Cut and paste invariants of manifolds via algebraic K-theory
Renee Hoekzema, Mona Merling, Laura Murray, Carmen Rovi, Julia Semikina
aa r X i v : . [ m a t h . A T ] J a n CUT AND PASTE INVARIANTS OF MANIFOLDS VIAALGEBRAIC K -THEORY RENEE HOEKZEMA, MONA MERLING, LAURA MURRAY,CARMEN ROVI AND JULIA SEMIKINA
Abstract.
Recent work of Jonathan Campbell and Inna Zakharevich has fo-cused on building machinery for studying scissors congruence problems via al-gebraic K -theory, and applying these tools to studying the Grothendieck ring ofvarieties. In this paper we give a new application of their framework: we constructa spectrum that recovers the classical SK (“schneiden und kleben,” German for“cut and paste”) groups for manifolds on π , and we construct a derived versionof the Euler characteristic. Contents
1. Introduction 1Conventions 4Acknowledgements 42. Scissors congruence groups for manifolds with boundary 52.1. SK-groups for manifolds with boundary 52.2. SKK-groups for manifolds with boundary 93. K -theory of categories with squares 113.1. Overview of Campbell and Zakharevich’s square K -theory 113.2. Category with squares from a Waldhausen category 134. K -theory of manifolds with boundary 164.1. The category with squares for manifolds with boundary 164.2. The computation of K (Mfd ∂n ) 175. The derived Euler characteristic for manifolds with boundary 205.1. The lift of the singular chain functor 205.2. Recovering the Euler characteristic on π Introduction
The classical scissors congruence problem asks whether given two polyhedra withthe same volume P and Q in R , one can cut P into a finite number of smaller polyhedra and reassemble these to form Q . Precisely, P and Q are scissors congruentif P = S mi =1 P i and Q = S mi =1 Q i , where P i ∼ = Q i for all i , and the subpolyhedra ineach set only intersect each other at edges or faces. There is an analogous definitionof an SK (German “schneiden und kleben,” cut and paste) relation for manifolds:Given a closed smooth oriented manifold M , one can cut it along a separatingcodimension 1 submanifold Σ with trivial normal bundle and paste back the twopieces along an orientation preserving diffeomorphism Σ → Σ to obtain a newmanifold, which we say is “cut and paste equivalent” or “scissors congruent” to it.We give a pictorial example of this relation:
1. Start with T
2. Cut along four copies of S
3. Paste back along boundaries
Figure 1.
Example of a cut and paste operationZakharevich has formalized the notion of scissors congruence via the notion of an assembler –this is a Grothendieck site with a few extra properties, whose topologyencodes the cut and paste operation. She constructs an associated K -theory spec-trum, which on π recovers classical scissors congruence groups [Zak17b]. Specificexamples of assemblers recover scissors congruence groups for polytopes and theGrothendieck ring of varieties, as π of their corresponding K -theory spectra. Thehigher K -groups encode further geometric information. Independently, Campbellhas introduced the formalism of subtractive categories , a modification of the def-inition of Waldhausen categories, to define a K -theory spectrum of varieties thatrecovers the Grothendieck ring of varieties on π [Cam19]. Though the approachesto encoding scissors congruence abstractly are different, the resulting spectra ofZakharevich and Campbell are shown to be equivalent in [CZ19a].The focus of Zakharevich and Campbell has been to construct and study a K -theory spectrum of varieties, and this spectrum level lift of the Grothendieck ring ofvarieties has led to a fruitful research program to better understand varieties. For UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 3 example, an analysis of K for the K -theory spectrum of varieties allowed Zakhare-vich to elucidate structure on the annihilator of the Lefschetz motive [Zak17a], andCampbell, Wolfson and Zakharevich use a lift of the zeta function for varieties toshow that π of the K -theory spectrum for varieties contains nontrivial geometric in-formation [CWZ19]. Studying cut and paste relations for manifolds via K -theoreticmachinery remains as of yet unexplored. We start this exploration in this paper.Unfortunately, the framework from [Zak17b, Cam19] does not directly apply tothe case of manifolds. The problem is that if one tries to find a common refinementof two different SK-decompositions of a manifold, one might have to cut boundariesand one gets manifolds with corners. This makes some of the axioms in both theassembler approach and the subtractive category approach break down. However,work in progress of Campbell and Zakharevich on “ K -theory with squares,” K (cid:3) , afurther synthetization of scissors congruence relations as K -theory that generalizesWaldhausen K -theory, does give the right framework to construct the desired scis-sors congruence spectrum for manifolds. Encompassing the manifold example wasalso one of the motivations behind Campbell’s and Zakharevich’s development of“ K -theory with squares”.The study of SK-invariants and SK-groups in [KKNO73] focuses on closed man-ifolds. However, in order for the K (cid:3) -theoretic scissors congruence machinery toapply, we need to work in the category of manifolds with boundary, since the piecesin an SK-decomposition have boundary. This is not well-explored classically, asmost of the existing work on SK-groups is for closed manifolds. We generalize thenotion of SK-equivalence to the case of manifolds with boundary and denote thecorresponding group by SK ∂n . Our definition of SK ∂n is different from the one men-tioned in [KKNO73] in that we insist that every boundary along which we cut getspasted, and this is crucial for the further application of the K -theoretic technology.We formulate a suitable notion of a category with squares Mfd ∂n , that fits intothe framework of the K -theory with squares framework, and whose distinguishedsquares exactly encode the “cut-and-paste” relations for n -dimensional manifoldswith boundary. We show that the Ω-spectrum obtained from the construction ofCampbell and Zakharevich, applied to Mfd ∂n , which we denote by K (cid:3) (Mfd ∂n ), recov-ers the SK ∂n as its zeroth homotopy group: Theorem A.
