aa r X i v : . [ m a t h . K T ] J a n Comparison of Higher Smooth Torsion
Christopher OhrtJanuary 10, 2019
Abstract
By explicitly comparing constructions, we prove that the higher torsioninvariants of smooth bundles defined by Igusa and Klein [8] via Morse the-ory agree with the higher torsion invariants defined by Badzioch, Dorabiala,Dwyer, Weiss, and Williams using homotopy theoretical methods ([2] and [6]).
Contents S -bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 The Whitehead Category . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 The Homotopy Type of F C ( E, ξ F ) and the category Q ( E ) . . . . . . 14 Kx E ( X ∆ • ) . . . . . . . . . . . . . . . . . . . 235.2 The Homotopy Type of K h E ( X ∆ • ) . . . . . . . . . . . . . . . . . . . 28 Higher torsion aims to provide an invariant classifying differential structures thatcan be put on a smooth bundle
F ֒ → E ։ B. Many different approaches have been1aken to define higher torsion invariants ([8], [10], [4], [6], [3]). In general a torsioninvariant will take as argument the bundle E and a finite, fiber-wise local system F → E (where the action of π B on H ∗ ( F ; F ) is sufficiently trivial) and producea cohomology class τ ( E ; F ) ∈ H k ( B ; R ) . To compare different torsion invariants,Igusa developed a system of axioms for such objects and used it to classify highertorsion in the case where F is trivial [9]. The author expanded this to the moregeneral “twisted” case of arbitrary (finite) F and finite π B [12]. Based on theseaxioms many comparisons were made [1], [7].Two particular constructions of higher torsion invariants were given by Igusaand Klein [8] and Badzioch, Dorabiala, Dwyer, Weiss, and Williams [2]. Given aparametrized, generalized Morse function f : E → R the former construct theirinvariant τ IK by carefully analyzing the evolution of critical Morse points in thefiber F x as x ∈ B varies and explicitly give a map in K-Theory along which theypull back the Borel regulator [5] to get the homology class τ IK ( E ; F ) ∈ H k ( B ; R ) . In contrast, the latter use homotopy theory to find a lift of the Becker-Gottliebtransfer p ! : B → Ω ∞ Σ ∞ ( E + ) into the fiber of the compositionΩ ∞ Σ ∞ ( E + ) assembly −−−−−→ A ( E ) linearization F −−−−−−−−→ K ( C )and also pull back the Borel regulator to define their smooth torsion homology class τ sm ( E ; F ) ∈ H k ( B ; R ) . Originally this was done for trivial local systems and thenthe author extended it to finite local systems [13].The explicit nature of Igusa-Klein torsion opens it up for calculations on S -bundles and many more calculations can be accessed via the axioms [8]. The naturaldefinition of smooth torsions makes this a very intuitive and universal tool, but isprohibitive to calculations. To the author’s knowledge, there are currently no non-trivial results. Using the axioms, Badzioch, Dorabiala, Klein, and Williams showedthat smooth torsion is a multiple of Igusa-Klein torsion related by a non-zero factor ifrestricted to trivial local systems. The exact value of said factor was not determined[1]. In [13] the author shows that for general (finite) local systems, smooth torsionstill satisfies almost all axioms, with the obstruction being the continuity axiomwhich requires an explicit calculation of smooth torsion for S -bundles.In this paper we will prove directly: Theorem 1.1.
Igusa-Klein and smooth torsion agree, whenever they are defined: τ IK ( E ; F ) = τ sm ( E ; F ) . As an immediate consequence we get many specific values for smooth torsionfrom the calculation of Igusa-Klein torsion [8] such as
Corollary 1.2.
For a linear S -bundle S ( ξ ) → B associated to the complex linebundle ξ → B with local system F ζ given by a root of unity ζ ∈ C ∗ we have τ smk ( S ( ξ ); F ζ ) = − n k L k +1 ( ζ ) ch k ( ξ ) . The function L k +1 is the real polylogarithm defined by L k +1 ( z ) = Re (cid:18) i k L k +1 ( z ) (cid:19) , here L k +1 is the complex polylogarithm L k +1 ( z ) = ∞ X m =1 z m m k +1 . Consequently smooth torsion satisfies the continuity axiom.
Besides this we can extend the definition of Igusa-Klein torsion: The originalconstruction only works if the fundamental group π B acts trivially on H ∗ ( F ; F ) , whereas for smooth torsion it is enough to say that said action is unipotent, i.e.there is a filtration 0 = V ⊂ V ⊂ . . . ⊂ V n = H ∗ ( F ; F ) where π B acts trivially onthe quotients V i +1 /V i . Now we can define τ IK := τ sm if the action is unipotent andsince τ sm satisfies all axioms this behaves naturally. In particular if π B is finite,higher torsion is defined for any finite local system F . Goodwillie and Igusa haverecently announced to be able to make this extension of the definition explicitly aswell.Our strategy for the proof is as follows: Let
W h F ( E ) be the fiber of the compo-sition Ω ∞ Σ ∞ ( E + ) → A ( E ) F −→ K ( C ) mentioned above. Then both the Igusa-Kleintorsion and the smooth torsion are defined by pulling back a certain homology classalong a map τ IK : B → W h F ( E ) and τ sm : B → W h F ( E ) respectively. Bothconstructions use very different models for the involved spaces, however. Inspiredby the unpublished manuscript [11], we give an explicit unifying model and use itto compare the two torsion maps. Outline:
In sections 2 and 3 we will recall the definitions of smooth and Igusa-Klein torsion respectively. In Section 4 we will provide the unifying model andcompare the torsions.
Acknoledgements:
The author wants to thank Kiyoshi Igusa for several veryhelpful conversations and for pointing towards the expansion categories, whichproved be the key to the comparison result.
This section repeats the constructions made in the beginning of [2] and [13]. Let X be a compact manifold. We will define a model for Ω ∞ Σ ∞ X + which (by abusingnotation a bit) we will call Q ( X + ) . It will be constructed as the direct limit understabilization of the Waldhausen K-theory spaces of certain categories of partitions.We will refrain from giving details as they can be found in [2].A partition of X × I is a (not necessarily smooth and possibly with corners)codimension 0 submanifold M ⊂ X × I that represents the lower half of a divisionof the interval in two parts and is somewhat standard around the boundary and onthe lower third. Part of the data is also a vector field transversal to the boundarywhich can be used to smoothen the partition. The set P k ( X ) consists of partitionsof X × I parametrized over ∆ k . These fit together in a simplicial set P • ( X ) . Thereis a stabilization map P • ( X ) → P • ( X × I ) defined by putting the non-trivial part of3 partition of X × I into the middle third (of the second interval) to get a partitionof ( X × I ) × I. We note that there is a partial monoid structure on P • ( X ) where weadd two partitions of X × I if they do not share any non-trivial parts. Stabilizationnow provides a monoid structure on colim n P • ( X × I n ) . The sets P k ( X ) can also be viewed as partially ordered sets by inclusion, andhence as categories. So we can apply the Waldhausen S • -construction (or rather theThomason variant thereof) [17] to get bisimplicial categories S • P • ( X ) . Recall thatthe objects of the category S n P ( X ) are ( n + 1)-tuples of partitions ( M i ) ni =0 with M ⊂ M ⊂ . . . ⊂ M n , together with identifications of any subquotients. Here M is required to be theinitial partition X × [0 , ] . Note that the space |S • P • ( X ) | is endowed with a canonicalbase point.By stabilization we get a space Q ( X + ) := Ωhocolim n |S • P • ( X × I n ) | ≃ Ω ∞ Σ ∞ X + . The weak equivalence on the right is rather intricate and was shown by Waldhausenin [16] and [18].Recall that the algebraic K-Theory of the space X is defined as A ( X ) = Ω |S • R hf ( X ) | where R hf ( X ) is the Waldhausen category of homotopy finite retractive spaces over X [17]. By “thickening up” this model for the algebraic K-theory of spaces one candefine a map α : Q ( X + ) → A ( X )that roughly takes a partition over X and views it as a retractive space over X. Thismap is a model for the assembly. See [2] for details.
