p))
+ ∼=−→ Hom(s−1 H ∂Wg,1(p) , LW(p))+
∼=−→ (s−(2n−2)LW(p))+ = (s−(2n−2)[LW(p), LW(p)])+
where the first isomorphism restricts to generators, the second isomorphism
evaluates at the fundamental class, and the final equality holds for degree rea-
sons. The latter is because elements of degree > 0 in s−(2n−2)LW(p) correspond
to elements of degree > 2n − 2 in LW(p), so are sums of brackets since this
Lie algebra is generated in degree n − 1 as HW is supported in degree n. The
bottom left vertical map is the restriction to positive degrees of the map
123
A homological approach to pseudoisotopy theory. I 1143
Der(LW(p))
∼=−→ Hom(s−1 H W(p), LW(p)) ∼=LW(p) ⊗ (s−1 H W(p))∨
∼=s−(2n−1)LW(p) ⊗ H W(p),
(46)
where the first isomorphism is given by restricting to generators, the second is
the canonical one, and the third is induced by the intersection form on s−1 HWg,1
(see Example B.1). By construction, the composition
Der(LW(p))
+ −→ (s−(2n−2)[LW(p), LW(p)])+ (47)
coincides with the evaluation at the class ω that represents the inclusion
∂Wg,1 ⊂ Wg,1, so it follows from Lemma 1 that this square commutes up
to a sign, since ω = ∑gi=1[ei , fi ] ∈ LWg,1 agrees up to a sign with the element
(76) from the appendix as e#i = fi and f #i = ei holds in the notation of the
appendix up to a fixed sign depending on n (which does not play a role in the
argument). As the bottom horizontal map is surjective as a consequence of
the graded Jacobi identity, the middle horizontal arrow is surjective as well,
and hence so is the top one. A consultation of the long exact sequence in
homotopy groups induced by the fibration (44) thus proves (i) since the ker-
nel of the middle horizontal arrow of the diagram is Derω(LW(p)) as (47) is
given by the evaluation at ω. This finishes the proof of (i). The proof of (ii) is
completely analogous, based on the fibration sequence obtained by applying
Maps∗(−, Vg) to the cofibration sequence ∂Wg,1 → Wg,1 → Wg instead of
(44). The bottom square of the diagram corresponding to (45) is now given by
Derι(LW(p), L
V
(p))
+ Derι(L∂Wg,1(p) , L
V
(p))
+
(
s−(2n−1)LV(p) ⊗ H W(p)
)+ (
s−(2n−2)[LV(p), LV(p)]
)+
evω∼= ∼=
[−,ι∗(−)]
(48)
whose vertical arrows are given by the composition
Derι(L∂Wg,1(p) , L
V
(p))
+ ∼=−→ Hom(s−1 H ∂Wg,1(p) , LV(p))+
∼=−→ (s−(2n−2)LV(p))+ = (s−(2n−2)[LV(p), LV(p)])+
and the restriction to elements of positive degrees of the composition
123
1144 M. Krannich
Derι(LW(p), L
V
(p))
∼=−→ Hom(s−1 H W(p), LV(p)) ∼=LV(p) ⊗ (s−1 H W(p))∨
∼=s−(2n−1)LV(p) ⊗ H W(p),
(49)
both completely analogous to the two compositions explained below (45). unionsq
Proof of Theorem 4.6 We consider the map of horizontal fibration sequences
Maps∂ (Vg, Vg) MapsD2n ((Vg, Wg,1), (Vg, Wg,1)) Maps∂ (Wg,1, Wg,1)
Maps∂ (Vg, Vg) Maps∗(Vg, Vg) Maps∗(Wg, Vg),
⊂
⊂
⊂ (−)◦ι
(50)
where the top right horizontal arrow is induced by restriction and the right-
most vertical arrow is given by extending a selfmap of Wg,1 relative to the
boundary over the complement of Wg,1 ⊂ Wg by the identity, followed
by postcomposition with the inclusion Wg ⊂ Vg. The induced morphism
π∗Mapsid∗ (Vg, Vg) → π∗Mapsι∗(Wg, Vg) is injective because its composition
with the morphism π∗Mapsι∗(Wg, Vg) → π∗Mapsι∗(Wg,1, Vg) induced by
restriction along the inclusion Wg,1 ⊂ Wg is a retract since the composition
∨g(Sn ∨ Sn)  Wg,1 ⊂ Wg ⊂ Vg  ∨g Sn
is a homotopy retraction. From the long exact sequence in homotopy groups
of the bottom fibration, we see that the monoid π0Maps∂(Vg, Vg) receives a
surjection from π1Mapsι∗(Wg, Vg), and this is a monoid homomorphism as it
agrees with the map induced on π0(−) by the homotopy fibre inclusion of the
fibre sequence of A∞-spaces
(
Mapsι∗(Wg, Vg)  hofibid(inc)
) −→ Maps∂(Vg, Vg) inc−→ Maps∗(Vg, Vg).
The group π1Mapsι∗(Wg, Vg) is abelian since we have
π1Mapsι∗(Wg, Vg) ∼= [S1 ∧ Wg, Vg]∗ ∼= [S1 ∧ (∨2g Sn ∨ S2n), Vg]∗
∼= πn+1(Vg)⊕2g ⊕ π2n+1(Vg),
using S1 ∧ Wg  S1 ∧ (∨2g Sn ∨ S2n) due to the fact that the attaching map
(43) of the top-dimensional cell in the usual CW-decomposition of Wg is a
sum of Whitehead-brackets and thus nullhomotopic after suspension. Being
surjected upon by an abelian group, the monoid π0Maps∂(Vg, Vg) is itself an
abelian group and hence agrees with π0hAut∂(Vg), as claimed in (i). To prove
(ii), we combine the injectivity we just observed with Lemma 4.8 (ii) and the
123
A homological approach to pseudoisotopy theory. I 1145
long exact sequence of the bottom row in (50) to obtain a short exact sequence
0 −→ Der(LV(p))+
(−)◦ι∗−→ Derιω(LW(p), LV(p))+
−→ π∗−1Mapsid∂ (Vg, Vg)(p) −→ 0 (51)
in the range ∗ < 2p − 3 − n. Combining this with Lemma 4.8 (i), we see that
the boundary map in the long exact sequence in homotopy groups of the upper
fibration of (50) fits in degrees ∗ < 2p − 3 − n into a commutative diagram
π∗Mapsid∂ (Wg,1, Wg,1)(p) π∗−1Maps∂ (Vg, Vg)(p)
Derω(LW(p))
+ Derιω(LW(p), L
V
(p))
+ Derιω(LW(p),LV(p))+
Der(LV
(p))
+
∼=
∂
∼=
ι∗◦(−) π
(52)
where π is the quotient map and the right vertical map is induced by (51). To
finish the proof of (ii), we thus need to show that the bottom composition of
(52) fits as the left vertical arrow in a diagram as in (ii). To see this, we first
combine the bottom square of (45) with the compatible square (48) to obtain
a commutative diagram with exact rows
0 Derω(LW(p))
+ (s−(2n−1)LW(p) ⊗ H W(p))+ (s−(2n−2)[LW(p), LW(p)])+ 0
0 Derιω(LW(p), L
V
(p))
+ (s−(2n−1)LV(p) ⊗ H W(p))+ (s−(2n−2)[LV(p), LV(p)])+ 0.