There is an isomorphism K (cid:3) (Mfd ∂n ) ∼ = SK ∂n , where K (cid:3) (Mfd ∂n ) is π of a scissors congruence K -theory spectrum K (cid:3) (Mfd ∂n ) . For closed manifolds, there is a more refined notion of SKK-invariance whichdiffers from SK-invariance by a controlled correction term that is allowed to de-pend only on the gluing diffeomorphisms but not the cut submanifold pieces. TheSKK-groups can be interpreted as Reinhardt vector field bordism groups [KKNO73],
R. HOEKZEMA, M. MERLING, L. MURRAY, C. ROVI AND J. SEMIKINA which equivalently can be seen to be π of the Madsen-Tillman spectrum M T SO ( n )[Ebe13], or π of the cobordism category. We give a definition of SKK-groups formanifolds with boundary, and the conjecture, which we will investigate in futurework, is that they arise as π of K (cid:3) (Mfd ∂n ). This expectation is inspired by discus-sions with Inna Zakharevich and Jonathan Campbell and is reminiscent of resultson K that Zakharevich has obtained in other contexts.Scissors congruence invariants for manifolds (SK-invariants) are abelian groupvalued homomorphisms from the monoid of manifolds under disjoint union, whichfactor through the SK-group. It is well known classically that for closed manifoldsthe Euler characteristic and the signature, and linear combinations thereof, are theonly SK-invariants, and these are still SK-invariants of manifolds with boundary.In this paper, we show that the Euler characteristic as a map to Z , viewed as thezeroth K -theory group of Z , is the π level of a map of spectra from the scissorscongruence spectrum for manifolds with boundary that we define. In future work,we plan to also investigate the signature map to the zeroth L -theory group of Z . Theorem B.
There is a map of K -theory spectra K (cid:3) (Mfd ∂ ) → K ( Z ) , which on π agrees with the Euler characteristic for smooth compact manifolds withboundary. The paper is organized as follows. In Section 2 we introduce the definitions of SK-and SKK-groups for smooth compact manifolds with boundary and we prove thatthey are related to the classical SK and SKK groups for smooth closed manifolds viaexact sequences. In Section 3 we review the set-up of categories with squares andtheir K -theory as defined by Campbell and Zakharevich. In Section 4 we constructthe category of squares for smooth compact manifolds with boundary and proveTheorem A, and in Section 5 we prove Theorem B. Conventions.
All manifolds in this paper are smooth, compact and oriented. Wewill distinguish between closed manifolds and manifolds with boundary. We will usethe notation ¯ M for the manifold M with reversed orientation. Acknowledgements.
We are greatly indebted to Jonathan Campbell and InnaZakharevich for their generosity in sharing their work in progress on squares K -theory, which our project relies on, and for their extensive patience in explainingit to us and answering our questions. It is a great pleasure to also acknowledgethe contributions to this project arising from discussions with Jonathan Block, JimDavis, Greg Friedman, Soren Galatius, Herman Gluck, Fabian Hebestreit, Matthias UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 5 Kreck, Malte Lackmann, Wolfgang L¨uck, Cary Malkiewich, Peter Teichner, AndrewTonks, and Chuck Weibel.Finally, we thank the organizers of the Women in Topology III program and theHausdorf Research Institute for Mathematics for their hospitality during the work-shop. The WIT III workshop was supported through grants NSF-DSM 1901795,NSF-HRD 1500481 - AWM ADVANCE grant and the Foundation Compositio Math-ematica, and we are very grateful for their support. The second named author wassupported by grant NSF-DMS 1709461. The third named author was supportedby grant NSF-DMS 1547292. The fifth named author was supported by the MaxPlanck Society and Wolfgang L¨ucks ERC Advanced Grant “KL2MG-interactions”(no. 662400).2.
Scissors congruence groups for manifolds with boundary
SK-groups for manifolds with boundary.
We start by reviewing the defini-tions of the classical scissors congruence groups of smooth closed oriented manifolds,namely the SK n -groups introduced in [KKNO73]. The “scissors congruence” or “cutand paste” relation on smooth closed oriented manifolds is given as follows: cut an n -dimensional manifold M along a codimension 1 smooth submanifold Σ with trivialnormal bundle that separates M in the sense that the complement of Σ in M is adisjoint union of two components M and M , each with boundary diffeomorphic toΣ. Then paste back the two pieces together along an orientation preserving diffeo-morphism φ : Σ → Σ . We say M and M ∪ φ M are “cut and paste equivalent” or“scissors congruent.”Note that for a codimension 1 submanifold Σ with trivial normal bundle thatdoes not separate M (for example the inclusion of S × { } into S × S ) we cantake the union with a second copy of Σ embedded close to it, and the disjoint unionΣ ⊔ Σ then separates M . Definition 2.1.
Two smooth closed manifolds M and N are SK -equivalent (or scissors congruent or cut and paste equivalent ) if N can be obtained from M by afinite sequence of cut and paste operations. Example 2.2.