Remark . Since the assembly map Q ( X + ) → A ( X ) has a homotopy left-inverse[16], we won’t need to fully understand this map, but merely know that it exists.For details compare the proof of proposition 2.4. Remark . We will often use the simplicially enriched model A ( X ∆ • ) for A ( X ) . The objects of R hf ( X ∆ n ) are ∆ n -families of retractive spaces over X, which can alsobe viewed as retractive spaces Y over X × ∆ n together with a projection Y → ∆ n fitting in the following commutative diagram Y ❇❇❇❇❇❇❇❇ / / X × ∆ n pr z z ttttttttt o o ∆ n We have A ( X ) ≃ A ( X ∆ • ) given by the inclusion of zero simplices. This simplicialenrichment is similar to the one used in the definition of Q ( X + ) , so we can view theassembly map as α : Q ( X + ) → A ( X ∆ • ) . emark . The following helps greatly in defining maps into Q ( X + ) (and A ( X )).Recall that there is a natural map |S W| × ∆ → |S • W| for any Waldhausen category W given by the inclusion of the 1-skeleton in the S • direction [17]. After taking the adjoint this gives a map |S W| → K W . Hence it is always enough to define a functor
C → W ∼ = S W to get a map |C| → K W for any small category C . Let S • ( B ) be the simplicial category of simplices σ : ∆ • → B with no non-trivialmorphisms. Clearly, we have | S • ( B ) | ≃ B. Let E → B be a smooth bundle. There is a transfer map p ! A : | S • ( B ) | → A ( E ∆ • )given by the functor that sends a simplex σ : ∆ n → B to the retractive space E × ∆ n ⊔ σ ∗ E ⇆ E × ∆ n . One can explicitly construct a lift p ! : | S • ( B ) | → Q ( E + ) such that α ◦ p ! ∼ p ! A are homotopic where α : Q ( X + ) → A ( E ∆ • ) [2]. Proposition 2.4.
The map p ! : | S • ( B ) | → Q ( E + ) has the homotopy type of theGottlieb-Becker transfer p BG : B → Ω ∞ Σ ∞ X + . Proof.
We adopt the proof from [2]. Let tr : A ( X ) → Ω ∞ Σ ∞ X + be Waldhausen’strace map [16], a right inverse to α with α ◦ tr ∼ id Q ( E + ) . It is know that thecomposition tr ◦ p ! A ∼ p BG . So we have p BG ∼ tr ◦ p ! A ∼ tr ◦ α ◦ p ! ∼ p ! . Remark . Because of the existence of the Waldhausen trace map as in the proofabove, we do not need to explicitly understand the transfer map p ! but rather only p ! A . We still follow [2] closely to define linearization maps. Let R be a ring and let Ch hf ( R ) be the Waldhausen category of homotopy finitely dominated chain com-plexes of projective R -modules. Recall that the Waldhausen K-theory of this cate-gory is just a model for the algebraic K-theory K ( R ) of R [17].5ow let X be a compact manifold and F a local system of R -modules on X. Then we get a functor R hf ( X ) → Ch hf ( R )by sending a retractive space X → Y → X to the relative singular chain complex C ∗ ( Y, X ; F ) . This induces a linearization map λ RF : A ( X ) → K ( R )and if we compose with the assembly α : Q ( X + ) → A ( X ) we get a map λ F : Q ( X + ) → K ( R ) . Let E → B be a bundle of compact manifolds and let F be a local system of R -modules on E. Similarly to before we can define a functor S • ( B ) → wCh hf ( R )(the w indicates that we are only looking at quasi-isomorphisms as morphisms.)In particular, this functor sends a simplex σ : ∆ k → B to the chain complex C ∗ ( σ ∗ E, F ) . Using Remark 2.3 this gives rise to a map c F : | S • ( B ) | → K ( R ) . Theorem 2.6.
Let E → B be a bundle of compact manifolds, R a ring, and F alocal system of R -modules on E. Then there is a preferred homotopy which makesthe following diagram commute: Q ( E + ) λ F (cid:15) (cid:15) | S • ( B ) | p ! sssssssss c F / / K ( R ) The homotopy is induced by the isomorphism H ∗ ( σ ∗ E ; F ) ∼ = H ∗ ( E ⊔ σ ∗ E, E ; F ) . Proof.
The composition λ F ◦ p ! ∼ λ RF ◦ p ! A sends the simplex σ : ∆ n → B to thechain complex C ∗ ( E ⊔ σ E , E ; F )which is homotopy equivalent to C ∗ ( σ ∗ E ; F )which is the image of σ under c F . This will be the starting point for us to define smooth parametrized torsion.6 .4 Unreduced and reduced smooth parametrized torsion
This section is where we depart slightly from [2] in that our results will be a littlebit more general than there. This also appears in [13]. The idea here is that if wecan show that the map c F : | S • ( B ) | → K ( R ) is homotopic to the constant map withvalue the 0 complex 0 ∈ K ( R ) then we get a lift | S • ( B ) | → hofib (cid:16) Q ( E + ) λ F −→ K ( R ) (cid:17) =: W h F ( E )where we call the codomain the Whitehead space of E. This will not always be thecase, but the following condition is almost sufficient:
Definition 2.7.
Let E → B be as before and let F be a complex local system on E. Let B be connected, b ∈ B be the basepoint, and let F be the fiber over b . We say π B acts unipotently on H ∗ ( F ; F ) if there exists a filtration of H ∗ ( F ; F ) by π B submodules 0 = V ( F ) ⊂ . . . ⊂ V k ( F ) = H ∗ ( F ; F )such that π B acts trivially on the quotients V i ( F ) /V i − ( F ) . Theorem 2.8.
Let E → B be a bundle, B path-connected, b ∈ B the basepoint, F b the fiber over the basepoint, F → E a complex local system such that π B acts unipotently on H ∗ ( F, F ) . Then there exists a preferred homotopy from the map c F : | S • ( B ) | → K ( C ) to the constant map with value the complex H ∗ ( F b , F ) ∈ K ( C ) (with trivial boundary maps).Proof. This can be found in [13] or adapted from [2].
Definition 2.9.
Let p : E → B be a compact manifold bundle with B connected.Let F b be the fiber over the basepoint and let F be a unipotent complex localsystem over E. We view the homology complex H ∗ ( F b ; F ) as an element in K ( C )and we define the unreduced Whitehead space W h F ( E, b ) := hofib( Q ( E + ) λ F → K ( C )) H ∗ ( F b ; F ) . The unreduced smooth torsion of p is the map e τ F : | S • ( B ) | → W h F ( E, b ) deter-mined by the transfer p ! and the homotopy ω F . We want to make this independent of the basepoint choice. The answer is thereduced torsion:
Definition 2.10.
For a compact manifold bundle p : E → B with base point b ∈ B and unipotent complex local system F on E we define the Whitehead space W h F ( E ) := hofib( Q ( E + ) → K ( C )) . The reduced smooth torsion τ F ( p ) is the map | S • ( B ) | → W h F ( E ) obtained from p ! by subtracting the element p ! ( b ) ∈ Q ( E + ) from the map p ! and the path ω F| b × I from the contracting homotopy ω F . emark . So far this only defines the torsion map. The cohomological torsion τ k F ( p ) ∈ H k ( B ; R ) is defined in the following way: If F is trivial, consider the finalmap Q ( E + ) → Q ( S ) . This lifts to a map on Whitehead spaces
W h ( E ) → W h ( ∗ ) . Since the middle term in the homotopy fibration
W h ( ∗ ) → Q ( S ) → K ( C )is rationally contractible, and the cohomology class b k (the Borel regulator) of K ( C )therefore gives a cohomology class b k ∈ W h ( ∗ ) . We then pull this back along thecomposition | S • ( B ) | → W h ( E ) → W h ( ∗ )to get the cohomological torsion.In the case where F is non-trivial, one replaces the point ∗ with the “equivariantpoint” BG (where G is a finite group). To do so, a manifold approximation to BG is needed. See [13]. In this section we will define the Igusa-Klein torsion. We will first give an intuitivedescription for the construction for S -bundles E → B. This will motivate theexplicit definitions of the categorical models for the Whitehead space. Then we willgeneralize these models to accommodate the definition of the Igusa-Klein torsion forany smooth bundle E → B. Lastly, we will explain why the models involved havethe correct homotopy type. S -bundles Let E → B be an S -bundles and F a local system on E that is completely deter-mined on the fiber (thus F can just be viewed as a root of unity). Assume that F is non-trivial so that the singular chain complex C ∗ ( S , F ) is acyclic. Now choosea fiber-wise generalized Morse function f : E → R (by [8] this is a contractiblechoice). This means that on every fiber S ∼ = E x over x ∈ B the function f restrictsto either a proper Morse function or a function that may only have critical pointsthat in local coordinates look like f ( x ) = c ± x (critical point of order 