(−)◦ι∗
[−,−]
ι∗⊗id ι∗
[−,ι∗(−)]
(53)
Next, writing
K := ker(ι∗ : H W → H V ),
we note that there is a chain of natural isomorphisms
Der(LV(p))∼=Hom(H V(p), LV(p))∼=LV(p)⊗(s−1 H V(p))∨∼=s−(2n−1)LV(p)⊗K(p) (54)
defined analogously to (and compatible with) (49), using that the isomorphism
(H W )∨ ∼= H W induced by the intersection form (neglecting grading shifts)
sends (H V )∨ ⊂ (H W )∨ to K ⊂ H W . Except for the equivariance claim, (ii)
now follows by combining the chain of isomorphisms (54) with the diagrams
(52)–(53) and the chain of isomorphisms
(
s−(2n−1)LV(p) ⊗ H W(p)
)
/
(
s−(2n−1)LV(p) ⊗ K(p)
)
∼= s−(2n−1)LV(p) ⊗
(
H W(p)/K(p)
) ∼= s−(2n−1)LV(p) ⊗ H V(p).
123
1146 M. Krannich
This uses that the inclusion hAutext∂ (Wg,1) ⊂ Maps∂(Wg,1, Wg,1) is 0-
coconnected and that we have hAut∂(Vg) = Maps∂(Vg, Vg) by (i). To see
the equivariance, note that all vertical maps in the diagram of (ii) are equiv-
ariant by construction. Since they are also surjective (see the discussion after
the statement), it suffices to show that the top row is equivariant. This is clear
for the second map in the top row, so we are left with showing equivariance
of the first map
π∗+1BhAut∂(Wg,1)(p) −→ s2n−1 H W(p) ⊗ LW(p). (55)
With respect to the canonical isomorphisms in positive degrees
π∗+1BhAut∂(Wg,1)(p) ∼= π∗hAut∂(Wg,1)(p) ∼= π∗Maps∂(Wg,1, Wg,1)(p),
the action on the domain of (55) is induced by conjugation. Going through the
proof, we see that (55) arises as a composition of the form
π∗Mapsid∂ (Wg,1, Wg,1)(p) −→ π∗Mapsid∗ (Wg,1, Wg,1)(p)
−→ Der(LW(p)) ∼= s2n−1 H W(p) ⊗ LW(p).
The first map relaxes the boundary condition, which is equivariant. The second
map is given by the isomorphism in Proposition 4.3, and its equivariance
follows from the naturality part of that proposition. The third map is provided
by the chain of isomorphisms (46), which is equivariant by the naturality of
the intersection form. This finishes the proof of (ii). To see (iii), note that all
restrictions on the degree in the proof of (ii) originated from the assumption
on the degree in Proposition 4.3. This proposition holds rationally without that
assumption, so the proof of 4.6 also applies to 4.6. unionsq
In a range of degrees, the particular shape of the p-local homotopy groups of
the space BhAutidD2n (Vg, Wg,1) ensured by Corollary 4.7 allows us to pass from
homotopy to homology groups, which is what we are actually interested in. In
the following statement, we consider the module Hn(Wg,1; Z(p)) as ungraded.
Corollary 4.9 For n ≥ 2, there is an injection of graded π0hAutD2n
(Vg, Wg,1)-modules
H˜∗(BhAutidD2n (Vg, Wg,1); Z(p)) ↪−→
(
Hn(Wg,1; Z(p))⊗3
)[n]
in degrees ∗ < min(2n − 1, 2p − 3 − n) for primes p.
Proof We may assume p > 3, since otherwise the claim has no con-
tent. As a result of Corollary 4.7, there is an injective map of graded
123
A homological approach to pseudoisotopy theory. I 1147
π0hAutD2n (Vg, Wg,1)-modules
π∗+1BhAutidD2n (Vg, Wg,1)(p) ↪−→ s−(2n−1)H W(p) ⊗ LW(p)
= s−(2n−1)H W(p) ⊗ L(s−1 H W(p))
in degrees 0 < ∗ < 2p − 4 − n. Using that we have
[s−1 H W(p), s−1 H W(p)] ⊂ (s−1 H W(p))⊗2
by antisymmetrisation and that H W(p) = H˜∗(Wg,1; Z(p)) is concentrated in
degree n, we obtain
s−(2n−1)H W(p) ⊗ L(s−1 H W(p)) ⊂ s−(2n−1)H W(p) ⊗ (s−1 H W(p))⊗2
= (Hn(Wg,1; Z(p))⊗3)[n − 1],
in degrees ∗ < 2n − 2, so the claim holds for homotopy instead of homology
groups. This leaves us with showing that the p-local Hurewicz homomorphism
π∗BhAutidD2n (Vg, Wg,1)(p) −→ H˜∗(BhAutidD2n (Vg, Wg,1)(p); Z(p))
is an isomorphism in degree ∗ < m := min(2n − 1, 2p − 3 − n). Since
submodules of free Z(p)-modules are free, it follows from the first part of the
proof that n-truncation induces a p-locally m-connected map of the form
BhAutidD2n (Vg, Wg,1) −→ K (A, n)
where A is a free Z(p)-module, so it suffices to show that K (A, n) has trivial
Z(p)-homology in the range n < ∗ < m. Since A is free, it is enough to show
that H∗(K (Z(p), n); Z(p)) ∼= H∗(K (Z, n); Z(p)) vanishes in this range, which
is certainly true rationally, so we may instead prove that H∗(K (Z, n); Fp)
vanishes for n + 1 < ∗ < m + 1. As the natural map
HF∗p(HZ) = limnH∗+n(K (Z, n); Fp) −→ H∗+n(K (Z, n); Fp)
is an isomorphism in degrees ∗ < n, this follows from showing that the
spectrum cohomology HF∗p(HZ) vanishes in degrees 0 < ∗ < min(n, 2p −
2−2n). But HF∗p(HZ) is a quotient of the mod p Steenrod algebra HF∗p(HFp)
by an ideal containing the Bockstein, so HF∗p(HZ) vanishes in degrees 0 <∗ < 2p − 2 and we conclude the assertion. unionsq
123
1148 M. Krannich
5 The proof of Theorem A
This section is devoted to the proof of the following refinement of Theorem A.
Theorem 5.1 For n > 3, there is a nilpotent space X and a zig–zag
BC(D2n) φ−→ X ψ←− ∞+10 K(Z)
such that p-locally, φ is min(2n − 4, 2p − 4 − n)-connected and ψ is
min(2n, 2p − 4)-connected.