In Figure 2 we can see that T ♯ T ⊔ S is SK-equivalent to T ⊔ T .Let M n be the monoid of diffeomorphism classes of smooth closed oriented n -dimensional manifolds [ M ] under disjoint union. The SK n -group from [KKNO73] isdefined to satisfy the universal property that any abelian valued monoid map from M n which respects SK-equivalence (also called an SK-invariant) factors through it. R. HOEKZEMA, M. MERLING, L. MURRAY, C. ROVI AND J. SEMIKINA φ ↓ ψ ↓≃ ∈ SK Figure 2.
Example of an SK-relation
Definition 2.3.
The scissors congruence group SK n for smooth closed oriented n -dimensional manifolds is the quotient of the Grothendieck group Gr( M n ) by theSK-equivalence relation.Explicitly, SK n is the free abelian group on diffeomorphism classes [ M ] modulothe following relations:(1) [ M ⊔ N ] = [ M ] + [ N ];(2) Given compact oriented manifolds M , M and orientation preserving diffeo-morphisms φ, ψ : ∂M → ∂M ,[ M ∪ φ ¯ M ] = [ M ∪ ψ ¯ M ] , where ¯ M is M with reversed orientation.We note that in order to define a scissors congruence spectrum, we need to work ina category of manifolds with boundary since the pieces in the cut and paste relationare manifolds with boundary. Therefore, we introduce a definition of SK-groupsfor manifolds with boundary; these are the groups which we will recover as π of ascissors congruence K -theory spectrum.We define the “cut and paste relation” on smooth compact manifolds with bound-ary analogously to that on closed manifolds: cut an n -dimensional manifold M alonga codimension 1 smooth submanifold Σ with trivial normal bundle, which separates M , and for which Σ ∩ ∂M = ∅ . Then paste back the two pieces together alongan orientation preserving diffeomorphism φ : Σ → Σ . We emphasize that we do notallow boundaries to be cut, and we require that all boundaries which come fromcutting to be pasted back together, leaving the existing boundaries of a manifolduntouched by the cut and paste operation.
UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 7 Definition 2.4.
Two smooth compact manifolds with boundary will be called SK -equivalent if one can be obtained from the other via a finite sequence of cut andpaste operations in the sense described above.
Remark 2.5.
Our definition of the cut and paste relation for manifolds with bound-ary is different than the one in [KKNO73, Chapter 5], where M ∪ φ M ∼ M ⊔ M .Namely, they allow pieces that are cut to not be pasted back together. In orderto apply the K -theoretic machinery to obtain the SK ∂n -group as π of a K -theoryspectrum, it is important to use our definition of SK ∂n . Definition 2.6.
Let M ∂n be the monoid of diffeomorphism classes of smooth com-pact oriented n -dimensional manifolds with boundary under disjoint union. The scissors congruence group SK ∂n for smooth compact oriented manifolds is the quo-tient of the Grothendieck group Gr( M ∂n ) by the SK-equivalence relation.Explicitly, SK ∂n is the free abelian group on diffeomorphism classes of smoothcompact oriented n -dimensional manifolds (with or without boundary) modulo thefollowing relations:(1) [ M ⊔ N ] ∼ [ M ] + [ N ];(2) Given compact oriented manifolds M , M , closed submanifolds Σ ⊆ ∂M and Σ ′ ⊆ ∂M , and orientation preserving diffeomorphisms φ, ψ : Σ → Σ ′ ,[ M ∪ φ ¯ M ] = [ M ∪ ψ ¯ M ] . Example 2.7.
In Figure 3 we see an example of an SK ∂n -relation. φ ↓ ψ ↓≃ ∈ SK ∂n Figure 3.
Example of an SK ∂ -relationWe now relate our definition of SK ∂n with the classical SK n via an exact sequence.Denote by C n the Grothendieck group of the monoid of diffeomorphism classes ofsmooth closed oriented n -dimensional nullcobordant manifolds under disjoint union. Theorem 2.8.
For every n ≥ the following sequence is exact R. HOEKZEMA, M. MERLING, L. MURRAY, C. ROVI AND J. SEMIKINA −−−−→ SK n α −−−−−−→ [ M ] [ M ] SK ∂n β −−−−−−→ [ N ] [ ∂N ] C n − −−−−→ . Proof.
Note that the map α : SK n → SK ∂n taking a class of manifolds in SK n to aclass containing the same manifolds in SK ∂n is well-defined, since every relation fromthe definition of SK n is also a relation in the definition of SK ∂n . The map β that takesa class of manifolds to the diffeomorphism class of the boundary is well-defined, sincethe equivalence relation from the definition of SK ∂n preserves the boundary.We show exactness at the middle term. It is clear from the definition that Im α ⊆ ker β. Let us show the reverse inclusion. Let x ∈ ker β. Every element of SK ∂n canbe written in the form x = [ M ] − [ N ] , where M, N are compact smooth oriented n -manifolds with boundary (not necessarily connected).Let ¯ M be the copy of M with the opposite orientation and let DM be the doubleof M, i.e. DM = M ∪ id ¯ M .
Note that DM is a closed manifold. Since C n − is afree abelian group and β ( x ) = [ ∂M ] − [ ∂N ] = 0 we conclude that the ∂M and ∂N are diffeomorphic. Hence we may glue ¯ M to N along the boundary.We will call this gluing diffeomorphism φ (it does not have to be unique, we justpick one) and denote by L the closed manifold, which is the result of this gluing.Therefore, DM = M ∪ id ¯ M , and L = N ∪ φ ¯ M .