2)or f ( x ) = c ± x (critical point of order 3) . The set of points x ∈ B over which the generalized Morse function f gives criticalpoints of order 3 on E x forms a codimension 1 submanifold of B called the bifurcationset.Now imagine two points x, y ∈ B outside off the bifurcation set and a pathconnecting x and y by crossing the bifurcation set. This means that as we movefrom x to y either two critical points of E x come together in a critical point of degree3 and cancel each other out or two critical points in E y are created from a critical8 B = I b b b y x y x = y x y x y ••• x y x y x y Figure 1: A maximum x and minimum y coming together at a birth death point.point of degree 3. See figure 1 for an example of an S -bundle over the interval I. The idea of Igusa-Klein torsion is to codify and track information about criticalpoints and the Morse complexes of E x as x varies in the base space B and use thisinformation to define a torsion invariant.To do so let S • ( B ) again be the simplicial category of simplices in B with nomorphisms. We will encode the above information as a functor S • ( B ) → W • ( C ) , where |W • ( C ) | is the Whitehead space. Following [8] we will first give an explicitmodel for the simplicial category W • ( C ) and later show that it has the desiredhomotopy type. Guided by the varying Morse complexes of the fibers E x over x ∈ B ,we see that the main feature this category should have is that its 0-simplices areMorse complexes, whereas its 1-simplices enable a connection between the Morse-complexes on different sides of the bifurcation set of B. Here is the formal definition:
Definition 3.1 (2.1.1 and 2.1.7 of [8]) . The simplicial category W • ( C , n ) is givenby the following: • Its objects in degree p are pairs ( C ∗ , P ) where – P = P ⊔ P is a graded partially ordered set where P and P haveexactly n elements with grading 0 or 1 respectively, – C ∗ is a p + 1-tuple of upper triangular (in the partial ordering) of isomor-phisms f ( i ) : C P → C P viewed as acyclic chain complexes 0 → C P → C P together with chain isomorphisms E ( i, j ) : f ( i ) → f ( j ) for i ≤ j, homotopies E ( j, k ) E ( i, j ) → E ( i, k ) , and higher homotopies. • morphisms ( C ∗ , P ) → ( C ′∗ , P ′ ) are given by a closed bijection P → P ′ and amonomial chain morphisms over it (see [8] for details). • Face and degeneracy maps are given by deleting and repeating terms in theobject tuplesThe idea behind this definition is that a Morse function f : S → R with exactly2 n critical points of degree 0 and 1 (ordered by the Morse function) gives a 0-simplexof W ( C , n ) via forming the Morse complex. This is not enough to treat our bundles9 → E → B as we do not expect every fiber to have the same amount of criticalpoints. Hence, we need to stabilize: Definition 3.2 (2.5.1-2.5.3 in [8]) . We make the following definitions: • Let ( C ∗ , P ) be an object of W p ( C , n ) . An expansion pair is a pair of elements x − ∈ P and x + ∈ P such that x − < x + , they are unrelated to any otherelements, and f ( x + ) = gx − with g ∈ U (1) . • Let ( C ∗ , P ) ∈ W p ( C , n ) and ( C ′∗ , P ′ ) ∈ W p ( C , n ′ ) . An expansion ( C ∗ , P ) → ( C ′∗ , P ′ ) is a degree 0 poset embedding P → P ′ such that P ′ \ P is a union ofexpansion pairs, together with a chain monomorphism over said inclusion. • The simplicial category W • ( C ) has as objects any objects of W • ( C , n ) for any n and morphism the morphisms in W • ( C , n ) together with the expansions.This can also be considered a bicategory.We are now ready to define the functor S • ( B ) → W • ( C ) . Let S → E → B be a smooth bundle and let ζ ∈ U (1) be a non-trivial root of unity (playing therole of a local system F → E ). Choose a generalized parametrized Morse function f : E → R . It is clear that such a map should assign any 0-simplex x ∈ B theMorse complex of E x with coefficients ζ . However, the main difficulty arises in howto define the functor on 1-simplices that cross the bifurcation set as the two ends ofsuch a simplex do not have to have the same amount of critical points. The solutionto this are “ghost” points:After a pair of critical points of the fiber of E → B meet and cancel over thebifurcation set, they remain detectable (in a neighborhood) as an inflection point onwhich the second derivative of f : E x → R vanishes. We call these points ghosts andwe can choose our simplices small enough such that over any simplex where thereis a pair of critical points dissolving into a ghost the corresponding ghost does notdissolve. We do not change the homotopy type of | S • ( B ) | ≃ B by only consideringsuch small simplices. We will only call an inflection point in the fiber over a givenpoint a ghost over a given small simplex, if there is a point within that simplexwhere the ghost develops into two critical points. By the discussion, over any pointin a given simplex the number of critical points plus twice the number of ghostswill be the same. With this we define the functor S • ( B ) → W ( C ) on 1-simplices bysending the path γ : ∆ → B to ( C ∗ , P ) where P = { critical points of degree 0, ghosts } ⊔ { critical points of degree 1, ghosts } and C ∗ is given by the Morse complexes of E γ (0) and E γ (1) (with expansion pairs forghosts) together with a chain isomorphism given by their connection. The functorcan be defined similarly on higher simplices. For details see [8]. Remark . Upon careful inspection, one notices that the so defined S • ( B ) →W • ( C ) is not simplicial. However, this can be alleviated by introducing a weakequivalence C • ( B ) → S • ( B ) and a simplicial functor C • ( B ) → W • ( C ) . This is doneexplicitly in [8]. By abuse of notation, we will continue to write S • ( B ) instead of C • ( B ) . .2 The Whitehead Category In the previous section we defined the Igusa-Klein torsion for any S -bundle E → B by explicitly constructing a functor S • ( B ) → W ( C ) . Now we turn our attentionto the more general case: Let F → E → B be a smooth manifold bundle. Weagain wish to define its Igusa-Klein torsion. Let F → E be a local system (trivialon B ) and choose a generalized parametrized Morse function f : E → B. TheMorse complexes of E x ∼ = M will not be concentrated in two degrees alone anymore,so to define the torsion functor we need a target category encoding general chaincomplexes as 0-simplices, together with homotopies and higher homotopies as highersimplices: Definition 3.4 (3.1.1, 3.2, 3.6 in [8]) . Let n ∗ = ( n , . . . , n k ) be a tuple of naturalnumbers. The simplicial category W h • ( C , n ∗ ) is defined as follows: • An object in degree q is a pair ( C ∗ , P ) where – P is a partially ordered graded set with n i elements in each degree i – C ∗ is a q + 1-tuple of chain complexes where the i th entry of the l thchain complex (1 ≤ l ≤ q + 1) is C ∗ ( l ) i = C P i (the boundary maps canbe different for each l ), together with homotopies and higher homotopiesconnecting the entries in the q + 1-tuple as if they were corners of a q -simplex. • Morphism are given by closed bijections P → P ′ and sufficiently coherentcollections of chain morphisms over them.This can again be stabilized via expansion pairs to obtain the category W h • ( C ) . Thefull subcategory
W h h • ( C ) is given by only considering objects comprised of acyclicchain complexes.Notice that this was completely analogous to the definition of W • ( C ) and containsthe former as a subcategory. Proposition 3.5 ([8]) . Let M → E → B be a smooth fiber bundle and F → E afinite local system as above such that the singular complex C ∗ ( M, F ) is acyclic. Thenthe (contractible) choice of a generalized parametrized Morse function f : E → R defines a functor S • ( B ) → W h h ( C ) analogously to the previously defined functor S • ( B ) → W ( C ) for S -bundles.Remark . Only the category
W h h • ( C ) has the desired homotopy type of the White-head space and not W h • ( C ) . Hence we only define the torsion functor for acyclicfibers and not more generally. We will consider a slightly more general case insubsection 3.2.2.