Remark 5.2 As remarked in the introduction, our proof is independent of
Waldhausen’s work on pseudoisotopy theory. If one is willing to invest Dwyer–
Weiss–Williams’ index theorem [12] (which relies in parts of Waldhausen’s
work), then it takes little effort to also explicitly identify X and ψ in terms of
well-known infinite loop spaces, see Section 5.4.
As a first step towards proving Theorem 5.1, we replace BC(D2n) by an
equivalent space that is more convenient to compare to the various automor-
phism spaces of high-dimensional handlebodies Vg = g Dn+1×Sn we studied
in the previous sections.
Lemma 5.3 For d ≥ 5, there exists a homotopy equivalence
BC(Dd)  ˜Diff Dd (Dd+1)/Diff Dd (Dd+1)
= hofib(BDiff Dd (Dd+1) → B˜Diff Dd (Dd+1))
Proof A choice of identification Dd × [0, 1] ∼= Dd+1 by smoothing corners
induces an equivalence BC(Dd)  BDiff Dd (Dd+1), so the claim is equiv-
alent to showing that the space of block diffeomorphisms ˜Diff Dd (Dd+1) is
contractible. This follows from Cerf’s result that every concordance of a man-
ifold of dimension at least 5 is isotopic to the identity [11], together with the
isomorphisms πk(˜Diff Dd (Dd+1); id) ∼= π0Diff Dd+k (Dd+k+1) ∼= π0C(Dd+k)
which is most easily seen by using the combinatorial description of the homo-
topy groups of the Kan complex ˜Diff Dd (Dd+1)• (see Section 1.4). unionsq
The alternative point of view on BC(D2n) as the homotopy fibre
˜Diff D2n (D2n+1)/Diff D2n (D2n+1) = ˜Diff D2n (V0)/Diff D2n (V0)
is advantageous as it makes a stabilisation map of the form
BC(D2n)  ˜Diff D2n (V0)/Diff D2n (V0) −→ ˜Diff D2n (Vg)/Diff D2n (Vg), (56)
123
A homological approach to pseudoisotopy theory. I 1149
apparent, which is induced by iterating the stabilisation maps for BDiff D2n (Vg)
and its block analogue explained in Section 3.5. It is a consequence of Morlet’s
lemma of disjunction that this map is highly connected:
Lemma 5.4 The stabilisation map (56) is (2n − 4)-connected.
Proof Taking vertical homotopy fibres in the diagram (see Section 3.1 for the
notation)
of fibre sequences whose diagonal arrows are given by the iterated stabilisation
maps results in a map of fibre sequences of the form
˜Diff∂ (Vg)
Diff∂ (Vg)
˜Diff D2n (Vg)
Diff D2n (Vg)
˜Diff
ext
∂ (Wg,1)
Diffext∂ (Wg,1)
˜Diff∂ (Wg,1)
Diff∂ (Wg,1)
˜Diff∂ (D2n+1)
Diff∂ (D2n+1)
˜Diff D2n (D
2n+1)
Diff D2n (D2n+1)
˜Diff
ext
∂ (D2n )
Diffext∂ (D2n )
˜Diff∂ (D2n)
Diff∂ (D2n)
,


whose inner diagonal map is the map in question and whose rightmost equiv-
alences follow from another application of Cerf’s result mentioned in the
previous proof. As the manifolds Vg and Wg,1 are (n − 1)-connected, the two
outer diagonal maps are (2n − 4)-connected by a form of Morlet’s lemma of
disjunction [6, p. 29, Cor. 3.2], so the claim follows from the induced ladder
of long exact sequences in homotopy groups. unionsq
Combining the previous two lemmas results in a (2n − 4)-connected map
BC(D2n) → ˜Diff D2n (V∞)/Diff D2n (V∞) := hocolimg˜Diff D2n (Vg)/Diff D2n (Vg) (57)
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 connectiv-
ity properties between this homotopy colimit and the zero component of the
once looped algebraic K -theory space of the integers ∞+10 K(Z), which we
model as the plus-construction3
∞+10 K(Z)  BGL∞(Z)+
3 We take all plus-constructions with respect to the unique maximal perfect subgroup, i.e. the
intersection of the terms in the transfinite derived series. It follows from maximality that this
subgroup is automatically normal.
123
1150 M. Krannich
of the homotopy colimit BGL∞(Z) := hocolimgBGLg(Z) over the stabilisa-
tion maps induced by the usual block inclusions GLg(Z) ⊂ GLg+1(Z). The
zig–zag we construct arises as part of a commutative zig–zag of horizontal
homotopy fibre sequences of the form
˜Diff D2n (V∞)/Diff D2n (V∞) BDiffsfrD2n (V∞; 0) B˜Diff
sfr
D2n (V∞; 0)
X BDiffsfr(V∞; 0)+ BGL∞(Z)+
0BGL∞(Z)+ ∗ BGL∞(Z)+〈1〉,
2©
3© 1©
(58)
which we explain now. Denoting the tangential structure encoding stable fram-
ings by sfr : EO → BO, the upper right corner is defined as
B˜Diff
sfr
D2n (V∞; 0) := hocolimgB˜Diff
sfr
D2n (Vg; 0)
along the stabilisation maps explained in Section 3.5, and the space
BDiffsfrD2n (V∞; 0) is defined analogously, using the unstable (2n + 1)-
dimensional tangential structure induced by sfr : EO → BO, which we denote
by the same symbol (see Section 1.8.1). The upper right horizontal map is the
homotopy colimit of the comparison map
BDiffsfrD2n (Vg; 0) −→ B˜Diff
sfr
D2n (Vg; 0)
whose homotopy fibre at the base point is canonically equivalent to
˜Diff D2n (V∞)/Diff D2n (V∞) in view of Lemma 1.8 and the fact that homo-
topy fibres commute with sequential homotopy colimits. Using a functorial
model of the plus-construction (see e.g. [4, VII.6.2]), the upper right square of
(58) is induced by the homotopy colimit of the composition
BDiffsfr(Vg; 0) → B˜Diffsfr(Vg; 0) → BGL(HVg ) = BGL(Hn(Vg; Z)) ∼= BGLg(Z) (59)
along the stabilisation maps (see Section 3.5), where the second map is induced
by the action on the middle homology of Vg. The space X is defined as the
homotopy fibre of the map BDiffsfr(V∞; 0)+ → BGL∞(Z)+, and it receives
a map from the top left corner induced by the commutativity of the upper
right square. The bottom row is induced by the inclusion of the basepoint
in the universal cover BGL∞(Z)+〈1〉 whose homotopy fibre agrees with the
base point component of BGL∞(Z)+. This explains the diagram (58), aside
from the map of fibre sequences from the bottom to the middle row, which
is induced by the universal cover BGL∞(Z)+〈1〉 → BGL∞(Z)+ and the
basepoint inclusion of BDiffsfr(V∞; 0)+ .
