MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS SAMUEL MU NOZ-ECH ANIZ ABSTRACT .We prove that the mapping class group is not an h-cobordism invariant of high-

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MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS

SAMUEL MU ˜

NOZ-ECH ´

ANIZ

ABSTRACT.

We prove that the mapping class group is not an

-cobordism invariant of high-

dimensional manifolds by exhibiting

-cobordant manifolds whose mapping class groups have

different cardinalities. In order to do so, we introduce a moduli space of "

-block" bundles and

understand its difference with the moduli space of ordinary block bundles.

1. INTRODUCTION

1.1. The main result. Automorphism groups of manifolds have been subject to extensive

research in algebraic and geometric topology. Inspired by the study of how different h-cobordant

manifolds can be (see e.g. [

JK15

JK18

]), in the present paper we investigate the question of how

automorphism groups of manifolds can vary within a ﬁxed h-cobordism class. Namely: given

an h-cobordism Wd+1between (closed) smooth1manifolds Mand M′of dimension d≥0, how

different can the homotopy types of the diffeomorphism groups Diff(M) and Diff(M′) be?

Certain analogues of this question have led to invariance-type results. Dwyer and Szczarba

[

DS83

, Cor. 2] proved that when d

=4, the rational homotopy type of the identity component

Diff0

(M)

⊂Diff

(M) does not change as M

varies within a ﬁxed homeomorphism class of

smooth manifolds. Krannich [

Kra19

, Thm. A] gave another instance of such a result, showing

that when d=2k

≥

6 and M

is closed, oriented and simply connected, the rational homology of

BDiff+(M) in a range is insensitive to replacing Mby M#Σ, for Σany homotopy d-sphere.

Our main result is, however, that the homotopy types of the diffeomorphism groups of

h-cobordant manifolds can indeed be different in general. Let Γ(M) denote the mapping class

group of M—the group of isotopy classes of diffeomorphisms of M, i.e., Γ(M) :=

π0

(

Diff

(M)).

The block mapping class group

(M) is the quotient of Γ(M) by the normal subgroup of those

classes of diffeomorphisms which are pseudoisotopic to the identity.

Theorem A. In each dimension d =12k

−

≥

0, there exist d-manifolds M

(see Theorem 4.6)

h-cobordant to the lens space L =L12k−1

7(r1:· · · :r6k), where

r1=· · · =rk=1,rk+1=· · · =r2k=2, . . . r5k+1=· · · =r6k=6 mod 7,

such that

(i) e

Γ(L)and e

Γ(M)are ﬁnite groups with cardinalities of different 3-adic valuations,

(ii) Γ(L)and Γ(M)are ﬁnite groups with cardinalities of different 3-adic valuations.

Remark 1.1.For an oriented connected manifold M, there are orientation preserving mapping

class groups Γ

(M) and

Γ+

(M), which have index one or two inside the whole mapping class

groups Γ(M) and

(M), respectively. Therefore, the conclusions of Theorem Aalso hold for

Γ+(−) and Γ+(−).

Remark 1.2.Theorem A(i) is the best possible result in the following sense: let

Diff

(M)

denote the (geometric realisation of the) semi-simplicial group of block diffeomorphisms of M

(cf. [

BLR06

, p. 20] or [

ERW14

, Defn. 2.1]), whose p-simplices consist of diffeomorphisms

∆

p∼

−→

∆

which are face-preserving (i.e. for every face

σ⊂

∆

restricts

2020 Mathematics Subject Classiﬁcation. 57R80, 55R60, 57S05, 57N37, 57Q10.

Key words and phrases. h-cobordism, mapping class group, diffeomorphism group, block diffeomorphism, Whitehead

torsion, pseudoisotopy.