The above construction proves to be somewhat unwieldy as the simplicial structureof
W h • ( C ) is quite complicated. Instead we will use the so called “multiple mappingcylinder” to turn an object of W h • ( C ) - that is a system of chain complexes andhigher homotopies - into a single filtered chain complex. We first define the latter:11 efinition 3.7 (4.1.1 and 4.1.2 in [8]) . Let P be a (partially ordered, graded) set. • A Λ P -module is a C -vector space M together with subspaces M A for all A ⊂ P such that – M P = M and – M A ∩ B = M A ∩ M B . The Λ P -modules naturally form a category. • A Λ P -filtered C -complex ( E, λ ) is a chain complex in the category of Λ P -modules together with a cohomology class λ A ( x ) ∈ H deg x ( E A ⊔{ x } , E A ; C )for all pairs A ⊂ P and x ∈ P such that – E ∅ = 0 – E { x } + E A = E A ∪{ x } – H deg x ( E A ⊔{ x } , E A ; C ) ∼ = C via the map induced by λ A ( x ) and this relativehomology vanishes in all other degrees. – The cohomology classes λ A ( x ) are compatible. Remark . One can think of a Λ P -filtered chain complex as a chain complextogether with a basis P i (elements of P in degree i ) for its i th homology for all i. Definition 3.9 (4.1.3 [8]) . There is a multiple mapping cylinder construction turn-ing an object ( C ∗ , P ) ∈ W h q ( C ) into a Λ P -filtered chain complex Z q ( C ∗ ) . This isdone by assembling all the homotopy information from C ∗ into a large chain com-plex. If q = 0 then Z ( C ∗ ) = C ∗ is itself already a filtered complex. The filtering onhigher q ’s is similar. Remark . As the name suggests, the idea of the multiple mapping cylinders is totake subsequent mapping cones: For example, let ( C ∗ , P ) ∈ W h ( C ) be a 1-simplex.That is it is completely represented by a chain complex homotopy equivalence f :( C P ) → ( C P ) . To retain all the information of this map, while still condensingthe structure into a single chain complex, we can take the mapping cone cone( f ) ∈ Ch ( C ) , which naturally has the structure of a filtered chain complex over P. Clearlywe have homotopy equivalences ( C P ) ≃ cone( f ) ≃ ( C P ) . For higher simplices onecan subsequently form cones of the connecting maps and homotopies.Next we define the category of filtered chain complexes. Recall that the classi-fying space BU (1) can be viewed as the geometric realization of the simplicial setwith BU (1) k = U (1) k . Let ξ be the universal line bundle over BU (1) . Definition 3.11 (5.2.2 in [8]) . Let
F C ( BU (1) • , ξ, n ∗ ) be the following simplicialcategory: • An object in degree q is a triple ( E, P, γ ) , where12 P is a partially ordered graded set with n i elements in each degree i – γ : P → BU (1) k = U (1) k is a map of sets and – E is a Λ P -filtered chain complex with cohomology classes λ A ( x ) givingisomorphisms H deg x ( E A ⊔{ x } , E A ) ∼ = ξ ( λ ( x )) ∼ = C . • A morphism is given by a closed bijection α : P → P ′ and a sufficientlycoherent chain complex morphism E → E ′ above it.This can be stabilized via extension pairs to a stable category F C ( BU (1) • , ξ ) . Thefull subcategory
F C h ( BU (1) • , ξ ) is given by only considering acyclic chain com-plexes. Proposition 3.12 (5.3.4 and 5.3.5 in [8]) . The multiple mapping cylinder construc-tion gives weak homotopy equivalences
W h • ( C ) ≃ F C ( BU (1) • , ξ ) and W h h • ( C ) ≃ F C h ( BU (1) • , ξ ) . Remark . The maps γ : P → U (1) k for an object ( E, P, γ ) in
F C ( BU (1) , ξ ) areneeded to encode morphism and expansion structures in W h ( C ) . Observation . Let M → E → B be a smooth bundle with acyclic local system F → E and generalized Morse function f : E → R . Instead of defining the torsionfunctor S • ( B ) → W h ( C ) we can also directly define the torsion functor S • ( B ) → F C ( BU (1) , ξ ) by composing with the multiple mapping cylinder construction. Remark . Definition 3.11 can be generalized by replacing BU (1) with any sim-plicial set X with a functor ξ : simp X → Vect C . We call the resulting category
F C ( X, ξ ) . Observation . Again let M → E → B be a smooth bundle with acyclic localsystem F → E and generalized Morse function f : E → R . Then the local systemdefines a functor ξ F : simp E → Vect C and we can factorize the torsion functorthrough F C h ( E, ξ F ) → F C h ( BU (1) , ξ ) to get Definition . The construction above gives the Igusa-Klein torsion as a map S • ( B ) → F C h ( E, ξ F ) . At this point we only defined Igusa-Klein torsion for 1-dimensional acyclic localsystems
F → E. We will briefly indicate how to remedy these shortcomings:Analogously to our definition of
F C ( BU (1) , ξ ) one can define F C ( BG, ξ ) for anygroup G together with a representation G → U ( n ) . This creates a natural target forthe torsion of any bundle M → E → B wit acyclic but not necessarily 1-dimensionallocal system F → E. Furthermore this can also be lifted to a torsion functor S • ( B ) → F C h ( E, ξ F ) . M → E → B is a smooth bundle with local system F → E such that π ( B ) acts trivially on the homologies of the fiber with coefficients F . However, F does not have to be acyclic anymore. As mentioned in 3.14, this stillgives a functor (after choosing a Morse function f : E → R ) S • ( B ) → F C ( E, ξ F )but this functor will not factor through F C h anymore. However, if π B acts triviallyon the homology of the fiber, we can after stabilization form the alternating mappingcone which will define a map S • ( B ) → F C h ( E, ξ F ) . We will take this map as the definition of the torsion in the non-acyclic case. Detailsare to be found in chapter 4.6 of [8].
F C ( E, ξ F ) and the category Q ( E ) We continue to summarize the constructions of [8]. So far, we defined the Igusa-Klein torsion functor S • ( B ) → F C h ( E, ξ F ) for smooth bundles E → B with localsystem F → E, but so far we have not yet established that F C h ( E, ξ F ) has thecorrect homotopy type of the Whitehead space. Hence, in this section we show that F C h ( E, ξ F ) can be identified as the homotopy fiber of the composition Q ( E + ) → A ( E ) → K ( C ) . After this the cohomological Igusa-Klein torsion is defined just asthe smooth torsion as pull-back of the Borel regulators.To identify
F C h ( E, ξ ) as the homotopy fiber we use the Waldhausen fibrationtheorem [17]. We will introduce a category K with two kinds of weak equivalences( w -equivalences and h -equivalences). Then Waldhausen gives a homotopy fibrationsequence (recall that we write Kw ( − ) to indicate Ω | w S • ( − ) | for the Waldhausen S • -construction) Kw K h → Kw K → Kh K and we identify Kw K h ≃ F C h ( E, ξ F ) , Kw K ≃ Q ( E + ) , and Kh K ≃ K ( C ) . Beforedefining K , we will define a simpler category Q ( E + ) with K Q ( E + ) ≃ Q ( E + ) whichwill facilitate the middle equivalence. Definition 3.18.
Let X • be a simplicial set, then the category Q ( X • ) of finitegraded poset over X • is defined as follows: • An object in degree q is graded poset P together with a map γ : P → X k . As part of the data, there is a subset of identified expansion pairs x − , x + in P over the same point in X k with deg x + = 1 + deg x − and x − < x + . • Morphisms are pointed maps over P. • A morphism P → Q is a cofibration if it is an order preserving monomorphism. • A morphism f : P → Q is a w -equivalence if its kernel ker f = f − ( ∗ ) is aunion of expansion pairs and f : P \ ker f → Q is a bijection.14his forms a Waldhausen category. Remark . In [8] Igusa considers a slightly different category Q I ( X • ) in whichthere are no identified expansion pairs in the objects P (and thereby w -equivalencesare just bijections).Let Q I ( X • ) be the subcategory of graded posets over X • wit null ordering. Thenthe retraction Q I ( X • ) → Q I ( • ) given by forgetting the ordering is a deformationretract. But points of different degrees don’t interact in Q I ( X • ) , and we get Q I ( X • ) = Y n ≥ Q I,n ( X • ) , where Q n denotes the subcategory of isolated degree n. Segal established Kw Q n ( X • ) ≃ Q ( | X • | ) [15].This means that Q I ( X • ) does not have the correct homotopy type. However,we can take the nerve along elementary expansions to get e • Q I ( X • ) . An object of e k Q I ( X • ) is a sequence P → P ∨ S → . . . P ∨ S k , where the S k are increasing sets of expansion pairs. This mends together the differentcopies and we have Kwe • Q I ( X • ) ≃ Q ( | X • | ) . For a more detailed discussion compare [8] 5.6.5 and following.
Observation . Notice that for any simplicial set X we have | we • Q I ( X • ) | ≃ | w Q ( X • ) | :The left hand side is the geometric realization of the bicategory with objects gradedposets over X , vertical morphisms expansions, and horizontal morphisms bijections.The right hand side is the geometric realization of the category with objects gradedposets over X and morphisms being compositions of bijections and collapses ofexpansion pairs. Consequently we have Kwe • Q I ( X • ) ≃ Kw Q ( X • ) . Corollary 3.21. Kw Q ( X • ) ≃ Q ( | X • | ) Remark . As outlined in Remark 3.19, forgetting the orderings does not changethe homotopy type, so from now on we will only work with graded sets. Furthermorewe will continue to work with Q instead of Q I . Definition 3.23.
Let X • be a simplicial set and ξ : simp X → Vect C a functor. • The simplicial category K ( X • , ξ ) has similar objects to F C ( X • , ξ ), that ispairs ( E, P ) where P is a graded set and E is a filtered chain complex over P. Additionally there should be set of identified expansion pairs x − , x + in P andwe demand that E splits as E P \{ x − ,x + } ⊕ E { x − ,x + } , where the latter is given by C in degree deg x − and deg x + connected by theidentity. 15 Morphisms ( f, α ) : (
E, P ) → ( E ′ , P ′ ) are again given by morphisms α ofgraded sets (so morphisms in Q ( X • )) and chain morphisms f over them. • Cofibrations in K ( X • , ξ ) are cofibrations in Q ( X • ) covered by chain isomor-phisms. • A morphism ( f, α ) is a w -equivalence if α is a w -equivalence in Q ( X • ) . • A morphism ( f, α ) is an h -equivalence if f is a chain homotopy equivalence. Remark . Again in [8] Igusa defines K I ( X • ) without identification of the expan-sion pairs. As in Remark 3.19, one can then form e • K I ( X • ) and this yield the sameresults as our K ( X • ) . Observation . Since for any object (
E, P ) of K ( X • , ξ ) the graded set P actsas a “homological basis” for E, it is clear that every w -equivalence is also an h -equivalence.Based on this observation and Igusa’s work showing that K I ( X • , ξ ) has a mappingcylinder construction satisfying Waldhausen’s cylinder axioms we get immediately: Theorem 3.26 (Based on Waldhausen [17]) . The sequence Kw K ( X • , ξ ) h → Kw K ( X • , ξ ) → Kh K ( X • , ξ ) is a homotopy fibration with canonical contracting homotopy given by the unique nat-ural transformation from the composition w K ( X • , ξ ) h → h K ( X • , ξ ) to the constantfunctor on the final object. Here the superscript − h indicates h -trivial objects. There is an obvious forgetful functor K ( X • , ξ ) → Q ( X • ) which respects the w -equivalences. Proposition 3.27 ([8]) . The induced functor map Kw K ( X • , ξ ) → Kw Q ( X • ) is a weak equivalence. Furthermore we get a functor K ( X • , ξ ) → Ch ( P hf ( C )) (1)by forgetting the filtrations. Proposition 3.28.
