Smooth parametrized torsion -- a manifold approach
SSMOOTH PARAMETRIZED TORSIONA MANIFOLD APPROACH
BERNARD BADZIOCH, WOJCIECH DORABIA(cid:32)LA, AND BRUCE WILLIAMS
Abstract.
We give a construction of a torsion invariant of bundles of smoothmanifolds which is based on the work of Dwyer, Weiss and Williams on smoothstructures on fibrations. Introduction . Recently there has been considerable interest and activity related to the con-struction and the computations of parametrized torsion invariants. The goal hereis the development of a generalization of Reidemeister torsion – which is a classicalsecondary invariant of CW-complexes – to bundles of manifolds. One approach tothis problem was proposed by Bismut and Lott [2]. Another, using parametrizedMorse functions resulted from the work of Igusa [6] and Klein [8].In [5] Dwyer, Weiss, and Williams presented yet another definition of torsionof bundles whose main feature is that it is described entirely in terms of algebraictopology . As a result their construction is quite intuitive. Given a smooth bundle p : E → B we can consider the Becker-Gottlieb transfer map p ! : B → Q ( E + ).If ρ : M → E is a locally constant sheaf of R -modules we can construct a map c ρ : B → K ( R ) which assigns to a point b ∈ B the point of K ( R ) represented bythe singular chain complex C ∗ ( F b , ρ | F b ) of the fiber of p over b with coefficient in ρ . One of the results of [5] implies that there exists a map λ : Q ( E + ) → K ( R ) suchthat the diagram Q ( E + ) λ (cid:15) (cid:15) B p ! (cid:60) (cid:60) (cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121) c ρ (cid:47) (cid:47) K ( R )commutes up to a preferred homotopy. If the sheaf ρ is such that all chain complexes C ∗ ( F b , ρ | F b ) are acyclic, then the map c ρ is canonically homotopic to the constantmap. Thus we obtain a lift of p ! to the space Wh ρ ( E ) which is the homotopy fiberof λ over the trivial element of K ( R ). This lift τ ρ ( p ) is the smooth torsion of thebundle p .In order to see what kind of information about a bundle is carried by its torsionlets assume first that we are given two smooth bundles p i : E i → B , i = 1 , f : E → E . Let A ( E i ) denote the Waldhausen A -theory of thespace E i . We have the assembly maps a : Q ( E i + ) → A ( E i ) (see Section 3) which Date : 11/11/2007. a r X i v : . [ m a t h . A T ] N ov B. BADZIOCH, W. DORABIA(cid:32)LA, AND B. WILLIAMS fit into a commutative square Q ( E ) f ∗ (cid:47) (cid:47) a (cid:15) (cid:15) Q ( E ) a (cid:15) (cid:15) A ( E ) f ∗ (cid:47) (cid:47) A ( E )Let p Ai : B → A ( E i ) be the composition p Ai = ap ! i . One of the implications of [5]is that if f is a fiberwise homotopy equivalence then we can construct a homotopy ω f : B × I → A ( E ) joining f ∗ p A with p A . Moreover, this homotopy can be liftedto a homotopy joining f ∗ p !1 with p !2 in Q ( E ) provided that f is homotopic to adiffeomorphism of bundles. The problem of lifting ω f defines then an obstructionto replacing f by a diffeomorphism. This obstruction closely resembles the classicalWhitehead torsion of a homotopy equivalence of finite CW-complexes.Lets return now to the case of a single bundle p : E → B with fiber M . Themap λ : Q ( E ) → K ( R ) factors though the assembly map so we get a commutativediagram Q ( E + ) a (cid:15) (cid:15) A ( E ) (cid:15) (cid:15) B p ! (cid:67) (cid:67) p A (cid:60) (cid:60) (cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121) c ρ (cid:47) (cid:47) K ( R )If we would have a homotopy equivalence f : M × B → E from the product bundleover B to p then the homotopy ω f would give us a contraction of p A (and thus alsoa contraction of c ρ ) to a constant map. The construction of the smooth torsiontakes advantage the fact that under some homological conditions the map c ρ iscontractible even if we do not have a homotopy equivalence f . In this case thecontraction of c ρ can be still interpreted as a way of relating the bundle p tothe product bundle on the level