Hence in SK ∂n , [ N ] + [ DM ] = [ N ∪ id ( ∂N × [0 , M ∪ id ¯ M ]= [ N ∪ φ ¯ M ] + [( ∂N × [0 , ∪ φ M ]= [ L ] + [ M ] . Consequently, x = [ M ] − [ N ] = [ DM ] − [ L ] ∈ Im α. See Figure 4 for an illustration of such an element.Finally, let us show injectivity of the map α . Let R n be the subgroup of Gr( M ∂n )generated by the SK-relation [ M ∪ φ ¯ M ] − [ M ∪ ψ ¯ M ], so that SK n = Gr( M ∂n ) /R n .Note that the set of elements that generate this relation is closed under summation, (cid:0) [ M ∪ φ ¯ M ] − [ M ∪ ψ ¯ M ] (cid:1) + (cid:0) [ M ′ ∪ φ ′ ¯ M ′ ] − [ M ′ ∪ ψ ′ ¯ M ′ ] (cid:1) = [( M ⊔ M ′ ) ∪ φ ⊔ φ ′ ( ¯ M ⊔ ¯ M ′ )] − [( M ⊔ M ′ ) ∪ ψ ⊔ ψ ′ ( ¯ M ⊔ ¯ M ′ )] . Thus R n is precisely the set of elements of this form, and similarly for the subgroup R ∂n of Gr( M ∂n ), which generates the SK-relation for manifolds with boundary. Then UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 9 it is clear that R ∂n ∩ Gr( M ∂n ) = R n , and injectivity of α follows. (cid:3) ≃ ∈ SK ∂n + + N DM N ∪ φ ¯ M = L ∂N × I ∪ φ M = M ≃ ∈ SK ∂n − − M N DM L − ∈ Im α M N
Figure 4.
Example of an element in Im( α : SK n → SK ∂n )2.2. SKK -groups for manifolds with boundary. One can define a more refinedrelation than that of cutting and pasting called SKK (“scheiden und kleben, kontrol-lierbar”=“controllable cutting and pasting”) in which we keep track of the gluingdiffeomorphisms. The resulting SKK n -groups obtained by modding out by the SKK-equivalence relation have been interpreted as Reinhardt vector field bordism groups[KKNO73], which have also been shown to arise as π of the Madsen-Tillman spectra M T SO ( n ) [Ebe13].In this subsection we review the definition of the group SKK n and we define aversion for manifolds with boundary, which we fit into an exact sequence with theclassical SKK-group for closed manifolds. We conjecture that the SKK-group formanifolds with boundary that we define arises as π of the scissors congruence K -theory spectrum we define in Section 3. We will investigate this connection in futurework. Again, let M n be the monoid of diffeomorphism classes of smooth closed oriented n -dimensional manifolds [ M ] under disjoint union. An SKK-invariant is an abelianvalued monoid map λ from M n for which the difference λ ( M ∪ φ ¯ M ′ ) − λ ( M ∪ ψ ¯ M ′ )only depends on the orientation preserving diffeomorphisms φ, ψ : ∂M → ∂M ′ , andnot on the manifolds M and M ′ . Clearly the SK-invariants are those SKK-invariantsfor which this difference is 0. The SKK n -group from [KKNO73] is defined to satisfythe universal property that any SKK-invariant factors through it. Definition 2.9.
The controllable scissors congruence group
SKK n for smooth closedoriented n -dimensional manifolds is the quotient of the Grothendieck group Gr( M n )by the relation[ M ∪ φ ¯ M ′ ] − [ M ∪ ψ ¯ M ′ ] = [ M ∪ φ ¯ M ′ ] − [ M ∪ ψ ¯ M ′ ]for compact oriented manifolds M , M ′ and M , M ′ such that ∂M = ∂M and ∂M ′ = ∂M ′ , and orientation preserving diffeomorphisms φ, ψ : ∂M → ∂M ′ . Example 2.10.
Figure 5 provides an example of an SKK-relation. φ ↓ ↓ ψ φ ↓ ↓ ψ Figure 5.
Example of an SKK-relationWe introduce a definition of SKK n -groups for n -dimensional smooth compactoriented manifolds with boundary analogously to our definition of SK ∂n , and we showwe can measure the difference to the classical definition for closed manifolds givenabove via an exact sequence. As above, we let M ∂n be the monoid of diffeomorphismclasses [ M ] of smooth compact oriented n -dimensional manifolds with boundaryunder disjoint union. Definition 2.11.
The controllable scissors congruence group
SKK ∂n for smooth com-pact oriented n -dimensional manifolds is the quotient of the Grothendieck groupGr( M ∂n ) by the relation[ M ∪ φ ¯ M ′ ] − [ M ∪ ψ ¯ M ′ ] = [ M ∪ φ ¯ M ′ ] − [ M ∪ ψ ¯ M ′ ]for compact oriented manifolds M , M ′ , M , M ′ , and Σ ⊆ ∂M i and Σ ′ ⊆ ∂M ′ i closedsubmanifolds for i = 1 ,
2, and φ, ψ : Σ → Σ ′ orientation preserving diffeomorphisms. Theorem 2.12.
For every n ≥ the following sequence is exact UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 11 −−−−→ SKK n α −−−−−−→ [ M ] [ M ] SKK ∂n β −−−−−−→ [ N ] [ ∂N ] C n − −−−−→ . Proof.