The induced map Kh K ( X • , ξ ) → KhCh ( P hf ( C )) = K ( C ) is a weak equivalence. It is clear that w K ( X, ξ ) h ≃ F C h ( X • , ξ ) . So the last ingredient to finish the characterization of
F C h as a homotopy fiber isthe following: Proposition 3.29 ([8]) . The natural map (given by Remark 2.3) | w K ( X • , ξ ) | → Kw K ( X • , ξ ) is a weak equivalence. A combinatorical model for the Becker-Gottliebtransfer
In the previous sections we defined the smooth and Igusa-Klein torsion of a smoothmanifold bundle E → B with local system F → E. Both were given as maps intothe Whitehead space
W h F ( E ) : The smooth torsion was given as a lift of the Becker-Gottlieb transfer p ! : | S • ( B ) | → Q ( E + ) whereas the Igusa-Klein torsion is directlyconstructed as a map | S • ( B ) | → W h ( E ) . Composition with the inclusion of the fiberwill give a transfer map p ! IK : | S • ( B ) | → Q ( E + ) , and –of course– the Igusa-Kleintorsion map is a lift of this. We will show that these two transfer maps have thesame homotopy type, ultimately leading to a proof of Theorem 1.1.First of all, recall that the model used in the previous section is | F C h ( E, ξ F ) | ≃ W h F ( E ) . According to Observation 3.14 the Igusa-Klein torsion τ IK : | S • ( B ) | → | F C h ( E, ξ F ) | is given by sending a simplex σ : ∆ k → B to the pair ( P, C ) where P is the gradedposet of critical and twice the birth-death points of a chosen fiber-wise generalizedMorse-function on E, the defining map is given by the map P × ∆ k → E obtainedby lifting σ : ∆ k → B to the level of critical and birth-death points (this is notnecessarily injective as two critical points can meet in a birth-death point), andfinally the filtered chain complex C is given by taking the multiple mapping cylinderconstruction of the Morse-complexes over σ with coefficients ξ. The Igusa-Kleintransfer p IK : | S • ( B ) | → Kw K ( E ∆ • , ξ F )is then given in view of Theorem 3.26 as the composition of τ IK with the maps | F C h ( E ∆ • , ξ F ) h | ≃ | w K ( E ∆ • , ξ F ) h | Remark 2.3 −−−−−−→ Kw K ( E ∆ • , ξ F ) h ֒ → Kw K ( E ∆ • , ξ F ) . By Remark 2.3 we can regard this composition as induced by the concrete functordescribed above.We will give an alternate model for Q ( E + ) and use this to connect the Igusa-Kleintransfer p IK to the Becker-Gottlieb transfer p ! given in Section 2 as a lift Q ( E ∆ • + ) (cid:15) (cid:15) | S • ( B ) | p ! ssssssssss p A / / A ( E ∆ • )where p ! A is given by sending σ : ∆ k → B to the retractive space E × ∆ k ⊔ σ ∗ E × ∆ k → E × ∆ k . We begin with the following definition: 17 efinition 4.1.
Let X be a topological space. We define the expansion category E ( X ∆ • ) as follows: • An object in degree k of E ( X ∆ k ) is a triple ( P, Y, r ) , where – P is a graded poset over X ∆ k with identified expansion pairs. – Y is a k -parameter family of relative cell complexes with cells indexed by P. In particular Y = X × ∆ k ⊔ G p ∈ P I deg p × ∆ k / ∼ , where no cell is attached to a cell of equal or higher order. – Every expansion pair corresponds to two cells in canceling position, di-rectly attached to X. – r : Y → X × ∆ k is a retraction respecting the data above. • A morphism is a pair ( α, f ) : (
P, Y, r ) → ( P ′ , Y ′ , r ′ ) where α : P → P ′ is amorphism in Q ( X ∆ k ) and f : Y → Y ′ is a morphism above respecting all thedata. This is completely determined by α if such an f exists. • A morphism ( α, f ) is a cofibration of α is a cofibration in Q ( X ∆ k ) . • A morphism ( α, f ) is a x -equivalence if α is a w -equivalence in Q ( X ∆ k ) and f sends every cell in ker α into X × ∆ k . • A morphism ( α, f ) is an h -equivalence if f is a homotopy equivalence.Altogether, this defines a simplicial Waldhausen category.There is a map x E ( X ∆ • ) → w Q ( X ∆ • )given by forgetting about the cells. Igusa and Waldhausen showed [11]. Proposition 4.2.
This map gives a weak equivalence Kx E ( X ∆ • ) ≃ Kw Q ( X ∆ • ) ≃ Q ( X + ) . Proof.
This was originally proved in [11]. We reproduce the proof in the appendix5.1.Furthermore there is a map h E ( X ∆ • ) → h R hf ( X ∆ • )given by forgetting the graded posets. Proposition 4.3.
This gives a weak equivalence Kh E ( X ∆ • ) ≃ A ( X ∆ • ) ≃ A ( X ) . roof. Again, this was proved in [11] and can be found in the appendix 5.2.One can see that every x -equivalence of E ( X ∆ • ) is also an h -equivalence, andhence we get a map Kx E ( X ∆ • ) → Kh E ( X ∆ • ) . We will use this map to compare the Becker-Gottlieb transfer and the Igusa-Kleintransfer as maps into A ( X ) . Remark . The fiber of the above map can be identified as Kx E ( X ∆ • ) h ≃ W h
P L ( X ) . Doing so was the original purpose of the Igusa-Waldhausen paper [11]. Since Wald-hausen found an alternate proof in [17], this paper was ultimately never published.
In this section we work to compare p ! IK and p ! from the previous sections. Much willbe guided by the following homotopy commutative diagram (the disconnected parton the right indicates the homotopy type of every model in the corresponding rows,all horizontal maps are weak equivalences). As always consider a smooth bundle E → B with local system F → E. Kw K ( E ∆ • , ξ F ) Kx E ( E ∆ • ) MMC ξ o o (cid:15) (cid:15) Ωhocolim n |S • P • ( E × I n ) | (cid:15) (cid:15) Q ( E + ) Assembly (cid:15) (cid:15) Kh E ( E ∆ • ) / / Kh R hf ( E ∆ • ) A ( E ) |S • | ( B ) | p ! IK ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) p ! M ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ p ! A,M ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ p ! A ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ p ! $ $ We already defined all the spaces involved as well as the transfers p ! IK and p ! A . The strategy to introduce and use the rest is as follows: First we define the Morsetransfer p ! M : |S • | → Kx E ( E ∆ • ) and the map M M C ξ The transfer p ! A,M is simply thecomposition of p ! M with the inclusion Kx E ( E ∆ • ) → Kh E ( E ∆ • ) . Recall that we havean explicit description of p ! A from Section 2 which we now can compare explicitly tothe composition of p ! A with the forgetful inclusion Kh E ( E ∆ • ) → R hf ( E ∆ • ) (whichis a weak equivalence as we will show in Appendix 5.2). Lastly, p ! and p ! IK are bothlifts of p ! A and p ! A,M , so in the end we leverage the section of the assembly map tolift our comparison. Definition 4.5.