of the K -theory. Viewed from this perspective thesmooth torsion τ ρ ( p ) is an obstruction to the existence of a diffeomorphism between p and the product bundle. In effect we can think of it as a linearized Whiteheadtorsion. This parallels the way the classical Reidemeister torsion is interpreted inthe setting of CW-complexes.While the idea of the construction of smooth torsion is simple to explain thetechnical details are rather involved. One of the main problems is that the targetof the transfer p ! and the domain of the map λ as described in [5] are different andare only linked by a zigzag of weak equivalences. As a result the torsion of a bundleis not really defined uniquely but rather up to a contractible space of choices. Thismakes this construction of torsion rather inaccessible to direct computations. Infact, at present there are no examples of bundles for which the smooth torsion doesnot vanish, even though such examples abound for Bismut-Lott and Igusa-Kleintorsions, and intuitively the smooth torsion of Dwyer-Weiss-Williams should be amore delicate invariant. The last sentence points out another problem with theDwyer-Weiss-Williams construction: at present there are no known results relatingit to the other definitions of torsion of smooth bundles. MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 3
An additional difficulty with the construction of torsion as described above isthat it depends on existence of the local system of coefficients ρ yielding acyclicityof fibers of p . Such systems of coefficient are not easy to construct.One the goals of this note is to show how these problems can be resolved. First,we substantially simplify the construction of smooth torsion using Waldhausen’smanifold approach to the Q -construction [10], [11]. This idea is not entirely original– in [5] the authors sketch it briefly at the very end of the paper and attribute itto Waldhausen. Our aim, however, is to develop it to the extent which wouldpermit us to study properties of the smooth torsion. In addition we show thatsmooth torsion can be defined even if the fibers of p are not acyclic, as long as thefundamental group of the base space acts trivially (or even unipotently – see 6.2)on the homology groups of the fibers. This demonstrates that smooth torsion existsfor a broad class of bundles.We also aim to bring the smooth torsion of Dwyer-Weiss-Williams closer to theBismut-Lott and the Igusa-Klein constructions. The starting point here is thepaper [7] of Igusa which describes a set of axioms for torsion of smooth bundles.Igusa shows that any notion of torsion satisfying these axioms must coincide withthe Igusa-Klein torsion up to some scalar constants. In Igusa’s setting, torsion isan invariant defined for all smooth unipotent bundles – the condition which as wementioned above turns out to be satisfied by the smooth torsion. This invariantis supposed to take its values in the cohomology groups H k ( B, R ) of the basespace of the bundle. We show that the smooth torsion can be reduced to such acohomological invariant. What remains to be verified is that the cohomology classeswe produce satisfy Igusa’s axioms. This is the goal which the present authors incollaboration with John Klein plan to complete in a future paper.1.2 . Organization of the paper. As we have mentioned above our main tool inthis paper is the construction of the space Q ( X + ) using the language of “partitions”given by Waldhausen in [10], [11]. We start by summarizing this construction inSection 2. In § Q ( X + ) to Waldhausen’s A -theory of the space X . While this map is notour main interest here, we will use it in Section 4 to show that a certain mapwe construct there coincides with the Becker-Gottlieb transfer p ! : B → Q ( E + ) ofa bundle p : E → B. In Section 5 the assembly map is used again to constructthe linearization map λ : Q ( B + ) → K ( R ). In § § E → B defines certain cohomology classes in H k ( B ; R ) , for k > Waldhausen’s manifold approach . Partitions.