This proof will be a more elaborate version of the proof of Theorem 2.8. Itfollows as before that the maps α, β are well-defined.We show exactness at the middle term. It is clear from the definition that Im α ⊆ ker β. Let us show the inverse inclusion. Let x ∈ ker β. Every element of SKK ∂n canbe written in the form x = [ M ] − [ N ] , where M, N are compact smooth oriented n -manifolds (not necessarily connected). As before, since β ( x ) = 0 we have that ∂M and ∂N are diffeomorphic. Hence we may replace M by a diffeomorphic manifoldwith boundary ∂N ; we will still denote this replacement by M. Let ¯ M be the copy of M with the opposite orientation and let DM be the doubleof M, i.e. DM = M ∪ id ¯ M .
The same way we define ¯ N and DN . Using the factthat ∂M ∼ = ∂N we get that the following equalities hold in the group SKK ∂n : (cid:16) [ M ] + [ DN ] (cid:17) − (cid:16) [ M ∪ id ¯ N ] + [ N ] (cid:17) = (cid:16) [ M ∪ id ( ∂M × [0 , N ∪ id ¯ N ] (cid:17) − (cid:16) [ M ∪ id ¯ N ] + [ N ∪ id ( ∂N × [0 , (cid:17) = (cid:16) [ M ∪ id ¯ M ] + [ N ∪ id ¯ N ] (cid:17) − (cid:16) [ M ∪ id ¯ N ] + [ N ∪ id ¯ M ] (cid:17) = (cid:16) [ DM ] + [ DN ] (cid:17) − (cid:16) [ M ∪ id ¯ N ] + [ N ∪ id ¯ M ] (cid:17) , where the middle equality follows from the equivalence relations applied to M = M ⊔ N, M ′ = ( ∂M × [0 , ⊔ ¯ N , and M = M ⊔ N, M ′ = ¯ M ⊔ ¯ N .
Therefore, [ M ] − [ N ] = [ DM ] − [ N ∪ id ¯ M ] ∈ Im α. The injectivity of the map α can be shown using the same argument as the one usedfor the injectivity statement in Theorem 2.8, since again the sets of defining relationsfor SKK n and SKK ∂n are subgroups of the corresponding Grothendieck groups. (cid:3) K -theory of categories with squares Overview of Campbell and Zakharevich’s square K -theory. This sub-section is an exposition of the definitions and results that we need from [CZ19b].
Definition 3.1.
A category with squares is a category C equipped with a choice ofbasepoint object O , two subcategories c C and f C of morphisms referred to as cofi-brations (denoted ) and cofiber maps (denoted ), and distinguished squares A B (cid:3)
C D satisfying the following conditions:1) C has coproducts and distinguished squares are closed under coproducts.2) Distinguished squares are commutative squares in C and compose horizontallyand vertically.3) Both c C and f C contain all isomorphisms of C .4) If a commutative square satisfies the property that either both horizontal mapsor both vertical maps are isomorphisms, then the square is distinguished.Campbell and Zakharevich developed the framework of categories with squaresin order to describe a generalized construction of K-theory spectra, inspired by theWaldhausen construction.We review their construction of K -theory for a category with squares from[CZ19b]. Let [ k ] denote the category 0 → → · · · → k . Definition 3.2.
Let C be a category with squares. Define C ( k ) to be the subcategoryof Fun([ k ] , C ) whose objects are sequences of cofibration maps C C · · · C k , and whose morphisms are natural transformations in which every commutativesquare is distinguished.Varying over k by composing cofibrations and distinguished squares, we get asimplicial category, denoted C • . The squares K -theory of C is defined, analogouslyto the definition for Waldhausen categories, as follows: Definition 3.3.
Let C be a category with squares. The squares K-theory space of C is K (cid:3) ( C ) ≃ Ω O | N q C • | where Ω O is the based loop space, based at the distinguished object O ∈ N C (0) .Campbell and Zakharevich prove that this K -theory space is an infinite loopspace using a form of the additivity theorem for categories with squares. By abuseof notation, we will refer to the resulting K -theory spectrum also as K (cid:3) ( C ). A mapof categories of squares, which is a functor that preserves distinguished basepointobjects and distinguished squares, induces a map of K -theory spectra. UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 13 Theorem 3.4 ([CZ19b]) . Let C be a category with squares. The space K (cid:3) ( C ) is aninfinite loop space. They also give an explicit description of the K -group for certain categories withsquares, which we record here. Lemma 3.5 ([CZ19b]) . Let C be a category with squares with basepoint O satisfying:(1) O is initial or terminal in c C .(2) O is initial or terminal in f C .(3) For all objects A, B ∈ C , there exists some object X ∈ C and distinguishedsquares: O A (cid:3)
B X O B (cid:3)
A X
Then K (cid:3) ( C ) ∼ = Z { ob C} / ∼ where ∼ is the equivalence relation generated by(1) [ O ] = 0 (2) [ A ] + [ D ] = [ B ] + [ C ] for every distinguished square A B (cid:3)
C D . Category with squares from a Waldhausen category.
Campbell and Za-kharevich prove that square K -theory is indeed a good generalization of the Wald-hausen construction, in the sense that given a Waldhausen category C one can asso-ciate to it a category with squares such that the Waldhausen and square K -theoriesagree. For our purposes in Section 5, we need to associate a slightly different cate-gory with squares to a Waldhausen category than that defined in [CZ19b]. We willshow that the Waldhausen K -theory and squares K -theory are also compatible inthis case; the proof is completely analogous to the proof given in [CZ19b], but weinclude the version for our particular case here for completeness. We comment onour choices in Remark 5.3 below. Definition 3.6.