The Morse transfer p ! M : | S • ( B ) | → Kx E ( E ∆ • )19s given by sending a simplex σ : ∆ k → B to the pair ( P, Y ) , where P is the gradedposet of the critical and twice the birthdeath points over σ (as before this can beviewed as a set over E ∆ k ) and Y = E × ∆ k ⊔ Y ′ . Here Y ′ ≃ σ ∗ E is viewed as a parametrized cell complex via the generalized Morsefunction. More precisely Y ′ = G p ∈ P I deg p × ∆ k / ∼ . The equivalence relation does not only identify attachments of the boundary of cells,it also identifies whenever two critical points join together at a birth-death point.Altogether Y gives a parametrized retractive space of E × ∆ k where the retractionis given by inclusion of the Morse skeleton in the first component Y = Y × ∆ k ⊔ Y ′ → Y × ∆ k . There is a functor
M M C ξ : x E ( E ∆ • ) → w K ( E ∆ • , ξ )Constructed in the following way: An object of x E ( E ∆ k ) is a pair ( P, Y ) where P is a graded poset with an inclusion P × ∆ k → E and Y = E × ∆ k ⊔ G p ∈ P I deg p × ∆ k / ∼ can be viewed as a parametrized ∆ k family of relative cell complexes over E withcells indexed and attached according to the poset order of P. In particular everyvertex [ l ] ∈ ∆ k gives a cell complex Y ( l ) and every edge [ l, l ′ ] ⊂ ∆ k gives a simplehomotopy equivalence Y ( l ) → Y ( l ′ ) . Higher faces of ∆ k will give homotopies andhigher homotopies between these simple homotopy equivalences. So we can form M M C ξ ( P, Y ) by setting
M M C ξ ( P, Y ) :=
M M C (( P, C )) , where C is the k -tuple of chain complexes given by C ( l ) = C ∗ ( Y ( l ) , E ; ξ ) (where1 ≤ l ≤ k ) together with homotopy equivalences and higher homotopies between Y ( l ) and Y ( l ′ ) given by the simple homotopy equivalences and higher homotopiesfrom above. The functor M M C is the multiple mapping cylinder. Notice that thiscan be done functorially.We get a functor in K -theory and it follows directly that p ! IK = M M C ξ ◦ p ! M : | S • ( B ) | → Kw K ( E ∆ • , ξ ) . Proposition 4.6.
The map
M M C ξ : Kx E ( E ∆ • ) → Kw K ( E ∆ • , ξ ) is a weak equivalence. roof. Both the forgetful map x E ( E ∆ • ) → w Q ( E ∆ • )and the composition with the forgetful map Kx E ( E ∆ • ) MMC ξ −→ Kw K ( E ∆ • , ξ ) → w Q ( E ∆ • )agree by inspection (they only care about the first component). Furthermore, bothforgetful maps are homotopy equivalences. Corollary 4.7.
The maps p ! IK and p ! M have the same homotopy type viewed as maps B → Q ( E + ) . Consequently, it is enough to show that p ! M has the homotopy type of the Becker-Gottlieb transfer. Our goal now is to show that p ! and p ! M have the same homotopy type. Recall that p ! was given as an explicit lift to p ! A : | S • ( B ) | → A ( E ∆ • )defined by sending σ : ∆ k → B to the retractive space E × ∆ k ⊔ σ ∗ E → E × ∆ k . On the other hand p ! M was explicitly constructed as a geometric realization p ! M : | S • ( B ) | → Kx E ( E ∆ • ) . Furthermore there is the inclusion Kx E ( E ∆ • ) → Kh E ( E ∆ • ) ≃ A ( E ∆ • ) . We denote the composition of this with p ! M by p ! A,M . Instead of comparing maps onthe level of Q ( E + ) we will compare p ! A and p ! A,M on the level of A ( E ) . First we need
Lemma 4.8.
The map Kx E ( E ∆ • ) → Kh E ( E ∆ • ) has the homotopy type of the assembly map a : Q ( E + ) → A ( E ) Proof.
All of these maps can be viewed as maps of ring spectra. If E = ∗ there isonly the initial map Q ( S ) → A ( S ) since the sphere spectrum Q ( S ) is the initialobject in that category. The lemma now follow from universality arguments for any E. Theorem 4.9.
The transfers p ! and p ! M have the same homotopy type as maps B → Q ( E + )21 roof. First notice that there is a homotopy equivalence between a ◦ p ! = p ! A and a ◦ p ! M = p ! A,M both seen as maps | S • ( B ) | → Kh R hf ( E ∆ • ) ≃ A ( E ∆ • )(for p ! A,M we need to compose with the inclusion Kh E ( E ∆ • ) → Kh R hf ( E ∆ • )): Onthe simplex σ : ∆ k → B the equivalence is given by the natural transformation com-ing from including the Morse-skeleton Y ′ into σ ∗ ( E ) giving an homotopy equivalenceinclusion ( a ◦ p ! M )( σ ) = E × ∆ k ⊔ Y ′ ֒ → E × ∆ k ⊔ σ ∗ E = p ! A ( σ )Furthermore, Waldhausen showed [16] that there is a homotopy right inverse to a given by the trace map tr : A ( E ) → Q ( E + ) , so tr ◦ a ∼ id (this map gives thesplitting of the fibration sequence W h
P L ( E ) → Q ( E + ) → A ( E )). So finally we have p ! ∼ tr ◦ a ◦ p ! ∼ tr ◦ a ◦ p ! M ∼ p ! M . Corollary 4.10.
So we established p ! ∼ p ! IK . So far we established that the two different transfer maps p ! , p ! IK : | S • ( B ) | → Q ( E + )agree. But to prove Theorem 1.1, we are interested in comparing their lifts τ sm , τ IK : | S • ( B ) | → W h ξ ( E ) , which are uniquely determined by their underlying maps p ! and p ! IK together with a homotopy H sm : λ ◦ p ! → const and H IK : λ ◦ p ! IK → const where λ : Q ( E + ) → K ( C ) is the assembly map followed by linearization.To prove that τ sm and τ IK are homotopic, we need to provide a homotopy ofhomotopies H : H IK ◦ H → H sm , where H : p ! → p ! IK is the homotopy found above.So H is a homotopy of homotopies of maps Q ( E + ) → K ( C ) . One can view it as theinside of the triangle-diagram λ ◦ p ! H sm $ $ ■■■■■■■■■■ λ ◦ H / / λ ◦ p ! IKH IK y y ssssssssss const , where all corners are maps Q ( E + ) → K ( C ) . To be precise we will be working with p ! M : | S • ( B ) | → Kx E ( E ) instead of p ! IK . We need to be careful because the lift of p ! (and p ! IK ) is explicitly given. However,by theorem 3.26, it corresponds to the contracting homotopy H M of the composition | S • ( B ) | → Kx E ( E ) → K ( C )given by the natural transformation to the final functor const : S • ( B ) → K ( C ) . The homotopy H sm come from theorem 2.8 and if the homology of the fiber isacyclic it is just given by the same final map [13]. So it is clear that the diagramabove commutes on the nose. This completes the proof of Theorem 1.1.22 emark . So far this only works if the fiber F of E → B is acyclic with respectto the local system ξ. If it is not, the definition of Igusa-Klein torsion asks thatthe fundamental group π ( B ) act trivially on the homology H ∗ ( F ; ξ ) (the smoothtorsion only asks for this action to be unipotent). In this case, to define the smoothtorsion we subtract the constant functor H ∗ ( F ; ξ ) from the construction in the loopspace structure of K ( C ) (compare Definition 2.10). For the Igusa-Klein torsion, oneforms a certain mapping cone as done in [8]. Both amount to the same outcomeand both torsions are still going to be equivalent. We present the proofs for Propositions 4.2 and 4.3. These already appeared in [11],but remained unpublished and not publicly accessible. We merely reproduce theresults.
K x E ( X ∆ • ) Let X be a topological space. We aim to prove Kx E ( X ∆ • ) ≃ Q ( X • ) . We will need an auxiliary category D . Definition 5.1.
The simplicial category D ( X ∆ • ) is the same as the category E ( X ∆ • )without identified expansion pairs. Explicitly we define D ( X ∆ k ) as follows: • An object in degree k of D ( X ∆ k ) is a triple ( P, Y, r ) , where – P is a graded poset over X ∆ k (without identified expansion pairs). – Y is a k -parameter family of relative cell complexes with cells indexed by P. In particular Y = X × ∆ k ⊔ G p ∈ P I deg p × ∆ k / ∼ , where no cell is attached to a cell of equal or higher order. – r : Y → X × ∆ k is a retraction respecting the data above. • A morphism is a pair ( α, f ) : (
P, Y, r ) → ( P ′ , Y ′ , r ′ ) where – α : P → P ′ is a pointed set map that is closed as a poset map – f : Y → Y ′ is a morphism above it respecting all the data such that – if A is closed subset of P then f ( Y A ) ⊂ ( Y ′ ) α ( P ) where Y A ⊂ Y is theset of all elements over A. • A morphism ( α, f ) is a cofibration if α is an order preserving monomorphism(making f an embedding of a parametrized subcomplex)23 A morphism ( α, f ) is a weak equivalence if α is a bijection (and thus f is aparametrized cellular homeomorphism).Altogether, this defines a simplicial Waldhausen category. Let D ( X ∆ • ) be the fullsubcategory with cofibrations of D ( X ∆ • ) of objects ( Y, P, r ) where P only has thetrivial ordering. Intuitively, this means that in D all the cells are attached at once. Remark . A morphism ( f, α ) in D ( X ∆ • ) is completely determined by α if it exists.Again let Q ( X ∆ • ) be the simplicial categories of finite sets (neither graded norordered) over X ∆ • . Then recall from Remark 3.19 K Q ( X ∆ • ) ≃ K Q ( X ∆ • ) ≃ Ω ∞ Σ ∞ X + . Notice that the weak equivalences of D ( X ∆ • ) are exactly the isomorphismsbecause everything has the trivial ordering. Furthermore every object in D ( X ∆ k )splits uniquely as a sum of objects with each only having cells in one given dimension. Lemma 5.3.