Let I denote the closed interval [0 , X with boundary ∂X a partition of X × I is a triple ( M, F, N ) where M and N arecodimension 0 submanifolds of X × I such that M ∪ N = X × I and F = M ∩ N is a submanifold of codimension 1. Moreover, we assume that M contains X × { } and is disjoint from X × { } , and finally, that for some open neighborhood U of ∂X in X the intersection of F with U × I coincides with U × { t } for some t ∈ I .While this description reflects the basic properties of partitions we will make somefurther technical assumptions which will make it easier to work with them: • we will assume that the value of t if fixed once for all partitions (say, t = ); B. BADZIOCH, W. DORABIA(cid:32)LA, AND B. WILLIAMS • we will assume that X × [0 , ] ⊆ M for any partition ( M, N, F ).In the language of [10] partitions satisfying the last two conditions are called lowerpartitions.While the idea is all components -
M, N, F - of a partition should be smooththis condition is too rigid for the constructions we will
XM NF0t1 want to perform. Following Waldhausen [10, Appendix]we will assume that these are just topological manifoldsbut we also assume that there is a preferred smooth vectorfield s on X × I which is normal to F in the followingsense. Given any smooth chart of X × I containing x ∈ F there are constants c, C > | r | ≤ C thedistance function satisfies the inequality d ( x + rs ( x ) , F ) ≥ c | r | This just means that the line passing through x and going in the direction of s ( x )stays well away from F .The existence of such a vector field s implies that the manifold F admits asmoothing which is obtained by sliding points of F along the integral curves of s .Moreover, the space of smoothings of F which can be obtained in this way is con-tractible, so we can think of the quadruple ( M, N, F, s ) as describing an essentiallyunique smooth partition. We again list some additional technical assumptions: • we will assume that on U × I the vectors of s are the unit vectors pointingupward, in the direction of I ; • we will also assume that for x ∈ X × { } the component of s ( x ) tangentto the interval I is a non-zero vector pointing upward.Clearly a partition ( M, N, F, s ) is determined by the manifold M and the vectorfield s . In order to simplify notation we will write ( M, s ) instead of (
M, N, F, s ).For a manifold Y we have the notion of a locally trivial family of partitions of X × I parametrized by Y . By this we mean a pair ( ¯ M , ¯ s ) where ¯ M ⊆ X × I × Y and s is a smooth vector field on X × I × Y such that • for each y ∈ Y the pair ( ¯ M ∩ X × I × { y } , ¯ s | X × I ×{ y } ) is a partition of X × I × { y }• for every y ∈ Y there is an open neighborhood y ∈ V ⊆ Y , a partition( M, s ) of X × I , and a diffeomorphism ϕ : X × I × V → X × I × V such that p V ◦ ϕ = p V where p V : X × I × V → V is the projection map, ϕ is the identity map on an appropriate neighborhood of ∂ ( X × I ) × V , ϕ ( M × V ) = ¯ M ∩ ( X × I × V ) and Dϕ ( s ) = ¯ s .If ( ¯ M , ¯ s ) is a partition of X × I parametrized by Y and f : Z → Y is a smoothfunction then we obtain the induced partition f ∗ Y parametrized by Z .Denote by P k ( X ) the set of all partitions parametrized by the k -simplex ∆ k .These sets can be assembled to form a simplicial set P • ( X ).2.2 . Stabilization. The set P ( X ), which can be identified with the set of allpartitions of X × I , has a partial monoid structure defined as follows: given twopartitions ( M , s ) and ( M , s ) in P ( X ) such that M ∩ M = X × [0 , ] we set( M , s ) + ( M , s ) := ( M ∪ M , s ). We can extend this definition to P k ( X ) for all k ≥ P • ( X ). In