123
A homological approach to pseudoisotopy theory. I 1151
In the two following subsections, we continue the preparations of the proof
of Theorem 5.1 by analysing the vertical maps 1© and 2©.
5.1 The stable homology of BDiff D2n(Vg) and the map 1©
Botvinnik and Perlmutter [10] have computed the stable homology of
BDiffθD2n (Vg; 0) in homotopy theoretical terms for all tangential structures
θ : B → BSO(2n+1) whose space B is n-connected. For us, their main result
[10, Cor. 6.8.1, Prop. 6.14] is most conveniently expressed as an identification
of the group completion of the disjoint union
Mθ := ∐g≥0 BDiffθD2n (Vg; 0),
which becomes a homotopy commutative topological monoid under bound-
ary connected sum when choosing an appropriate point-set model (see [10,
Prop. 6.11, Prop. 6.14]).
Theorem 5.5 (Botvinnik–Perlmutter)For n > 3, a tangential structure
θ : B → BSO(2n + 1) for which B is n-connected, there is a homotopy
equivalence of the form
BMθ  ∞∞+ B.
The equivalence they produce factors as a composition
BMθ −→ BC∂θ −→ ∞∞+ B (60)
whose intermediate terms is the group-completion of a topological category C∂θ
of bordisms between 2n-manifolds with θ -structures, possibly with boundary.
This category was studied by Genauer [17], who established the final equiv-
alence in (60) as a parametrised form of the Pontryagin–Thom construction.
Botvinnik and Perlmutter showed that Mθ can be seen as a submonoid of
the endomorphism monoid C∂θ (D2n, D2n) of the closed disc D2n with some
θ -structure and that the chain of inclusions Mθ ⊂ C∂θ (D2n, D2n) ⊂ C∂θ is an
equivalence upon taking classifying spaces (see [10, Thm 6.3, Prop. 6.14]).
As Mθ is homotopy commutative, Randal-Williams’ elucidation of the
group completion theorem [36, Cor. 1.2] moreover provides an equivalence
BDiffθD2n (V∞; 0)+  0BMθ , (61)
which leads to the following consequence of Theorem 5.5 when specialised
to the tangential tangential structure encoding stable framings.
123
1152 M. Krannich
Corollary 5.6 The space BDiffsfrD2n (V∞; 0)+ is nilpotent and p-locally
min(2n, 2p − 4)-connected as long as n > 3.
Proof Connected H -spaces are nilpotent, so (61) settles the nilpotency claim.
Regarding the connectivity part of the statement, note that the unstable (2n +
1)-dimensional tangential structure (sfr)2n+1 induced by the stable structure
sfr : EO → BO (see Section 1.8.1) is equivalent to the inclusion O/O(2n +
1) → BO(2n+1)of the homotopy fibre of the stabilisation BO(2n+1) → BO,
so Theorem 5.5 and the discussion preceding this corollary show that the space
in question is equivalent to ∞0 ∞+ O/O(2n + 1). As O/O(2n + 1) is 2n-
connected, the homotopy groups of ∞0 ∞+ O/O(2n + 1) agree in positive
degrees less than 2n +1 with the stable homotopy groups of spheres which are
free of p-torsion in degrees less than 2p − 3 by a result of Serre [38, p. 498,
Prop. 5], so the claim follows. unionsq
Said differently Corollary 5.6 shows that the base point inclusion 1© in (58)
is p-locally min(2n, 2p − 4)-connected for n > 3.
5.2 The homology action and the map 2©
As a consequence of Proposition 3.7 (ii), the map on fundamental group
induced by 2© is infinite abelian, so the map 2© is as far from being highly
connected as possible, even p-locally for any p. Nevertheless, it turns out that
it does induce an isomorphism on p-local homology groups in a range, which
we shall prove by separately studying the effect on homology of the two maps
in the factorisation
B˜Diff
sfr
D2n (Vg; 0) −→ BGextg, −→ BGL(HVg ), (62)
of the second map in (59). Here Gextg, ⊂ GL(HWg,1) is the subgroup considered
in Sect. 3.3.
Lemma 5.7 For n ≥ 3 and any prime p, the induced map
H∗
(
B˜Diff
sfr
D2n (Vg; 0); Z(p)
) −→ H∗(BGextg,; Z(p))
is an isomorphism for ∗ < min(2n − 1, 2p − 3 − n) and a surjection in that
degree.
Proof The claim is vacuous for p = 2, so we assume otherwise. Consider the
factorisation of the map in question
123
A homological approach to pseudoisotopy theory. I 1153
B˜Diff
sfr
D2n (Vg; 0) → B˜hAut
∼=
D2n (Vg, Wg,1)
→ Bπ0˜hAut
∼=
D2n (Vg, Wg,1) → BGextg,,
(63)
where the first map is that of Corollary 2.4 applied to the triad (Vg; D2n, Wg,1),
the second map is induced by taking path components, and the third is given
by acting on the homology of Wg,1 ⊂ ∂Vg (see Section 3.3). It follows from
the corollary just mentioned that the first map induces an isomorphism in
homology with Z(p)-coefficients in degrees ∗ < 2p − 3 − n and a surjec-
tion in that degree since Vg is obtained from D2n by attaching n-handles and
the triad (Vg; D2n, Wg,1) satisfies the π -π -condition as Wg,1 and Vg are sim-
ply connected for n ≥ 2. To study the remaining maps, we may replace the
space of block homotopy automorphisms by its equivalent non-block analogue
BhAut∼=D2n (Vg, Wg,1) (see Section 1.5), so the E2-page of the p-local Serre
spectral sequence of the second map has the form
E2k,l ∼= Hk
(
π0hAut
∼=
D2n (Vg, Wg,1); Hl
(
BhAutidD2n (Vg, Wg,1); Z(p)
))
. (64)
To compute E2k,l , we employ the Serre spectral sequence of the extension
0 −→ Lg −→ π0hAut∼=D2n (Vg, Wg,1) −→ Gextg, −→ 0, (65)
with coefficients in Hl(BhAutidD2n (Vg, Wg,1); Z(p)), where Lg is the kernel as
indicated. The E2-page of this spectral sequence has the form
E2s,t ∼= Hs
(
Gextg,; Ht
(
Lg; Hl(BhAutidD2n (Vg, Wg,1); Z(p))
))
.
The kernel Lg of the extension (65) is a subgroup of the kernel of the analogous
extension without the -subscripts, and as the latter is finite and p-torsion free
for n < 2p − 4 by Lemma 3.10, so is the former. The group Lg thus has no
nontrivial homology in positive degrees with coefficients in a Z(p)[Lg]-module
if n < 2p − 4, so in this case the spectral sequence E2s,t is concentrated in the
bottom row and we conclude that
E2k,l ∼= Hk
(
Gextg,; Hl
(
BhAutidD2n (Vg, Wg,1); Z(p)
)
Lg
)
for n < 2p−4, (66)
where (−)Lg stands for taking coinvariants. By Lemma 3.9, the group Gextg,
contains the negative of the identity, which is central and acts on the coefficients
in (66) by −1 as long as 0 < l < m := min(2n − 1, 2p − 3 − n) as a result
of Corollary 4.9, since it acts by −1 on Hn(Wg,1; Z)⊗3 as (−1)3 = (−1).
123
1154 M. Krannich
This allows us to apply the “centre kills”-trick4 to conclude that the E2-page
E2k,l is 2-torsion for 0 < l < m (which implies n < 2p − 4) and is therefore
trivial as p was assumed to be odd. In this range, the spectral sequence (64) is
therefore concentrated at the bottom row, so the second map in (63) induces
an isomorphism on Z(p)-homology in degrees ∗ < m and a surjection in that
degree. This leaves us with arguing that the final map in (63) also has this
property. But we already observed that the kernel in (65) is p-locally acyclic
for n < 2p − 4, so in this case the third map is in fact a p-local homology
isomorphism. As the statement is vacuous unless 1 < m and thus n < 2p −4,
this finishes the proof. unionsq
In contrast to the first map in (62), the second map might not induce a p-
local homology isomorphism in a range of degrees, but we will see below that
its homotopy colimit BGext∞, → BGL(HV∞) with respect to the stabilisation
maps explained in Section 3.5 does.
Lemma 5.8 For odd primes p, the induced map
H∗
(
BGext∞,; Z(p)
) −→ H∗(BGL(HV∞); Z(p))
is an isomorphism.
Proof By an application of the Serre spectral sequence of the extension
0 −→ Msfrg −→ Gextg, −→ GL(HVg ) −→ 0
established in Proposition 3.7 (ii), it suffices to show that the colimit induced
by stabilisation
colimgH∗
(
GL(HVg ); H˜∗(Msfrg ; Z(p))
)
∼= colimgH∗
(
GL(HVg ); H˜∗(Msfrg ⊗ Z(p); Z(p))
)
vanishes; the isomorphism can be viewed as being induced by the p-
localisation of K (Msfrg , 1), see Section 1.2. Recall from Section 3.3 that
the GL(HVg )-module Msfrg ⊂ (HVg ⊗ HVg )∨ consists of all even (−1)n+1-
symmetric bilinear forms μ ∈ (HVg ⊗ HVg )∨, so as p is assumed to be
odd, its p-localisation Msfrg ⊗ Z(p) is isomorphic to the dual of the sym-
metric square Sym2(HVg )∨ ⊗ Z(p) if n is odd and the dual of the exterior
4 Given an element g ∈ G in a group and a Z[G]-module M , the multiplication map
g(−) : M → M is equivariant with respect to the conjugation map cg : G → G. By the
same argument as for the trivial module, the pair (cg, g(−)) induces the identity on H∗(G; M),
so if g is central and thus cg = id, then (id, (1 − g)(−)) annihilates H∗(G; M). In particular,
if g acts on M by multiplication by k ∈ Z, then H∗(G; M) is (1 − k)-torsion.
123
A homological approach to pseudoisotopy theory. I 1155
square 2(HVg )∨ ⊗Z(p) if n is even. In particular, by antisymmetrisation, the
GL(HVg )-module Msfrg ⊗ Z(p) is a direct summand of (HVg ⊗ HVg )∨ ⊗ Z(p).
Choosing a basis HVg ∼= Zg compatible with the stabilisation maps, this shows
that it suffices to show that the stable GLg(Z)-homology with coefficients in
Hk((Zg ⊗ Zg)∨; Z) ∼= k(Zg ⊗ Zg)∨ vanishes for k > 0. Pulling back this
module along the automorphism of GLg(Z) given by taking transpose inverse,
we see that we may replace this module by its dual k(Zg ⊗ Zg) whose
GLg(Z)-homology for large g with respect to k does indeed vanish by an
application of a result due to Betley [1, Thm 3.1]. unionsq
Corollary 5.9 Let n ≥ 3 and p a prime. The map on homology induced by
the map 2©,
H∗(B˜Diff
sfr
D2n (V∞; 0); Z(p)) −→ H∗(BGL∞(Z)+; Z(p)),
is an isomorphism for ∗ < min(2n − 1, 2p − 3 − n) and a surjection in that
degree.
Proof This is free of content if p is even and follows for p odd from a com-
bination of Lemmas 5.7 and 5.8, using that the canonical map BGL∞(Z) →
BGL∞(Z)+ is acyclic. unionsq
5.3 Proof of Theorem 5.1
As in the previous proofs, we assume p > 2; the claim is vacuous otherwise.
Our goal is to demonstrate that the precomposition of the left column in (58)
with the (2n − 4)-connected map (57) provides a zig–zag as promised by
Theorem 5.1. As a result of the discussion in Section 5.1, the space X is
the homotopy fibre of a map out of an infinite loop space which is moreover
surjective on fundamental groups as a result of Proposition 3.7, so it follows
from Lemma 1.4 that X is nilpotent. That the map 3© is p-locally min(2n, 2p−
4)-connected is a consequence of 1© having this property by Corollary 5.6
and the 1-connected cover BGL∞(Z)+〈1〉 → BGL∞(Z)+ being a p-local
equivalence asπ1BGL∞(Z)+ ∼= Z/2 and p is odd. This leaves us with showing
that the composition
BC(D2n) −→ ˜Diff D2n (V∞)/˜Diff D2n (V∞) −→ X
is p-locally min(2n −4, 2p −4−n)-connected, which we can test on p-local
homology groups by Lemma 1.1 as source and target are nilpotent. The first
map in this composition is (2n−4)-connected by Lemma 5.4, so we may focus
on the second and show that it induces an isomorphism in p-local homology in
123
1156 M. Krannich
the required range. Since plus-constructions do not affect homology groups and
2© is an isomorphism in homology with Z(p)-coefficients in degrees less than
min(2n − 1, 2p − 3 − 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. [19, Thm. 3.2]) to the map of Serre spectral sequences induced by the
first two rows of (58), provided we ensure that the actions of the fundamental
groups of the bases of both fibre sequences on the Z(p)-homology of the
respective fibres are trivial in this range. For the second row, this follows from
the p-local high connectivity of the map 3© established in the first part of
the proof, using that the canonical action of π1BGL∞(Z)+ on the homology
of BGL∞(Z)+ is trivial as BGL∞(Z)+  0B(∐g BGLg(Z)) is a loop
space (in fact, an infinite one). For the first row, the triviality of the action is
a consequence of Lemma 5.4 together with the observation that any element
in π0Diff D2n (Vg) can be represented by a diffeomorphism that fixes D2n+1 =
V0 ⊂ Vg pointwise, so commutes with any diffeomorphism in the image of the
iterated stabilisation map C(D2n)  Diff D2n (D2n+1) → Diff D2n (Vg). This
finishes the proof of Theorem 5.1, which in particular implies Theorem A.
5.4 A reformulation
Although not necessary for the proof of Theorem 5.1 or Theorem A, we shall
explain in Section 5.4.2 below how an instance of the Dwyer–Weiss–Williams
index theorem [12] shows that the plus constructed stable homology action
BDiffsfrD2n (V∞)
+
 −→ BGL∞(Z)+ (67)
featuring in (58) agrees with respect to the equivalence BDiffsfrD2n (V∞)
+
 
∞0 ∞+ O/O(2n+1) explained in Section 5.1 with the map obtained by apply-
ing ∞0 (−) to the composition
∞+ O/O(2n + 1)
pr−→ S ι−→K (Z) (−1)
n
−→ K (Z), (68)
where pr is the projection and ι the unit. This identifies the homotopy fibre X
in (58) as the space
X  ∞+ hofib
(
∞+ O/O(2n + 1)
pr−→ S ι−→ K (Z)),
so we may postcompose the map φ from Theorem 5.1 with the (2n + 1)-
connected map hofib(ι◦pr) → hofib(ι) induced by the projection to arrive at a
cleaner formulation of Theorem A: there is a p-locally min(2n−4, 2p−4−n)-
connected map
BC(D2n) −→ ∞0 hofib(S ι−→ K (Z)). (69)
123
A homological approach to pseudoisotopy theory. I 1157
5.4.1 Relation to Waldhausen’s work
It is reasonable to expect that the map (69) agrees up to equivalences with the
composition
BC(D2n) −→ ∞0 WhDiff(∗) = ∞0 hofib
(
S → K (S))
−→ ∞0 hofib
(
S → K (Z)) (70)
of the map known from Waldhausen’s work [42] with the map induced by
linearisation. While it is not hard to show that the map (69) does indeed factor
over the second map in (70), a convincing comparison between (69) and (70)
appears to be more laborious and we will not go into this matter at this point
(however, see the final remark of the introduction).
5.4.2 An application of the index theorem
To justify the claimed identification of (67) with the map resulting from
applying ∞0 (−) of (68), one can argue as follows: firstly, it suffices to
show that these maps agree when precomposed with an arbitrary map B →
BDiffsfrD2n (V∞)
+
 that classifies a smooth Vg-bundleπ : E → B for some g ≥ 0
together with a trivial D2n-subbundle and stably framed vertical tangent bun-
dle Tπ E . Tracing through the equivalences featuring in Section 5.1, one finds
that the composition
B −→ BDiffsfrD2n (V∞)+  ∞0 ∞+ (O/O(2n + 1)) (71)
represents the class
[Tπ E] ◦ BG(π) − χ(Vg) ∈ [∞+ B, ∞+ (O/O(2n + 1)]
whereχ(−) is the Euler characteristic, BG(π) ∈ [∞+ B, ∞+ E] is the Becker–
Gottlieb transfer of the bundle π described in terms of the Pontryagin–Thom
construction [2,3], and the class [Tπ E] ∈ [∞+ E, ∞+ O/O(2n+1)] is induced
by the vertical tangent bundle of π together with its stable framing. It thus
suffices to show that the class
(−1)n(ι ◦ pr)∗(BG(π) − χ(Vg)) ∈ [∞+ B, K (Z)] (72)
agrees with the class represented by the composition
B −→ BDiffsfrD2n (V∞)+ −→ BGL∞(Z)+  ∞0 K (Z).
123
1158 M. Krannich
The latter can be identified with the class
[Hn(π)] − g ∈ [∞+ B, K (Z)] (73)
where Hn(π) is the local system over B of fibrewise middle-dimensional
homology groups of π . Dwyer–Weiss–Williams’ improved Riemann–Roch
theorem [12, p. 2] gives
(ι ◦ pr)∗BG(π) = 1 + (−1)n · [Hn(π)] ∈ [∞+ B, K (Z)],
so using χ(Vg) = 1 + (−1)ng it follows that (72) and (73) indeed agree.
Appendix A. Stable tangential bundle maps are stable bundle maps
Fix a d-dimensional vector bundle ξ over a space X , a stable vector bundle
{ψk → Bk}k≥0, subcomplexes A, C ⊂ X , and a bundle map 0 : ξ |A ⊕ εk →
ψd+k covering a map ¯0 : X → Bd+k for some k ≥ 0. In addition to the
notation for various types of bundle maps in Section 1, we abbreviate
˜MapA(X, B; ¯0)• := colimm≥k˜MapA(X, Bm; ¯0)•,
where ˜MapA(X, Bm; ¯0)• is the semi-simplicial set whose p-simplices are
block maps p × X → p × Bm which agree with idp × ¯0 on p × A.
The colimit is taken over the maps induced by post-composition with the maps
Bm → Bm+1 underlying the structure maps of ψ . The sub semi-simplicial set
of maps p × X → p × Bm over p is denoted by MapsA(X, B; ¯0)• ⊂
˜MapA(X, B; ¯0)•.
Lemma A.1 If the base X of ξ is a finite CW-complex, then the maps
˜BunA(ξ s, ψ; 0)τ• −→ ˜MapA(X, B; ¯0)• and
˜hAutA(ξ s; C)τ• −→ ˜hAutA(X; C)•
induced by forgetting bundle maps are Kan fibrations.
Proof For ψ = ξ s and 0 = inc the inclusion, the second map agrees
with the pullback of the first map along the inclusion ˜hAutA(X; C)• ⊂
˜MapA(X, X; inc)•, so it suffices to show that the first map is a Kan fibra-
tion. We shall do so in the case where A is empty; the argument for general A
123
A homological approach to pseudoisotopy theory. I 1159
is similar. The upper horizontal map in a lifting problem
(
p
j )• ˜Bun(ξ s, ψ)τ•
(p)• ˜Map(X, B)•,
induces bundle maps
φi : τpi × ξ ⊕ ε
k −→ τpi × ψd+k
for i = j that agree on their faces and cover the restriction of the map φ¯ : p ×
X → p × Bd+k induced by the bottom horizontal arrow to pi × X . By
replacing ξ with its stabilisation ξ ⊕ εk , we may assume k = 0. There is an
extension of φi to τp |pi given by the composition
τp |pi × ξ
∼=−→ ε ⊕ τpi × ξ
idε⊕φi−−−−→ ε ⊕ τpi × ψd
∼=−→ τp |pi × ψd
whose outer isomorphisms are induced by the differential of the diffeomor-
phism ci,
 of Section 1.3. The condition (4) in the definition of tangential
bundle maps is made precisely such that these extensions agree on their inter-
sections, so they assemble to a bundle map
φpj
: τp |pj × ξ −→ τp |pj × ψd
that covers the restriction of φ¯ to pj × X and we are left to argue that this
bundle map is the restriction of a p-simplex in Bun(ξ s, ψ)τ• covering φ¯. As
the inclusion pj × X ⊂ p × X is a trivial cofibration, obstruction theory
provides an extension of φpj to a bundle map
φp : τp × ξ −→ τp × ψd
that covers φ¯, but this extension might violate condition (4) on the j th face,
i.e. the map
φ j : ε ⊕ τpj × ξ −→ ε ⊕ τpj × ψd
obtained by restricting φp to pj × X and using the isomorphism τp |pj ∼=
ε ⊕ τpj induced by the derivative of c j,
 need not have the form idε ⊕ φ j for
123
1160 M. Krannich
a bundle map
φ j : τpj × ξ −→ τpj × ψd .
Nevertheless, its restriction to ∂pj × X does have this form and we will argue
that after adding a trivial bundle of sufficiently large dimension, say n, we can
alter φ j by a homotopy of 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 changeφp⊕idεn to a p-simplex in Bun(ξ⊕εn, ψd⊕εn)τ•
that covers φ¯ and extends φpj ⊕idεn and thus provides a lift as required. To this
end, we denote by Iso(ν, η) → Y for vector bundles ν and η over a space Y the
fibre bundle whose sections correspond to bundle morphisms ν → η over the
identity, so the fibre over y ∈ Y is the space of isomorphisms νy → ηy between
the fibres of ν and η. Abbreviating ν := τpj × ξ and η := φ¯
∗(τpj × ψd), we
consider the diagram
∂
p
j × X Iso(ν ⊕ εn, η ⊕ εn) Iso(ε ⊕ ν ⊕ εn, ε ⊕ η ⊕ εn)

p
j × X pj × X pj × X
idε⊕−
in which the solid diagonal arrow is induced by φ j ⊕ idεn and the upper left
horizontal one by its restriction to ∂pj × X , which makes the subdiagram
of solid arrows commute strictly. From this point of view, the task we set us
is equivalent to constructing a dashed arrow for some n that makes the upper
leftmost triangle commutes strictly and the one formed by the two diagonal
arrows up to homotopy relative ∂pj × X . Using that X is a finite CW complex
and that the connectivity of the map on vertical homotopy fibres of the right
square increases in n as it agrees with the inclusion O(p + d + n) ⊂ O(p +
d + n + 1) up to equivalence, this follows from obstruction theory. unionsq
Remark We learnt the “stabilisation trick” of the previous proof from
Appendix D of [8], which contains results similar to those of this appendix.
Corollary A.2 If the base X of ξ is a finite CW-complex, then the following
semi-simplicial sets satisfy the Kan property.
(i) ˜BunA(ξ s, ψ; 0)τ•
(ii) BunA(ξ s, ψ; 0)τ•
(iii) ˜hAutA(ξ s; C)τ•
(iv) hAutA(ξ s; C)τ•
123
A homological approach to pseudoisotopy theory. I 1161
Proof It is straight-forward to see that ˜MapA(X, B; ¯0)• is Kan, so the first
part follows from 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 BunA(ξ s, ψ; 0)τ• → MapsA(X, B; ¯0)• is a Kan
fibration because it is the pullback of the first map of Lemma A.1 along the
inclusion MapsA(X, B; ¯0)• ⊂ ˜MapA(X, B; ¯0)•. Also the semi-simplicial
sets hAutA(X; C)• and ˜hAutA(X; C)• are easily seen to be Kan (see Sec-
tion 1.5), so the remaining claims can be proved in the same way. unionsq
Lemma A.3 If the base X of ξ is a finite CW-complex, then the inclusions
BunA(ξ s, ψ; 0)τ• ⊂˜BunA(ξ s, ψ; 0)τ• and
hAutA(ξ s; C)τ• ⊂ ˜hAutA(ξ s; C)τ•
are equivalences.
Proof These inclusions are pullbacks of the inclusions
MapsA(X, B; ¯0)• ⊂ ˜MapA(X, B; ¯0)• and
hAutA(X; C)• ⊂ ˜hAutA(X; C)•
(74)
along the two Kan fibrations discussed in Lemma A.1, so the claim follows
from showing that the inclusions (74) are equivalences. As already mentioned
in the previous proof, it is straight-forward to show that these semi-simplicial
sets are Kan. Using the combinatorial description of their homotopy groups, the
claim follows from the contractibility of the mapping space Maps∂p(p,p)
(cf. Section 1.5). unionsq
Lemma A.4 If the base X of ξ is a finite CW-complex, then the extension
maps of (7)
BunA(ξ s, ψ; 0)• → BunA(ξ s, ψ; 0)τ• and
hAutA(ξ s; C)• → hAutA(ξ s; C)τ•
are equivalences.
Proof The proof for the two maps are essentially identical; we restrict our
attention to the first map. Its source and target have the Kan property (the
source because it is the singular complex of a space, the target by Corol-
lary A.2), so we may test the claim on semi-simplicial homotopy groups.
The two semi-simplicial sets involved have the same 0-simplices, so the map
is clearly surjective on path components. To show that it is surjective on
homotopy groups in positive degrees, we fix a semi-simplicial base point in
123
1162 M. Krannich
BunA(ξ s, ψ; 0)• by choosing a 0-simplex  : ξ ⊕ εl → ψd+l and taking
products with the tangent bundles τp . Let
φ : ξ ⊕ εk × τp −→ ψd+k × τp
be a p-simplex that represents a class inπp(BunA(ξ s, ψ; 0)τ•; ). Up to chang-
ing φ within its homotopy class, we may assume that the underlying map
φ¯ : X × p → Bd+k × p satisfies the collaring condition from Section 1.3
for some 0 < 
 < 1/2. By replacing ξ with ξ ⊕ 
k , we may assume k = 0.
Fixing a trivialisation F : τp ∼= Rp × p, our candidate for a preimage in
πp(BunA(ξ s, ψ; 0)•; ) is the class defined by the composition
ξ ⊕ ε p × p idξ×F
−1
−→ ξ × τp φ−→ ψd × τp
idψd ×F−→ ψd ⊕ ε p × p −→ ψd+p × p,
(75)
where the last arrow is induced by the structure map ofψ . To justify this, we will
show the existences of a (p + 1)-simplex in BunA(ξ ⊕ ε p ⊕ ε2, ψd+p+2; 0)τ•
which on the pth face agrees with the (p + 2)-fold stabilisation of φ, on the
(p + 1)st face with the image φF ∈ BunA(ξ ⊕ 
 p ⊕ 
2, ψd+p+2; 0)τp of the
2-fold stabilisation of the composition (75) and on the remaining faces with
the basepoint. To this end, we consider the linear map
 :
Rp × R × R × R p −→ Rp × R × R × R p
(x, u, v, y) −→
{
(y, u, v, x) if p is even
(y, v, u, x) if p is odd,
,
which has determinant 1 (this is the reason we introduced the additional (R ×
R)-coordinate), so there exists a path γ : [0, 1] → GL(R p × R × R × Rp)
from the identity to , which we may choose to be constant in a neighborhood
of [0, 
] ∪ [1− 
, 1]. In terms of this path, we define a homotopy Ht of bundle
automorphisms of ε p ⊕ τp covering the identity by
ε p ⊕ ε2 ⊕ τp
idR p×R2×F−−−−−−−→ Rp × R2 × Rp × p
γt×idp−−−−−→ Rp × R2 × Rp × p idR p×R2×F
−1
−−−−−−−−→ ε p ⊕ τp .