Let n ∈ N and let D n ( X ∆ • ) be the subcategory of D ( X ∆ • ) with onlycells of degree n. Then we have D n ( X ∆ • ) ≃ Q ( X ∆ • ) . Proof.
Let f : D n ( X ∆ k ) → Q ( X ∆ k ) be the forgetful functor with f ( Y, P, r ) := (
P, γ ) , where γ : P → X ∆ k is given by the attachments of the basepoints of the cells thatmake Y. Let j : Q ( X ∆ k ) → D n ( X ∆ k ) be the functor that is given by j ( P, γ ) := (
Y, P, r ) , where Y = X × ∆ k ⊔ G p ∈ P I n × ∆ k / ∼ with all cells attached at their basepoint via γ : P → X ∆ k . The retraction r issimply given by mapping the cells to their basepoint.Clearly, the composition f ◦ j : Q ( X ∆ k ) → Q ( X ∆ k ) is the identity. On theother hand, the composition j ◦ f : D n ( X ∆ • ) → D n ( X ∆ • ) is given by contractingall attachment maps to attachments at the basepoints of the cells. A homotopy j ◦ f ∼ id comes from the functors H : D n ( X ∆ k ) × ∆([ k ] , [1]) → D n ( X ∆ k )given by sending (( Y, P, r ) , α ) ( Y α , P, r α ) . Here Y α has the same cells as Y withdifferent attachment maps: Let η : ∂I n × ∆ k → X be the attachment map of a cellof Y (Notice η = r | ∂I n ), then the new attachment map is η α : ∂I n × ∆ k → X with η α ( s, t ) = r ( α ∗ ( t ) s, t ) , where α ∗ : ∆ k → ∆ = I is the induced map. The retraction r α : I n × ∆ k → X is given by r α ( s, t ) = r ( α ∗ ( t ) s, t ) . roposition 5.4. We have i S • D ( X ∆ • ) ≃ i S • D ( X ∆ • ) ≃ w S • D ( X ∆ • ) . Proof.
It will be enough to show i S • D ( X ∆ k ) ≃ i S • D ( X ∆ k ) and that w S • D ( X ∆ k )is homotopy equivalent to a simplicial subcategory of i S • D ( X ∆ k ) which contains i S • D ( X ∆ k ) . We will show the latter first.Let D ( X ∆ k ) be the subcategory of D ( X ∆ k ) of objects ( Y, P, r ) where the partialordering on P is as minimal as possible. While this is not a subcategory withcofibrations as it does not have all push-outs, it does have quotients and we can form i S • D ( X ∆ k ) ⊂ i S • D ( X ∆ k ) and this subcategory contains i S • D ( X ∆ k ) as demanded.Let g : w S • D ( X ∆ k ) → i S • D ( X ∆ k )be the functor given by sending ( Y, P, r ) to (
Y, P ′ , r ) where P = P ′ as sets and P ′ has the minimally necessary ordering. Let j : i S • D ( X ∆ k ) → w S • D ( X ∆ k )be given by the inclusion. Then we have gj = id and jg ∼ id as weak equivalencesin w D ( X ∆ k ) are given by set-bijections on the posets.It now suffices to show that i S • D ( X ∆ k ) ≃ i S • D ( X ∆ k ) . Let D n ( X ∆ k ) be the fullsubcategory of D ( X ∆ k ) in which cells are attached in no more than n layers. Wewill show inductively i S • D n ( X ∆ k ) ≃ i S • D n +1 ( X ∆ k ) . Let Z be the Waldhausen category with objects being pairs (( Y, P, r ) , z ) where( Y, P, r ) ∈ D n +1 ( X ∆ k ) and z : P → { , , } is a “height function” with • Every element of z − { , } is minimal and • the poset z − { , } does not contain any ( n + 1)-chains.The morphisms are the morphisms ( f, α ) : ( Y, P, r ) → ( Y ′ , P ′ , r ′ ) such that α takes z − { } into ( z ′− { } ) + and z − { } into ( z ′− { , } ) + . A cofibration is a heightperserving cofibration in D ( X ∆ k ) and a weak equivalence ( f, α ) : ( Y, P, r ) → ( Y ′ , P ′ , r ′ )induces an isomorphism Y → Y ′ so that α sends z − { } into z ′− { } . Let Z be be the full Waldhausen subcategory of Z with objects from D ( X ∆ k )and let E be the full Waldhausen subcategory of Z given by z − { } = ∅ and let E = E ∩ Z . Then E is exactly equivalent to the category of cofibrations A ֒ → B ։ B/A with A ∈ D ( X ∆ k ) , B ∈ D n +1 ( X ∆ k ) , and B/A ∈ D n ( X ∆ k ) and E is exactlythe category of cofibration sequences in D ( X ∆ k ) . By the additivity theorem [17]and induction we have i S • E ≃ i S • ( D ( X ∆ k ) × D n ( X ∆ k )) ≃ i S • ( D ( X ∆ k ) × D ( X ∆ k )) ≃ i S • E . We will consider the following map of fibration sequences i S • E / / (cid:15) (cid:15) w S • Z (cid:15) (cid:15) / / w S • S • ( E → Z ) (cid:15) (cid:15) i S • E / / w S • Z / / w S • S • ( E → Z ) . w S • Z ≃ w S • Z which in light of Lemma 5.5 provesthe proposition. We have w S • S • ( E → Z ) ≃ w S • Z k Z and w S • S • ( E → Z ) ≃ w S • Z k Z . Let j : Z → Z be the inclusion functor and q : Z → Z be the functor thatchanges the attachment maps ψ : ∂I n × ∆ k → X × ∆ k to r ◦ ψ where r : Y → X is theretraction, thereby attaching cells to the base directly. Clearly, we have qj = id Z . Consider the functor h : Z → Z given by sending ( Y, P, r ) to only its minimal cells.There clearly are natural transformations given by inclusionsid Z ← jh → jq. While the functor h is not exact, it still gives a morphism of bicategories h : S n Z k Z → w S n Z k Z and the tranformations above give homotopies between jq and id considered asfunctors h : S n Z k Z → w S n Z k Z . Lemma 5.5.
The forgetful functors
Z → D n +1 ( X ∆ k ) and Z → D ( X ∆ k ) induceweak equivalences w S • Z ≃ i S • D n +1 ( X ∆ k ) and w S • Z ≃ i S • D ( X ∆ k ) . Proof.
We will use Qillen’s Theorem A [14] to show that f m : w S m Z → i D n +1 ( X ∆ k )is a weak equivalence by showing that f m /P is contractible for every P = (( Y , P , r ) ֒ → . . . ֒ → ( Y m , P m , r m )) . We will do so by providing an initial object, first in the case m = 1 . In this case P is a single complex with poset P . We can give a heightfunction z : P → { , , } by z ( x ) := x is not minimal0 if x is minimal and belongs to a chain of length n + 1 elseThis provides the initial object. The case m > Z are similar.Finally we can prove: Theorem 5.6.
The simplicial forgetful functor E ( X ∆ • ) → Q ( X ∆ • ) induces a weakequivalence Kx E ( X ∆ • ) ≃ Kw Q ( X ∆ • ) ≃ Ω ∞ Σ ∞ X + . Proof.
For i ≤ j let E ji ( X ∆ • ) be the subcategory of cell complexes with cells onlywith degrees between i and j. Since expansion pairs require cells in different dimen-sions we have E ii ( X ∆ • ) ∼ = D i ( X ∆ • ) , where D i ( X ∆ • ) only contains cells in dimension i. From the discussion above we have x S • E ii ( X ∆ • ) ≃ w S • D i ( X ∆ • ) ≃ i S • D i ( X ∆ • ) ≃ i S • Q ( X ∆ • ) ≃ w SQ ( X ∆ • ) . We also have x S • E ( X ∆ • ) ≃ colim j x S • E j ( X ∆ • ) . So it suffices to show the following lemma:26 emma 5.7.
The inclusion induces a homotopy equivalence x S • E ii ( X ∆ • ) ≃ x S • E ji ( X ∆ • ) for all ≤ i ≤ j. Proof.