We will work in the smooth setting for notational preference, but all of the results in this paper are equally valid for

the topological and PL categories. See Remarks 4.11 and 5.4 for modiﬁed arguments when CAT =Top and PL.

arXiv:2210.06573v2 [math.AT] 17 May 2024

MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS 2

to a diffeomorphism of M

×σ

). Then we have

(M)=

π0

(

Diff

((M)). The restriction

map

ρM

Diff

(W)

−→ g

Diff

(M) is a ﬁbration with ﬁbre

DiffM

(W), the subspace of block

diffeomorphisms of Wwhich ﬁx pointwise a neighbourhood of M

⊂

W. By the s-cobordism

theorem (see Theorem 2.1 below), there exists some h-cobordism

−

Wfrom M

′

to Msuch that

W∪M′−W∼

=M×Iand −W∪MW∼

=M′×I. Then the group homomorphisms

IdW∪M′−:e

C(M′) :=g

DiffM′×{0}(M′×I)−→ g

DiffM(W),

Id−W∪M−:g

DiffM(W)−→ g

DiffM′(−W∪MW)∼

C(M′)

are easily seen to be homotopy inverse to each other. But the group

′

) of block concordances

of M

′

is contractible (cf. [

BLR06

, Lem. 2.1]), and therefore

ρM

induces an equivalence onto the

components that it hits, and similarly for

ρM′

. In other words, the classifying spaces B

Diff

(M)

and Bg

Diff(M′) share the same universal cover, so

πi(g

Diff(M)) ∼

=πi(g

Diff(M′)),i≥1.

The upshot is that the homotopy types of

Diff

(M) and

Diff

′

) can at most differ by their sets of

path-components, and Theorem A(i) provides an example showcasing this phenomenon.

1.2. Moduli spaces of h- and s-block bundles. Recall that for d

≥

5, the Whitehead group

(M) of a compact d-manifold M(see Section 2.1) classiﬁes isomorphism classes of h-

cobordisms starting at M. This group has an involution denoted

τ7→ τ

which, roughly speaking

and up to a factor of (

−

, corresponds to reversing the direction of an h-cobordism (see (2.4)).

In Section 3we will introduce the h- and s-block moduli spaces,

and

respectively,

whose vertices (as semi-simplicial sets) are the smooth closed d-manifolds, for some ﬁxed integer

≥

0. A path in the former (resp. latter) space between d-manifolds Mand M

′

is exactly

an h-cobordism W:M

⇝

′

(resp. an s-cobordism W:M

⇝

′

, i.e., an h-cobordism with

vanishing Whitehead torsion (see Section 2.2)). The s-block moduli space

is, somewhat in

disguise, a well-known object; in Proposition 3.6 we identify the path-component of M

with Bg

Diff(M), the classifying space for the group of block diffeomorphisms of M.

The second main result we state arises as part of the proof of Theorem A, but may be of

independent interest: there is a natural inclusion

Ms−→ f

which forgets the simpleness

condition. We identify the homotopy ﬁbre of this inclusion (i.e. the homotopical difference

between the h- and s-block moduli spaces) as a certain inﬁnite loop space.

Theorem B. Let M be a closed d-dimensional manifold, and let C

{

}

act on the Whitehead

group

(M)by t

·τ

:=(

−

d−1τ

. Write H

(M)for the Eilenberg–MacLane spectrum

associated to

(M), and let H

(M)

hC2

:=H

(M)

∧C2

(EC

)

stand for the homotopy

C2-orbits of HWh(M). For d ≥5, there is a homotopy cartesian square

Ω∞(HWh(M)hC2)f

{Md}f

Mh,

where the lower horizontal map is the inclusion of M as a point in f

Mh.

As we will explain in Section 3.3, this result is intimately tied to the Rothenberg exact sequence

[Ran81, Prop. 1.10.1].

Structure of the paper. Section 2serves as a reminder to the reader of the s-cobordism theorem

and some of the properties of Whitehead torsion.

In Section 3we prove Theorem B. The proof boils down to arguing that certain simplicial

abelian group F

alg

•

(A) corresponds to the spectrum HA

hC2

under the Dold–Kan correspondence

(see Theorem 3.10).

MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS 3

Section 4deals with part (i) of Theorem A, which is proved in Theorem 4.6. We analyse the

lower degree part of the homotopy long exact sequence associated to the homotopy pullback

square of Theorem B. The proof of Theorem A(ii) builds on part (i) and pseudoisotopy theory,

and comprises Section 5.

Appendix Ais an algebraic K-theory computation required for Sections 4and 5. Appendix B

explores the connection between Theorem Band the theory of Weiss–Williams [WW88].

Acknowledgements. The author is immensely grateful to his Ph.D. supervisor Prof. Oscar

Randal-Williams for suggesting this problem and for the many illuminating discussions that have

beneﬁted this work. The author would also like to thank John Nicholson for making him aware

of [ZTC19]. The EPSRC supported the author with a Ph.D. Studentship, grant no. 2597647.

2. NOTATION AND RECOLLECTIONS

All manifolds will be assumed to be compact and smooth (possibly with corners).

2.1. Whitehead Torsion. The Whitehead group of (

π,

w) [

Mil66

6], where

is a group and

w:π→C2={±1}is a homomorphism, is the abelian group

Wh(π, w) :=GL(Zπ)ab/(±π)

equipped with the following involution: the anti-involution on the group ring Zπgiven by

a=X

g∈π

ag·g7−→ a:=X

g∈π

w(g)ag·g−1,ag∈Z,

induces an involution on

(

π,

w) by sending an element represented by a matrix

τij

) to

its conjugate transpose

:=(

τji

). We will refer to this involution as the algebraic involution

(

π,

w). We will write

(

) for

(

π,

w) if wis the trivial homomorphism, or if we are

simply disregarding this involution. If Xis a ﬁnite CW-complex with a choice of basepoint in

each of its connected components, the Whitehead group of Xis

Wh(X) :=M

Xj∈π0(X)

Wh(π1(Xj)).

If X=Mis moreover a manifold, the algebraic involution on

(M) is that induced by

w=w1(M)∈H1(M;Z/2), the ﬁrst Stiefel–Whitney class of M.

Given a homotopy equivalence between ﬁnite pointed CW-complexes f:X

≃

−→

Y, we will

denote by

(f)

∈Wh

(X) its (Whitehead)torsion [

Mil66

7]. It only depends on fup to homotopy

[

Mil66

, Lem. 7.7]. Let us collect a few properties of the Whitehead torsion

(

−

) that we will use

throughout the paper:

•

Composition rule:

(

−

) is a crossed homomorphism in the sense that if f:X

≃

−→

Yand

g:Y≃

−→ Zare homotopy equivalences, then [Mil66, Lem. 7.8]

(2.1) τ(g◦f)=τ(f)+f−1

∗τ(g),

where f

∗

(X)

∼

−→ Wh

(Y) is the natural isomorphism induced by

π1

(f) :

π1

(X)

∼

−→ π1

(Y).

•

Inclusion-exclusion principle: if X=X

0∪

and Y=Y

0∪

, where X

X01 :=X0∩X1and Y01 :=Y0∩Y1are all ﬁnite CW-complexes, and

f0:X0

≃

−→ Y0,f1:X1

≃

−→ Y1,f01 =f0∩f1:X01

≃

−→ Y01,

are homotopy equivalences, then the torsion of the homotopy equivalence f=f

0∪

≃

−→

is [Coh73, Thm. 23.1]

(2.2) τ(f)=(i0)∗τ(f0)+(i1)∗τ(f1)−(i01)∗τ(f01)∈Wh(X),

where i0:X0−→ X,i1:X1−→ Xand i01 :X01 −→ Xare the inclusions.

MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS 4

•

Product rule:

(

−

) is multiplicative with respect to the Euler characteristic in the sense that

for any homotopy equivalence f:X

≃

−→

Yand any ﬁnite connected CW-complex Kwith

basepoint ∗ ∈ K[Coh73, Thm. 23.2],

(2.3) τ(f×idK)=χ(K)·i∗τ(f)∈Wh(X×K),

where i:X∼

=X× {∗} −→ X×Kis the inclusion.

A homotopy equivalence fas above is said to be simple, and denoted f:X

≃s

−→

Y, if

(f)=0.

We will write s

Aut

(X)

⊂

Aut

(X) for the topological submonoid (see (2.1)) of simple homotopy

automorphisms of X.

2.2. The s-cobordism theorem. Let M

be a smooth compact manifold of dimension d. A

cobordism from M rel

∂

Mis a triple (W;M

′

), also written as W:M

⇝

′

, consisting of a

(d+1)-manifold Wd+1with boundary

∂W∼

=M∪M′∪(∂M×[0,1])

so that M

∩

(

∂

1]) =

∂

× {

}

and M

′∩

(

∂

1]) =

∂

× {

}

(in particular

∂

′

∂

M). Cobordisms are often accompanied with an additional data of collars, i.e., open

neighbourhoods of Mand M

′

in Wdiffeomorphic to M

, 

) and M

′×

−,

1] (rel

∂

for some small

 >

0, but the choice of such is contractible. If

∂

M=Ø, this coincides with

the usual notion of a cobordism between closed manifolds. Such a cobordism is called an

h-cobordism if the inclusions i

: (M

, ∂

→

, ∂

I) and i

M′

: (M

′, ∂

′

)

→

, ∂

are homotopy equivalences of pairs. In such case we will write W:M

⇝

′

to emphasise that W

is an h-cobordism from Mto M′. The torsion of Wwith respect to Mis

τ(W,M) :=τ(iM)∈Wh(M).

M)=0, such an h-cobordism W:M

⇝

′

is said to be simple (or an s-cobordism), and

denoted W:M

⇝

′

. This deﬁnition does not depend on the direction of Wsince the torsion of

an h-cobordism satisﬁes the duality formula [Mil66,§10]

(2.4) τ(W,M′)=(−1)d(hW)∗τ(W,M).

Here hW:M≃M′is the natural homotopy equivalence

(2.5) hW:M W M′,

≃

rM′

≃

where rM′is some homotopy inverse to iM′(so hWis only well-deﬁned up to homotopy).

Due to the composition rule (2.1), the torsion of an h-cobordism is nearly additive with

respect to composition: namely if W:M

⇝

′

and W

′

′h

⇝

′′

are h-cobordisms, we write

′◦

W:M

⇝

′′

for the h-cobordism W

∪M′

′

, which can be made smooth by pasting along

collars. Then

(2.6) τ(W′◦W,M)=τ(W,M)+(hW)−1

∗τ(W′,M′).

Let h

Cob∂

(M) denote the set of h-cobordisms rel boundary starting at M, up to diffeomorphism

rel M. We will use the following a great deal [Maz63,Bar64].

Theorem 2.1 (s-Cobordism Theorem rel boundary).If d =

dim

≥

5, then there is a bijection

hCob∂(M)←→ Wh(M),(W:Mh

⇝M′)7−→ τ(W,M).

3. THE BLOCK MODULI SPACES OF MANIFOLDS

As explained in the introduction, we now present the h- and s-block moduli spaces of

manifolds, in which a path, i.e., a 1-simplex, is an h- or s-cobordism, respectively. To describe

what higher-dimensional simplices should be we give the next deﬁnition, which is inspired by

[HLLRW21,§2].

MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS 5

Deﬁnition 3.1. Fix once and for all some small

 >

0. A compact smooth manifold with corners

Wd+p⊂R∞×∆pis said to be stratiﬁed over ∆pif:

(i) W is a closed manifold if p =0,

(ii)

W is transverse to

R∞×σ

for every proper face

σ⊂

∆

and W

:=W

∩

(

R∞×σ

)is

a(d+dim σ)-dimensional manifold stratiﬁed over ∆dim σ∼

=σ,

(iii) W satisﬁes the -collaring conditions of [HLLRW21, Defn. 2.3.1(ii)].

We will write W

d+p⇒

∆

for such a manifold. A map f :W

→

V between manifolds stratiﬁed

over ∆

is said to be face-preserving, and denoted f :W

→∆

V, if for every face

σ⊂

∆

have f (W

)

⊂

and f satisﬁes certain collaring conditions (namely f must be the product

σ×Id

in the



-neighbourhood of the strata W

, where of f

:=f

|Wσ

). If moreover f

is a

homotopy equivalence, simple homotopy equivalence or diffeomorphism for all

σ⊂

∆

, we will

write f :W♠

−→∆V for ♠=≃h,≃sor ∼

=, respectively.

Notation 3.2. Let Λp

i⊂∆pdenote the i-th horn of ∆p(i =0,...,p).

•

If 0

≤

0<· · · <

r≤

p, we write

⟨

0,...,

r⟩ ⊂

∆

for the face spanned by the vertices

i0,...,ir∈∆p.

•If W is stratiﬁed over ∆p, we will often write ∂iW for W⟨0,...,bi,...,p⟩and Wifor W⟨i⟩. For

instance, ⟨0. . . ,b

i,...,p⟩ ≡ ∂i∆p⊂∆p.

•

If K

⊂

∆

is a simplicial sub-complex, we will write W

for W

∩

(

R∞×

K). In the

particular case that K = Λ

, we set Λ

(W) :=W

Λp

. For instance, if

σ⊂

∆

is some

face, Λi(σ)denotes the i-th horn of σ(i =0,...,dim σ).

•If f :W−→∆V is face-preserving, we will write ∂if for f∂i∆p=f|∂iW.

Example 3.3.A cobordism W

d+1

⇝

′

between closed manifolds Mand M

′

is always

diffeomorphic to a manifold W′⊂R∞×∆1stratiﬁed over ∆1with W′

0∼

=Mand W′

1∼

=M′.

Deﬁnition 3.4. Fix some integer d

≥

0. The h-block moduli space of d-manifolds is the

semi-simplicial set f

•with p-simplices

(3.1) f

p:=





Wd+p

⇓

∆p

:∃f:W≃h

−→∆W0×∆p



,

and with face maps given by restriction to face-strata

∂i:f

p−→ f

p−1,

Wd+p

⇓

∆p

7−→

∂iWd+p

⇓

∂i∆p∼

=∆p−1,

i=0,...,p.

The s-block moduli space of d-manifolds

•

is its simple analogue, where

≃h

in (3.1) is

replaced by

≃s

, and has a natural inclusion

•−→ f

•

. We will let

and

denote the

geometric realisations |f

•|and |f

•|, respectively.

Remark 3.5.Under the conditions of Deﬁnition 3.1, the semi-simplicial sets

•

and

•

are

Kan, as remarked right after [HLLRW21, Defn. 2.3.1].

The next two subsections are devoted to prove Theorem B. But ﬁrst, we study the s-block

moduli space

more closely. We recall that the classifying space B

Diff

(M) for the simplicial

group of block diffeomorphisms has a semi-simplicial model (see e.g. [

ERW14

]) in which the

p-simplices are

Diff(M)p=





Wd+p

⇓

∆p

:∃φ:W∼

−→∆M×∆p



,

and therefore there is a forgetful inclusion Bg

Diff(M)−→ f

Ms.

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MAPPINGCLASSGROUPSOFh-COBORDANTMANIFOLDSSAMUELMU˜NOZ-ECH´ANIZABSTRACT.Weprovethatthemappingclassgroupisnotanh-cobordisminvariantofhigh-dimensionalmanifoldsbyexhibitingh-cobordantmanifoldswhosemappingclassgroupshavedifferentcardinalities.Inordertodoso,weintroduceamodulispaceof"h-block"bundlesandunder...

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MAPPING CLASS GROUPS OF h-COBORDANT MANIFOLDS SAMUEL MU NOZ-ECH ANIZ ABSTRACT .We prove that the mapping class group is not an h-cobordism invariant of high-

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