order to define MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 5 addition for all partitions (and thus define an H -space structure on P • ( X )) we needto introduce stabilization of partitions (called lower stabilization in [10]). It is amap of simplicial sets σ : P • ( X ) → P • ( X × J ) where J = [0 , M, s ) ∈ P ( X ) we set σ ( M, s ) = ( σ ( M ) , σ ( s )) where σ ( M ) = X × J × [0 , ] ∪ M × [ , ]. X I X IJ s In order to define the vector field σ ( s ), fix a smooth vector field s (cid:48) on the interval J such that s (cid:48) is non-zero at the points and and is zero on some neighborhoodof ∂J . For ( x, t, t (cid:48) ) ∈ X × I × J we then set σ ( s )( x, t, t (cid:48) ) := s ( x, t ) + s (cid:48) ( t )We note that the vector field σ ( s ) does not really satisfy all assumptions of ourdefinition of a partition since it is not a unit vector field pointing in the directionof I when restricted to a neighborhood of ∂ ( X × J ) × I . This however can be easilyfixed.In a similar way we can define stabilization maps σ : P k ( X ) → P k ( X × J ) forall k > σ : P • ( X ) → P • ( X × J ). Noticethat given any two partitions it is always possible to slide their stabilizations awayfrom each other along J so that the sum is defined. In this way the partial monoidstructure becomes a monoid structure on colim m P • ( X × I m ).2.3 . Group completion. The Waldhausen manifold model for Q ( X + ) can be ob-tained as a group completion of the simplicial monoid colim m P • ( X × I m ). In orderto describe this group completion one can use Thomason’s variant of Waldhausen’s S • -construction (see [9], p.343). Let T n P ( X ) denote the category whose objectsare ( n + 1)-tuples { ( M i , s i ) } ni =0 such that ( M i , s i ) ∈ P ( X ), s i = s j for all i, j andthat we have inclusions M ⊆ M ⊆ · · · ⊆ M n In T n P ( X ) we have a unique morphism { ( M i , s i ) } → { ( M (cid:48) i , s (cid:48) i ) } if and only if M (cid:48) ∩ M i ⊆ M and M (cid:48) i = M i ∪ M (cid:48) for all i ≥
0. Analogously, for any n, k ≥ T n P k ( X ) whose objects are increasing sequences of length n in P k ( X ). For any fixed n the categories T n P k ( X ) assemble to give a sim-plicial category T n P • ( X ). We have functors d j : T n +1 P • ( X ) → T n P • ( X ) and s j : T n P • ( X ) → T n +1 P • ( X ) which are obtained by removing (or respectively re-peating) the j -th element of the sequence { ( M i , s i ) } . In this way T • P • ( X ) becomesa simplicial object in the category of simplicial categories. Let | T • P • ( X ) | denotethe space obtained by first taking nerve of each of the categories T n P • ( X ) and thenapplying geometric realization to the resulting bisimplicial set. Notice that we havea cofibration | T P • ( X ) | → | T • P • ( X ) | . One can check that the space | T P • ( X ) | iscontractible, and so we get a weak equivalence | T • P • ( X ) | (cid:39) | T • P • ( X ) | / | T P • ( X ) | B. BADZIOCH, W. DORABIA(cid:32)LA, AND B. WILLIAMS
By abuse of notation from now on we will denote by | T • P • ( X ) | the quotient spaceon the right. The advantage of this modification is that | T • P • ( X ) | has now acanonical choice of a basepoint.Since everything we did so far behaves well with respect to the stabilization mapswe obtain the induced maps of spaces σ : | T • P • ( X × I m ) | → | T • P • ( X × I m +1 ) | Passing to the homotopy colimit we get2.4.
Theorem (Waldhausen[11]) . There is a weak equivalence
Ω hocolim m | T • P • ( X × I m ) | (cid:39) Q ( X + )In view of this result from now on by Q ( X + ) we will understand the space onthe left hand side of the above equivalence.2.5. Remark.