Let
B ⊂ E ji ( X ∆ k ) be the subcategory of all ( Y, P, r ) such that P consists onlyof expansion pairs. Furthermore let k E ji ( X ∆ k ) be the subcategory of all cofibrationsin E ij ( X ∆ k ) with quotients in B . Let v E ji ( X ∆ k ) be the subcategory of x E ji ( X ∆ k )of all collapsing maps ( f, α ) such that ker α is a union of expansion pairs and α induces an isomorphism of graded posets when restricted to coim α. Notice that the v -weak equivalences are canonical left-inverses for the k -weak equivalences. This canbe used to show that the v -equivalences do in fact form a category of generalizedequivalences.Let u E ji ( X ∆ k ) be the subcategory of x E ji ( X ∆ k ) of all ( f, α ) where α is a bijection.This again is a category of weak equivalences. We conclude that there is a homotopyfiber sequence i S • B → u S • E ji ( X ∆ k ) → uv S • E ji ( X ∆ k ) , where the latter is the simplicial bicategory given by u S n E ji ( X ∆ k ) v S n E ji ( X ∆ k ) indegree n. We continue to identify the terms of this sequence. First of all, we see that B isequivalent as a category with cofibrations to ( D ) ji +1 ( X ∆ k ) . Consequently we have i S • B ≃ i S • ( D ) ji +1 ( X ∆ k ) . Let ε : E ji ( X ∆ k ) → D ji ( X ∆ k ) be the functor that unpairs all expansion pairs and j the inclusion. We get ε ◦ j = id and there is a natural u -equivalence jε ≃ id , sooverall we learn u S • E ji ( X ∆ k ) ≃ w S • D ji ( X ∆ k ) . Furthermore by Proposition 5.4 above we have w S • D ji ( X ∆ k ) ≃ i S • ( D ) ji ( X ∆ k ) ≃ i S • ( D ) ji +1 ( X ∆ k ) × i S • ( D ) ii ( X ∆ k ) . From the homotopy fiber sequence above we can conclude uv S • E ji ( X ∆ k ) ≃ i S • ( D ) ii ( X ∆ k ) ≃ x S • E ii ( X ∆ k ) . So it remains to show uv S • E ji ( X ∆ k ) ≃ x S • E ji ( X ∆ k ) . Notice that every x -equivalence in E ji ( X ∆ k ) splits naturally as the compositionof a u - and v -equivalence. This carries through to give a splitting x S n E ji ( X ∆ k ) ≃ u S n E ji ( X ∆ k ) v S n E ji ( X ∆ k ) ≃ uv S n E ji ( X ∆ k ) . .2 The Homotopy Type of K h E ( X ∆ • ) We move on to show Kh E ( X ∆ • ) ≃ A ( X ) . We still follow [11]. Again, we will use a supplemental category. However, beforedefining it, we need to lay some ground work defining mapping cylinders in variouscategories as we will be using Waldhausen’s approximation theorem [17].
Definition 5.8.
Given a morphism α : P → P ′ in Q ( X ∆ k ) we define its mappingcylinder T ( α ) := P ∨ P ′ ∨ Σ δP where δP deletes all expansion pairs and Σ increasesevery degree by 1. Furthermore we have z ≤ σx for z ∈ P ′ , σx ∈ Σ δP iff z ≤ α ( x ) . There are obvious maps P ∨ P ′ ֒ → T ( α ) and T ( α ) → P ′ . Definition 5.9.
Let ( f, α ) : (
Y, P, r ) → ( Y ′ , P ′ , r ′ ) be a morphism in E ( X ∆ k ) , then its mapping cylinder is given by ( T ( f ) , T ( α ) , r ′′ ) where T ( f ) is the topologi-cal reduced mapping cylinder and T ( α ) is the mapping cylinder from above. Theretraction r ′′ is given canoincally.One can verify that these define proper cylinder functors on the Waldhausencategories Q ( X ∆ k ) and E ( X ∆ k ) . Definition 5.10.
Let X be a space, we define the Waldhausen category M ( X ∆ k )in the following way: • The objects of M ( X ∆ k ) are ( Y, P, r ) – the same as for E ( X [ k ]) . • A morphism is a pair ( f, α ) : (
Y, P, r ) → ( Y ′ , P ′ , r ′ ) where – α : Λ P → Λ P ′ is a ∨ -preserving map and Λ P is the set of closed subsetsof P – f : Y → Y ′ is a continuous map fixing X × ∆ k and commuting with theretraction such that – f maps Y A into Y α ( A ) • A map ( f, α ) is a cofibration if – α is induced via the inclusion P → Λ P ( x x := { y | y ≤ x } ) by acofibration P → P ′ in Q ( X ∆ k ) – f is a homeomorphism of Y A onto Y α ( A ) for all A ∈ Λ P. • A weak equivalence ( f, α ) is an h -equivalence meaning that f is a homotopyequivalence. Remark . The two main differences between E ( X ∆ k ) and M ( X ∆ k ) are that thelatter has no expansion pairs and more morphisms, as every morphism ( f, α ) in E ( X ∆ k ) is completely determined – if existent – by α. Definition 5.12.
Let ( f, α ) : (
Y, P, r ) → ( Y ′ , P ′ , r ′ ) be a morphism in M ( X ∆ k ) . It’s mapping cylinder is given by ( T ( f ) , T ( α ) , r ′′ ) where T ( α ) = P ∨ P ′ ∨ Σ P. Theprojection T ( α ) → P ′ is given by sending a closed subset A ∨ B ∨ Σ C to α ( A ) ∪ B. Again, T ( f ) is the mapping cylinder. 28his defines a cylinder functor on M ( X ∆ k ) . Lemma 5.13.
We have h S • M ( X ∆ ) ≃ h S • R f ( X ) and hence Kh M ( X ∆ ) ≃ A ( X ) . Proof.
Let M ( X ∆ ) CW be the subcategory of M ( X ∆ ) , where all cells are attachedin order of degree (hence all objects are CW-complexes). This becomes a Wald-hausen subcategory with cylinder functor.We will use Waldhausen’s approximation theorem to show that both inclusions h S • M ( X ∆ ) CW → h S • R f ( X ) and h S • M ( X ∆ ) CW → h S • M ( X ∆ ) are weak equiv-alences. For the former this is straight forward.For the latter let ( f, α ) : ( Y, P, r ) → ( Y ′ , P ′ , r ′ ) be any morphism in M ( X ∆ )such that Y is a CW-complex. By CW-approximation there is a weak equivalence f ′ : ( Y ′′ , r ′′ ) → ( Y ′ , r ′ ) where Y ′′ is a CW complex. Let P be the graded posetof cells of Y ′′ . We define a morphism ( f ′ , α ) : ( Y ′′ , P ′′ , r ′′ ) → ( Y ′ , P ′ , r ′ ) by setting α ( A ) = P ′ for all A ∈ Λ P ′′ . Cellular approximation gives a homotopy approximation f h : ( Y, r ) → ( Y ′′ , r ′′ )to f : Y → Y ′ and we enrich it to ( f h , α h ) : ( Y, P, r ) → ( Y ′′ , P ′′ , r ′′ ) by setting α h ( x ) := { y | deg y ≤ deg x } . Now one can see that ( f, α ) factors as(
Y, P, r ) ֒ → ( T ( f h ) , T ( α h ) , r ) → ( Y ′ , P ′ , r ′ ) . Lemma 5.14.
The degenerate inclusion induces a weak equivalence h S • M ( X ∆ ) → h S • M ( X ∆ • ) . Proof.
It is enough to show h S • M ( X ∆ ) → h S • M ( X ∆ k )for all k. Call the degeneracy operator S : M ( X ∆ ) → M ( X ∆ k ) . We will use theapproximation theorem again and show that S satisfies the approximation property.Let ( f, α ) : ( Y, P, r ) → ( Z, Q, s ) be a morphism in M ( X ∆ k ) where ( Y, P, r ) isdegenerate. Denote the restriction to the first vertex by ( f , α ) . Let ( T ( f ) , T ( α ))be the mapping cone of ( f , α ) . Then ( f, α ) factors as( Y , P , r ) ֒ → ( T ( f ) , T ( α ) , r ) ≃ −→ ( Z , Q , s ) . Recall that Y = S ( Y ) = Y × ∆ k and P = P . Now we can cellularly expand theconstruction above to get a homotopy equivalence h : T ( f × ∆ k ) → Z. This is rooted in the fact that any Z B → ∆ k is a Serre fibration. Further noticethat ( T ( f × ∆ k ) , T ( α ) , r ) ∈ S ( M ( X ∆ k )) and this concludes the proof that S hasthe approximation property. 29 emma 5.15. The simplicial forgetful functor induces a homotopy equivalence h S • E ( X ∆ k ) → h S • M ( X ∆ k ) . Proof.
Denote the forgetful functor by ǫ k . We will again show that it has the ap-proximation property. Let ǫ k : ( Y, P, r ) → ( Z, Q, s ) be a morphism in M ( X ∆ k ) . Let A be the closed subset of T ( α ) given by deleting all expansion pairs from Σ P. Then T ( f ) A ≃ T ( f ) ≃ Z and the cofibration ǫ k ( Y, P, r ) ֒ → ( T ( f ) A , A, r )lifts to E ( X ∆ k ) . Altogether we have shown
Theorem 5.16.
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