Let C be a Waldhausen category with weak equivalences. Definean associated category with squares C (cid:3) in the following way. The horizontal mapsare the cofibrations in C , and the vertical maps are all maps. The distinguished squares are the squares A B (cid:3)
C D with the property that the unique map C ∪ A B ≃ −→ D is a weak equivalence. Thedistinguished basepoint object is the zero object. Proposition 3.7.
The category C (cid:3) satisfies the axioms of a category with squaresfrom Definition 3.1.Proof. We check the four axioms. For (1), C has coproducts because it is a Wald-hausen category. Suppose that C ∪ A B ≃ −→ D and C ′ ∪ A ′ B ′ ≃ −→ D ′ . Note that since pushouts and coproducts commute with each other, and since C ∪ A B ⊔ C ′ ∪ A ′ B ′ ≃ −→ D ⊔ D ′ by the gluing axiom, distinguished squares are closed under coproducs.To check axiom (2), suppose we compose two distinguished squares horizontally A B E (cid:3) (cid:3)
C D F
We have a chain of weak equivalences C ∪ A E ∼ = ( C ∪ A B ) ∪ B E ≃ −→ D ∪ B E ≃ −→ F, where the first weak equivalence is by the gluing axiom.Now suppose we compose two distinguished squares vertically A B (cid:3)
C D (cid:3)
E F
Similarly, we have E ∪ A B ∼ = E ∪ C ( C ∪ A B ) ≃ −→ E ∪ C D ≃ −→ F, where again the first weak equivalence is by the gluing axiom. UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 15 Axiom (3) is immediate since the isomorphisms are contained in the cofibrationsin a Waldhausen category, and we don’t have any restrictions on the vertical maps.To check axiom (4), suppose first that the two vertical morphisms in a commutingsquare in C A B (cid:3)
C D are isomorphisms. Then C ∪ A B ∼ = B ∼ = D and the square is a pushout square.Similarly, if the horizontal maps are isomorphisms, C ∪ A B ∼ = C ∼ = D , and again thesquare is a pushout. (cid:3) Proposition 3.8.
The Waldhausen K -theory K Wald ( C ) agrees with the K -theory K (cid:3) ( C (cid:3) ) of the associated category with squares from Definition 3.6.Proof. By definition, K (cid:3) ( C (cid:3) ) is the realization of the bisimplicial set with ( p, q )-simplexes given by A A · · · A p (cid:3) (cid:3) (cid:3) A A · · · A p (cid:3) (cid:3) (cid:3) ... ... . . . ... (cid:3) (cid:3) (cid:3) A q A q · · · A qp in which each square is distinguished. Thus it is the nerve of the category whoseobjects are sequences of cofibrations A A · · · A n and morphisms maps of such diagrams that satisfy the condition that for every i ≤ j the induced map A ′ i ∪ A i A j → A ′ j is a weak equivalence.Thus the above is precisely the bisimplicial set obtained by applying the nerve toThomason’s simplicial category wT q C defined in [Wal87, page 334]. By Thomason-Waldhausen, there is a zig-zag of equivalences via some intermedi-ate construction wT q C wT + q C ≃ o o ≃ / / wS q C . Therefore, via a zig-zag, we have an equivalence of K -theory spectra K (cid:3) ( C (cid:3) ) ≃ K Wald ( C ) . (cid:3) Remark 3.9.
The category with squares associated to the Waldhausen category C in Definition 3.6 is different from the category with squares associated to C in[CZ19b, Example 1.2.]. However, they have equivalent K -theories since they areboth equivalent to the usual Waldhausen K -theory K Wald ( C ). For the categorywith squares from [CZ19b, Example 1.2.], this is proved directly in [CZ19b, Lemma1.5.]. 4. K -theory of manifolds with boundary In this section we use the framework described in Section 3 in order to definea K -theory spectrum for the category of n -dimensional compact smooth manifoldswith boundary, which recovers as π the scissors congruence group SK ∂n .4.1. The category with squares for manifolds with boundary.
We start bydefining a category with squares structure on the category Mfd ∂n of smooth compact n -dimensional manifolds with boundary and smooth maps. Definition 4.1.
Let Mfd ∂n be the category of smooth compact n -dimensional man-ifolds with boundary and smooth maps. We define the subcategories c Mfd ∂n ofhorizontal maps (denoted ) and and f Mfd ∂n of vertical maps (denoted ֒ → ) to bothbe given by the morphisms in Mfd ∂n which are smooth embeddings of manifoldswith boundary f : M → N such that ∂M is mapped to a submanifold with trivialnormal bundle, and such that each connected component of the boundary ∂M iseither mapped entirely onto a boundary component or entirely into the interior of N . We define distinguished squares to be those commutative squares in Mfd ∂n N MM ′ M ∪ N M ′ . that are pushout squares, i.e. such that M ∪ N M ′ is a smooth manifold. The chosenbasepoint object is the empty manifold. Example 4.2.
Figure 6 gives pictorial examples of distinguished squares in Mfd ∂n . UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 17 (cid:3) (cid:3) Figure 6.
Two examples of distinguished squares
Lemma 4.3.
The category
Mfd ∂n with the structure from Definition 4.1 satisfies theaxioms of a category with squares from Definition 3.1.Proof. The coproduct in Mfd ∂n is given by disjoint union of manifolds, and thecollection of distinguished squares is closed under disjoint union. Pushout squaresare commutative and compose horizontically and vertically. Consider the diagram A BC D R. ij j ′ fi ′ f ′ l If j ′ is an isomorphism then we can define the map l uniquely as f j ′− ; similarlyif i ′ is an isomorphism. Therefore in both cases this is a pushout diagram. HenceMfd ∂n satisfies the definition of a category with squares. (cid:3) The computation of K ( Mfd ∂n ) . Using Lemma 3.5 for the category withsquares Mfd ∂n defined above, we show that the K (cid:3) -group agrees with the SK ∂n -group. Theorem 4.4.
For the manifold category with squares
Mfd ∂n from Definition 4.1, K (cid:3) (Mfd ∂n ) ∼ = SK ∂n . Proof.
The empty set is initial in both c Mfd ∂n and f Mfd ∂n . Moreover, for all objects M and N in Mfd ∂n , there exist pushout squares ∅ N (cid:3) M M ⊔ N ∅ M (cid:3) N M ⊔ N. Therefore Mfd ∂n satisfies the conditions of Lemma 3.5, which gives a description ofthe relations of the left hand side.First, assume that the relations from K (cid:3) hold. To show that these imply therelations in SK ∂n , we first need to check that the generating objects are compatible(note that SK ∂n is generated by diffeomorphism classes of manifolds, whereas K apriori is generated by manifolds). Consider a diffeomorphism M φ −→ M ′ . Then ∅ M (cid:3) ∅ M ′ φ is a distinguished square; and so the relations in K (cid:3) give that:[ M ] + [ ∅ ] = [ M ′ ] + [ ∅ ][ M ] = [ M ′ ]Next, consider the square ∅ M (cid:3) M ′ M ⊔ M ′ . This is a distinguished square, which means that[ M ] + [ M ′ ] = [ M ⊔ M ′ ] + [ ∅ ]= [ M ⊔ M ′ ] . For the other relation in SK ∂n , consider compact oriented manifolds M, M ′ , closedsubmanifolds Σ ⊆ ∂M and Σ ′ ⊆ ∂M ′ , and orientation preserving diffeomorphisms φ, ψ : Σ → Σ ′ . We want to show that[ M ∪ φ M ′ ] = [ M ∪ ψ M ′ ] . UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 19 Consider (Σ × ǫ ) where ǫ = [0 , ε ] for some small ε ≥
0. We can extend the maps φ, ψ by the identity to maps ˜ φ, ˜ ψ from (Σ × ǫ ) to (Σ ′ × ǫ ), which we consider inside M and M ′ respectively as collars of the boundary components. This is possible asthe boundary has trivial normal bundle. We have that M ∪ φ M ′ is diffeomorphic to M ∪ ˜ φ M ′ . Using the maps φ, ψ , consider the squares(Σ × ǫ ) M (cid:3) M ′ M ∪ φ M ′ ˜ φ (Σ × ǫ ) M (cid:3) M ′ M ∪ ψ M ′ . ˜ ψ The relation given by distinguished squares implies:[ M ∪ φ M ′ ] + [(Σ × ǫ )] = [ M ] + [ M ′ ]= [ M ∪ ψ M ′ ] + [(Σ × ǫ )]Thus, [ M ∪ φ M ′ ] = [ M ∪ ψ M ′ ].In the other direction, assume that the relations for SK ∂n hold. Consider relation(1) in Definition 2.6 applied to the following:[ ∅ ⊔ ∅ ] = [ ∅ ] + [ ∅ ][ ∅ ] = [ ∅ ]Thus, for ∅ , the initial object in our category with squares, we have [ ∅ ] = 0.Finally, for relation (2) of Definition 2.6, suppose the following is a distinguishedsquare: A B (cid:3)
C D
Define N := A ∩ cl ( B − A ) ⊆ ∂A , where cl ( B − A ) is the closure of the complementof A in B , i.e. N is the part of the boundary of A that is mapped to the interior of B . We define M := cl ( B − A ) ⊔ ( N × ǫ ) ,M ′ := A ⊔ C. Let id : N ⊔ N → N ⊔ N be the identity map; let τ : N ⊔ N → N ⊔ N be the twistmap. Note that M ∪ id M ′ ∼ = B ⊔ C and M ∪ τ M ′ ∼ = A ⊔ D . Then the fact that[ M ∪ id M ′ ] = [ M ∪ τ M ′ ] gives the relations[ B ⊔ C ] = [ A ⊔ D ][ B ] + [ C ] = [ A ] + [ D ] . (cid:3) The derived Euler characteristic for manifolds with boundary
The Euler characteristic map χ : M ∂n → Z from the monoid of diffeomorphismclasses of smooth compact manifolds is an SK-invariant, since χ ( M ∪ Σ N ) = χ ( M )+ χ ( N ) − χ (Σ); thus it factors through SK ∂n . We show that the Euler characteristicmap χ : SK ∂n → Z lifts to a map of spectra. The strategy will be to construct a mapof categories with squares from the category of smooth compact oriented manifoldswith boundary to the category with squares from Definition 3.6 associated to theWaldhausen category of perfect Z -chain complexes. The main theorem we prove inthis section is the following. Theorem 5.1.
There is a map of K -theory spectra K (cid:3) (Mfd ∂ ) → K ( Z ) , which on π agrees with the Euler characteristic for smooth compact manifolds withboundary. We first prove the propositions we need in the next section and give the proof ofthe theorem at the end of the final section.5.1.
The lift of the singular chain functor.
Let Ch perf Z be the Waldhausencategory of perfect chain complexes, i.e., those complexes that are quasi-isomorphicto a bounded finitely generated Z -complex,with cofibrations given by levelwise injective maps and weak equivalences givenby quasi-isomorphisms. Consider the associated category with squares (Ch perf Z ) (cid:3) asdefined in Definition 3.6.Consider the singular chain functor S : Mfd ∂n → Ch perf Z which sends a compact manifold with boundary to its singular chain complex. Thehomology of this complex is finitely generated in each degree and bounded since ourmanifolds are compact. Proposition 5.2.
The map S is a map of categories with squares S : Mfd ∂n → (Ch perf Z ) (cid:3) UT AND PASTE INVARIANTS OF MANIFOLDS VIA ALGEBRAIC K -THEORY 21 Proof.
Suppose we have a distinguished square
A B (cid:3)
C D in Mfd ∂ , and we apply S to it. In the resulting square in Ch perf Z , the horizontal mapsare levelwise injective, as required. So in order to show that it is a distinguishedsquare, it remains to show that the map S ( A ) ∪ S ( A ) S ( B ) → S ( D )is a quasisomorphism.Note that by our construction of distinguished squares in Mfd ∂ the union of theinteriors of B and C covers D. Let S n ( B + C ) be the subgroup of S n ( D ) consistingof n -chains that are sums of n -chains in B and n -chains in C . By the standardMayer-Vietoris argument, the following sequence is exact0 −−−−→ S n ( A ) −−−−−−→ x ( x, − x ) S n ( B ) ⊕ S n ( C ) −−−−−−−→ ( y,z ) y + z S n ( B + C ) −−−−→ . Hence the chain complex S ∗ ( B + C ) is a pushout S ∗ ( B ) ∪ S ∗ ( A ) S ∗ ( C ). On theother hand by [Hat02, Proposition 2.21], the inclusions S n ( B + C ) → S n ( D ) induceisomorphisms on homology groups, which finishes the proof. (cid:3) Remark 5.3.
The reason for the choices in our Definition 3.6 of a category withsquares associated to a Waldhausen category is precisely to make the above propo-sition work. The difference between the category with squares in Definition 3.6 andthat in [CZ19b, Example 1.2] is that we allow the Waldhausen category C to haveweak equivalences and not only isomorphisms, so we can apply it to the categoryof chain complexes, and we allow all maps as vertical maps as opposed to only thecofiber maps. This more relaxed definition of the distinguished squares is crucial inallowing us to show that distinguished squares in the category of manifolds map todistinguished squares in the category of chain complexes.5.2. Recovering the Euler characteristic on π . Lastly, we claim that K (Ch perf Z ) ≃ K ( Z ) via an isomorphism under which S ( M ) corresponds to χ ( M )on π for a smooth compact oriented manifold M .Denote by Ch b Z the category of bounded finitely generated Z -modules, so theperfect chain complexes are those that are quasi-isomorphic to complexes in Ch b Z .Clearly, Ch b Z ⊆ Ch perf Z and moreover by the discussion in [Wei13, V. 2.7.2] (or alternatively directly by the Waldhausen approximation theorem) this inclusion in-duces an isomorphism on K -groups. A similar argument for cohomology appears in[CWZ19, Lemma 2.8]. Proposition 5.4.
The map q : K (Ch perf Z ) → K (Ch b Z ) sending a perfect chain com-plex C ∗ to the class of the corresponding quasi-isomorphic chain complex H ( C ∗ ) ∈ Ch b Z is well-defined and is an isomorphism.Proof. The map is well-defined since quasi-isomorphic chain complexes have isomor-phic homology, and itis surjective because of the inclusion Ch b Z ⊆ Ch perf Z . On the other hand if Y = q ( X )vanishes in K (Ch b Z ) then we may identify X with Y in K (Ch perf Z ) and it will alsovanish there, because the set of defining relations (which we quotient out in thepresentation for K ) of K (Ch perf Z ) contains the defining relations of K (Ch b Z ) . (cid:3) Now, recall that the map φ : K (Ch b Z ) → K ( Z ) given by [ C ∗ ] χ ( C ∗ ) = P i ( − i [ C i ] is an isomorphism [Wei13, Proposition II.6.6.]. By an easy exer-cise using the additivity property, the Euler characteristic of a bounded complexonly depends on its homology and χ ( C ∗ ) = P i ( − i [ H i ( C ∗ )] . Thus the com-position q ◦ φ : K (Ch perf Z ) → K ( Z ) is also an isomorphism and maps [ C ∗ ] to χ ( C ∗ ) = P i ( − i [ H i ( C ∗ )] . Proof of Theorem 5.1.
From Proposition 5.2, the singular chain functor S : Mfd ∂n → (Ch perf Z ) (cid:3) is a map of categories with squares when the right hand side is given thestructure of a category with squares from Definition 3.6. This then induces a mapon K -theory spectra K (cid:3) (Mfd ∂ ) → K (cid:3) (cid:0) (Ch perf Z ) (cid:3) (cid:1) . By Proposition 3.8 the target is K ( Z ). By Proposition 5.4 and the discussion fol-lowing it, the map on π agrees with the Euler characteristic. (cid:3) References [Cam19] Jonathan A. Campbell,
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