Using H , we define a homotopy H˜ of bundle maps as the composition
ξ ⊕ ε p ⊕ ε2 × τp idξ×Ht−→ ξ ⊕ ε p ⊕ ε2 × τp φ−→ ψd ⊕ εp ⊕ ε2 × τp
123
A homological approach to pseudoisotopy theory. I 1163
idψd ×H−1t−→ ψd ⊕ εp ⊕ ε2 × τp → ψd+p+2 × τp ,
where the last map is induced by the structure map of ψ . Going through the
definitions, one checks that this is a homotopy from the stabilisation of φ to
φF and that its underlying homotopy of maps of spaces is constantly φ¯. Using
the canonical trivialisation of τ[0,1], this homotopy gives rise to a bundle map
H˜ : ξ ⊕ ε p ⊕ ε2 × τp×[0,1] → ψd+p+2 × τp×[0,1] which one checks to
descend uniquely to a dashed arrow making the diagram
ξ ⊕ ε p ⊕ ε2 × τp×[0,1] ψd+p+2 × τp×[0,1]
ξ ⊕ ε p ⊕ ε2 × τp+1 ψd+p+2 × τp+1
H˜
id
ξ⊕εp⊕ε2×dc idξ⊕εp⊕ε2×dc
H¯
commute, where c is the map
c : 
p × [0, 1] −→ p+1
((x0, . . . , x p), s) −→ (x0, . . . , x p−1, s · x p, (1 − s) · x p)
whose derivative is surjective (though not fibrewise). The resulting bundle map
H¯ has the correct behaviour on all faces, so almost provides a (p+1)-simplex
in BunA(ξ ⊕ε p ⊕ε2, ψd+p+2; 0)τ• as wished. The only problem is that it does
not satisfy the collaring condition (4) for i = p, but this can be rectified as
follows: choosing a block diffeomorphism α : p+1 → p+1 which agrees
with the identity on p+1p+1,δ for some δ > 0 (see Section 1.3 for the notation)
and makes the diagram
p+1
[0, 
) × p\pp,δ′
p+1
α
cp
c
commute for some δ′ > 
, the composition
ξ ⊕ ε p ⊕ ε2 × τp+1
id
ξ⊕εp⊕ε2×dα−→ ξ ⊕ ε p ⊕ ε2 × τp+1 H¯−→ ψd+p+2 × τp+1
id
ξ⊕εp⊕ε2×dα−1−→ ψd+p+2 × τp+1
123
1164 M. Krannich
defines a (p + 1)-simplex as required. This finishes the proof of surjectivity
of the map on homotopy groups and injectivity follows from a relative version
of the argument. unionsq
Appendix B. A lemma on symplectic derivations
This appendix serves to record a minor generalisation of Proposition 3.9 in
[8]. We adopt the notation and conventions introduced in Sections 4.1 and
4.2 and fix a principal ideal domain R with 2 ∈ R×, an integer k ∈ Z, and
a graded R-module V that is degreewise free, together with map λ : V ⊗
V → R[0] of degree −k such that the adjoint map s−k V → V ∨ given by
mapping v to λ(v,−) is an isomorphism. We moreover assume that λ is graded
antisymmetric, i.e. that λ(x, y) = −(−1)|x ||y|λ(y, x) holds for all x, y ∈ V .
Example B.1 Let M be a compact R-oriented d-manifold whose R-homology
is free. If ∂M  Sd−1, then the desuspension s−1H˜∗(M; R) of the reduced
homology qualifies as an example of V as above where k = d − 2 and λ
is induced by the intersection form. Concretely, λ is defined to vanish on
elements s−1x ⊗ s−1y unless |x | + |y| = d in which case it is given by
λ(s−1x ⊗ s−1y) = (−1)|x |〈D(x) ∪ D(y), [M, ∂M]〉, where D(−) denotes
the Poincaré duality isomorphism.
Following [8, Sec. 3.5], given a homogenous basis {ai } of V , we use the
dual basis {a#i } determined by λ(ai , a#j ) = δi j to define an element ω :=∑
i a
#
i ⊗ ai ∈ V ⊗ V . This element can be seen to be independent of the
chosen basis (see p. 91 loc.cit. for a proof for R = Q, which generalises) and
to be homogenous of degree k, since |a#i | = k −|ai |. We identify [V, V ] with a
subspace of V ⊗V via the antisymmetrisation map [V, V ] → V ⊗V mapping
[x, y] to x ⊗ y − (−1)|x ||y|y ⊗ x , which is injective since its composition with
the bracket is multiplication by 2 ∈ R×. Under this identification, one verifies
ω = 12
∑
i [a#i , ai ] ∈ [V, V ]. (76)
as in p. 91 loc.cit. Given a dg Lie algebra K over R and a map f : V → K of
graded R-modules, we consider the diagram of graded R-modules
Der f (L(V ), K ) s−k K
s−k K ⊗ V
evω
∼= [−, f (−)]
(77)
whose vertical isomorphism is given by the composition
123
A homological approach to pseudoisotopy theory. I 1165
Der f (L(V ), K )
∼=−→ Hom(V, K ) ∼=−→ K ⊗ V ∨
∼=−→ K ⊗ s−k V = s−k K ⊗ V,
where the first map is given by restriction to generators, the second map is
the canonical isomorphism, and the third map is induced by the inverse of the
adjoint map of λ.
The following lemma is a straight-forward extension of Proposition 3.9 of
[8], which corresponds to the case R = Q, K = L(V ), and f = incV⊂L(V ).
Lemma 1 The diagram (77) is commutative.
Proof Following the beginning of the proof of [8, Prop. 3.9], one sees that a
derivation θ ∈ Der f (L(V ), K ) evaluates on the special element ω as θ(ω) =∑
i [θ(a#i ), f (ai )]. We will use this to show the equivalent claim that the inverse
of the vertical map in (77) followed by evω agrees with [−, f (−)]. This inverse
assigns an element k ⊗ v ∈ s−k K ⊗ V the unique f -derivation θk,v that
extends the linear map λ(v,−) · k : V → K . Using that for v ∈ V , we have
v = ∑i λ(v, a#i ) · ai by the definition of the dual basis, we compute
θk,v(ω) =
∑
i
[θk,v(a#i ), f (ai )] =
∑
i
[λ(v, a#i ) · k, f (ai )]
= [k, f (
∑
i
λ(v, a#i ) · ai )] = [k, v],
which implies the claim. unionsq
Acknowledgements My thanks go to Oscar Randal-Williams for several valuable discussions
and to a referee for their constructive feedback. I was partially supported by O. Randal-Williams’
Philip Leverhulme Prize from the Leverhulme Trust and the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (grant agreement
No. 756444).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International
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