The following observation will be useful for constructing maps into Q ( X + ). Notice that we have a map | T P • ( X ) | × ∆ → | T • P • ( X ) | After stabilizing the right hand side and taking the adjoint we obtain a map | T P • ( X ) | → Q ( X + ). Thus any map into the nerve of the category T P • ( X )naturally yields a map into Q ( X + ).3. The assembly map
Waldhausen’s motivation for constructing the space Q ( X + ) in the way sketchedin the last section was to relate this space to A ( X ) – the A -theory of the space X .Since we will need to use this relationship later in this paper, we now describe amap a : Q ( X + ) → A ( X ) which we will call the assembly map.The simplest way of constructing the space A ( X ) is to start with the category R fd ( X ) whose objects are homotopy finitely dominated retractive spaces over X and whose morphisms are maps of retractive spaces. The category R fd ( X ) is aWaldhausen category in the sense of [9, Definition 1.2] with cofibrations given bySerre cofibrations and weak equivalences defined as weak homotopy equivalences.It follows that we can turn it into a simplicial category T • R fd ( X ) using again theThomason’s variant of Waldhausen’s S • -construction. We set A ( X ) := Ω( | T • R fd ( X ) | / | T R fd ( X ) | )In order to obtain a direct map from our model of Q ( X + ) this construction needsto be modified somewhat. First, for k ≥ R fdk ( X )whose objects are locally homotopy trivial families of retractive spaces over X parametrized by the simplex ∆ k . These categories taken together form a simplicialcategory R fd • ( X ). Analogously as we did in the case of categories of partitions wecan define stabilization functors σ : R fd • ( X ) → R fd • ( X × I )in the following way: if Y is a retractive space over X then σ ( Y ) := Y × [ 13 ,
23 ] ∪ X × [ , ] X × I MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 7
Applying the T • -construction to R fd • ( X ) we get a bisimplicial category T • R fd • ( X ).Define A p ( X ) := hocolim m | T • R fd ( X × I m ) | / | T R fd ( X × I m ) | Notice that if X is a smooth manifold and if ( M, s ) is a partition of X × I then a ( M ) := M ∩ ( X × [ , X . Theassignment ( M, s ) (cid:55)→ a ( M ) extends to a functor of simplicial categories a : P • ( X ) → R fd • ( X ) which commutes with the stabilization functors. This induces a map a : Q ( X + ) → A p ( X )Since the category R fd ( X ) is isomorphic to R fd ( X ) we have a functor i : R fd ( X ) → R fd • ( X ) which induces a map i : A ( X ) → A p ( X ). We have3.1. Theorem (Waldhausen, [10, Lemma 5.4]) . The map i : A ( X ) → A p ( X ) is ahomotopy equivalence. While this result says that we are not changing much by replacing A ( X ) with A p ( X ), it will be convenient to have an assembly map whose codomain is A ( X ). Inorder to get such a map define ˜ Q ( X + ) to be the homotopy pullback of the diagram(1) A ( X ) (cid:47) (cid:47) A p ( X ) Q ( X + ) a (cid:111) (cid:111) In view of Theorem 3.1 we have ˜ Q ( X + ) (cid:39) Q ( X + ), and ˜ Q ( X + ) comes equippedwith a map ˜ a : ˜ Q ( X + ) → A ( X ). 4. Transfer
Going back to the diagram on page 1 we see that if we want to describe thesmooth torsion of a bundle p : E → B we need to construct the transfer map p ! : B → Q ( E + ) (or rather p ! : B → ˜ Q ( E + )) and the linearization map λ : ˜ Q ( E + ) → K ( R ). We deal with the transfer in this section and with the linearization in thenext one.Let p : E → B be a smooth bundle of manifolds with B and E compact. Denoteby T v E the subbundle of the tangent bundle T E consisting of vectors tangentto the fibers of p . By choosing a Riemannian metric on T E we can identify thebundle p ∗ T B with the subbundle of
T E which is the orthogonal complement of T v E . Using this identification given the map p × id : E × I → B × I we can considerthe bundle ( p × id) ∗ T ( B × I ) as a subbundle of T ( E × I ). As a consequence anysection s : B × I → T ( B × I ) will define a section ( p × id) ∗ s of the bundle T ( E × I ).Assume for a moment that fibers of p are closed manifolds. In this case given apartition ( M, s ) ∈ P ( B ) the pair (( p × id) − M, ( p × id) ∗ s ) defines a partition of E × I , so we get a map of sets Q ( p ! ) : P ( B ) → P ( E ). Since this map preservesthe partial ordering of partitions we in fact obtain a functor T P ( B ) → T P ( E ).In the same way we can define functors T n P k ( B ) → T n P k ( E ) for all k, n ≥ Q ( p ! ) : | T • P • ( B ) | → | T • P • ( E ) | Since Q ( p ! ) is compatible with stabilization we get a map Q ( p ! ) : Q ( B + ) → Q ( E + ).If fibers of the bundle p are manifolds with boundary we need to modify the aboveconstruction slightly so that for a partition ( M, s ) ∈ P ( B ) the element Q ( p ! )( M, s )behaves nicely near ∂E × I . This can be done as follows. