AK-THEORY SPECTRUM FOR COBORDISM CUT AND PASTE GROUPS RENEE S. HOEKZEMA CARMEN ROVI AND JULIA SEMIKINA

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AK-THEORY SPECTRUM FOR COBORDISM CUT AND PASTE

GROUPS

RENEE S. HOEKZEMA, CARMEN ROVI, AND JULIA SEMIKINA

Abstract. Cobordism groups and cut and paste groups of manifolds arise from

imposing two diﬀerent relations on the monoid of manifolds under disjoint union.

By imposing both relations simultaneously, a cobordism cut and paste group SKn

is deﬁned. In this paper, we extend this deﬁnition to manifolds with boundary

obtaining the group SK∂

nand study the relationship of the group to an appro-

priately deﬁned cobordism group of manifolds with boundary. The main results

are the construction of a spectrum that recovers on π0the cobordism cut and

paste groups of manifolds with boundary, SK∂

n, and a map of spectra that lifts

the canonical quotient map SK∂

n→SK∂

Contents

1. Introduction 2

Conventions 5

Acknowledgements 5

2. Cobordism cut and paste groups of manifolds with boundary 5

2.1. Classical SK groups 5

2.2. Cobordism with trivial boundary 6

2.3. SK groups of manifolds with boundary 8

2.4. Fibrations of cobordism categories 9

3. Cubes of manifolds with boundary 12

3.1. The simplicial set X∗

0,0,012

3.2. Simplicial sets X∗

1,0,0and X∗

0,1,013

3.3. The simplicial set X∗

0,0,113

3.4. The simplicial set X∗

1,1,014

3.5. Simplicial sets X∗

1,0,1and X∗

0,1,114

3.6. The simplicial set X∗

1,1,115

3.7. General construction of X∗

•,•,•16

4. Recovering cobordism cut and paste groups as K

017

5. Kis a spectrum 25

6. Maps from BCob∂triv

n+1 and K□(Mfd∂

n)29

References 30

arXiv:2210.00682v2 [math.AT] 5 Sep 2024

2 R. S. HOEKZEMA, C. ROVI, AND J. SEMIKINA

1. Introduction

The classical notion of cut and paste equivalence for closed manifolds was ﬁrst

introduced by Karras, Kreck, Neumann, and Ossa [KKNO73]. The cut and paste

operation on closed n-dimensional manifolds is described by the following procedure.

Given a manifold, one cuts along a separating codimension 1 submanifold with trivial

normal bundle and glues the resulting two pieces back together along an orientation-

preserving diﬀeomorphism to obtain a new manifold. Manifolds Mand Nare called

SK-equivalent (SK stands for “schneiden und kleben”, German for “cut and paste”)

if Ncan be obtained from Mby a sequence of ﬁnitely many cut and paste operations.

Equivalence classes of n-manifolds up to SK-equivalence form the groups SKnof cut

and paste invariants of manifolds, with the operation given by disjoint union.

In [HMM+22] the notions of SK-equivalence and SK-groups were generalised to

the case of manifolds with boundary. Allowing boundary made it possible to for-

mulate a suitable notion of a pointed category with squares Mfd∂

n, that ﬁts into

the framework of K-theory with squares due to [CKMZ23]. Their construction is a

modiﬁed version of the Thomason’s construction of the K-theory spectrum for Wald-

hausen’s category, but it allows more ﬂexibility for the category. The distinguished

squares for the category Mfd∂

ncorrespond to pushout squares that glue manifolds

along a common codimension 0 submanifold, and these can be shown to encode

the SK∂-relations. The K-theory spectrum K□(Mfd∂

n) constructed in [HMM+22],

recovers the SK∂

nas its zeroth homotopy group.

A diﬀerent notion of equivalence of manifolds is cobordism. Two n-dimensional

manifolds Mand Nare cobordant if their disjoint union forms the boundary of

a (n+ 1)-dimensional manifold. Cobordism classes of n-manifolds again form a

group Ωnunder disjoint union. The cobordism cut and paste group SKnis given

by quotienting n-manifolds by both cobordism and SK-equivalence. For oriented

manifolds it was shown in [KKNO73] that the only cobordism cut and paste invariant

is the signature.

In the current paper we generalise the deﬁnition of the SK-groups to the case

of manifolds with boundary and denote the corresponding group SK∂

n. In order

to deﬁne this group we need to quotient by an appropriate notion of cobordism

for manifolds with boundary. The usual notion of cobordism for manifolds with

boundary allows for cobordisms with corners and free boundary components, but

in this setting any two manifolds of the same dimension are cobordant. To ﬁx

this issue we restrict to cobordisms that are trivial on the boundary, meaning that

when restricted to the boundary of the in- and outgoing manifolds, the cobordism

is diﬀeomorphic to a cylinder. We show that the groups SK∂

nare related to the

AK-THEORY SPECTRUM FOR COBORDISM CUT AND PASTE GROUPS 3

classical SKngroups via an exact sequence analogous to the one found for SK∂

nin

[HMM+22]

0−−−−→ SKn

−−−−−−→

[M]7→[M]SK∂

−−−−−−→

[N]7→[∂N ]Cn−1−−−−→ 0.

Here Cn−1is the group completion of the monoid of orientation preserving diﬀeo-

morphism classes of bounding (n−1)-manifolds under disjoint union. This is a free

abelian group.

Next we construct a K-theory spectrum that recovers the SK∂

nas its zeroth

homotopy group. For this we take inspiration from the K-theory with squares

construction of [CKMZ23]and generalise it in several ways. The input for the

K□-construction is a category with squares C, which is a pointed category with two

subcategories of morphisms referred to as horizontal (denoted →) and vertical maps

(denoted ↣), and distinguished squares

A B

□

C D

satisfying certain conditions (see [CKMZ23] and [HMM+22]).

Given a category with squares C, one associates to it a bisimplicial set N•C•,

where the set of (k, l)-simplices NkC(l)is given by the set of all k×ldiagrams

C00 C01 . . . C0l

□ □ □

C10 C11 . . . C1l

□ □ □

.....

□ □ □

Ck0Ck1. . . Ckl,

with face maps given by deleting a corresponding row or column and degeneracy

maps given by inserting the copy of the corresponding row or column. The squares

K-theory space of Cis

K□(C)≃ΩO|N•C•|,

where ΩOis the based loop space, based at the distinguished object O∈N0C(0).

For our purposes we generalise the construction as follows.

4 R. S. HOEKZEMA, C. ROVI, AND J. SEMIKINA

(1) Instead of having two simplicial directions and distinguished squares we will

have three directions(→,↣,⇝) and distinguished cubes as part of the data.

It is easy to see that the construction of [CKMZ23]has a straightforward

generalisation for the categories with distinguished m-cubes for any m≥2.

(2) The substantial modiﬁcation is that we do not require morphisms corre-

sponding to diﬀerent simplicial directions to come from the same category.

In our application objects are n-dimensional manifolds with boundary, but

the three classes of morphisms are of a diﬀerent nature: two simplicial direc-

tions, horizontal (→) and vertical (↣), are coming from a particular type

of embeddings of manifolds that we refer to as SK-embeddings; the third

(depth) simplicial direction (⇝) is given by cobordisms between manifolds

that are cylindrical on the boundary. We moreover have SK-embeddings

between the cobordisms as part of the data deﬁning a cube.

(3) We take the topology into account by means of a fourth simplicial direction,

in the spirit of [RS17]. As a result of this modiﬁcation, we obtain a topolog-

ical version of the cut and paste spectrum of manifolds (for the discrete one

see [HMM+22]), which also has SK∂

nas its π0.We denote it by K□(Mfd∂

n).

We construct a certain quadrisimplicial set X∗

•,•,•with (r, k, l, m)-simplices given

by k×l×mdiagrams consisting of distinguished cubes ﬁbred over ∆rand show

that its geometric realisation is an inﬁnite loop space. Since the resulting space is

connected, its π0does not say anything interesting, and we shift π0by looking at

the space of loops instead. We denote the resulting spectrum by K(Mfd∂

n), where

the notation reﬂects the three simplicial directions given by maps of manifolds

that are used in the construction; Mfd∂

nrefers to the nature of objects; and bar over

it is present to emphasize the topology captured in the fourth simplicial direction.

Let us denote by K

i(Mfd∂

n) the πiof the spectrum.

Theorem A. There is an isomorphism K

0(Mfd∂

n)∼

=SK∂

From our deﬁnition of the quadrisimplicial set X∗

•,•,•we automatically deduce a

topological analogue of the squares cut and paste spectrum, whose inﬁnite loop space

is obtained from a trisimplicial space X∗

•,•,0by taking the loop space of its geometric

realisation. We show that the inclusion of the trisimplicial into the quadrisimplicial

set provides a categoriﬁcation of the quotient map SK∂

n→SK∂

Theorem B. The map of spectra

K□(Mfd∂

n)→K(Mfd∂

coming from the inclusion X∗

•,•,0→X∗

•,•,•induces the canonical quotient map

SK∂

n→SK∂

on the zeroth homotopy groups.

AK-THEORY SPECTRUM FOR COBORDISM CUT AND PASTE GROUPS 5

The paper is organised as follows. In Section 2we introduce an appropriate trivial

boundary cobordism relation and corresponding cobordism groups Ω∂

nfor smooth

compact oriented n-manifolds with boundary. Then we deﬁne SK∂-groups (“cut

and paste cobordism groups”) for manifolds with boundary. We relate these new

groups to the classical ones for closed manifolds via exact sequences. In Section 3we

construct a quadrisimplicial set X∗

•,•,•in the spirit of the construction of [CKMZ23],

which gives rise to a spectrum K(Mfd∂

n). In Section 5we verify that |X∗

•,•,•|is

indeed an inﬁnite loop space. In Section 4we prove Theorem Aand ﬁnally in Section

6we prove Theorem B.

Conventions. All manifolds in this paper will be smooth, oriented and compact.

All maps between manifolds/cobordisms will be smooth and orientation preserving.

Acknowledgements. We would like to thank Jonathan Campbell, Jim Davis,

Johannes Ebert, Fabian Hebestreit, Manuel Krannich, Achim Krause, Wolfgang

L¨uck, Mona Merling, Thomas Nikolaus, Oscar Randal-Williams, George Raptis,

Jens Reinhold, Jan Steinebrunner, Wolfgang Steimle, Inna Zakharevich for helpful

discussions. We would also like to thank the anonymous referee for very help-

ful comments We thank the Mathematics and Statistics Department at Loyola

Chicago for hosting us. This research was supported through the program “Re-

search in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2020

and 2022, and through the program “Summer Research in Mathematics” by the

Mathematical Sciences Research Institute. The ﬁrst author was supported by the

Emerson Collective and by the Dutch Research Council (NWO) through the grant

VI.Veni.212.170. The second author was supported by a summer research grant

from Loyola University Chicago and by MPIM Bonn. The third author was funded

by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un-

der Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics M¨unster:

Dynamics–Geometry–Structure.

2. Cobordism cut and paste groups of manifolds with boundary

2.1. Classical SK groups. The cut and paste group SKnand the cobordism group

Ωnare both formed by quotienting the monoid of manifolds under disjoint union

by the cut and paste relation in the ﬁrst case, and by the cobordism relation in the

second. There are no natural maps from one group to the other in either direction.

They can be compared by quotienting the monoid of manifolds by both relations,

forming the cobordism cut and paste group SKn. The following pair of short exact

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AK-THEORYSPECTRUMFORCOBORDISMCUTANDPASTEGROUPSRENEES.HOEKZEMA,CARMENROVI,ANDJULIASEMIKINAAbstract.Cobordismgroupsandcutandpastegroupsofmanifoldsarisefromimposingtwodifferentrelationsonthemonoidofmanifoldsunderdisjointunion.Byimposingbothrelationssimultaneously,acobordismcutandpastegroupSKnisdefined....

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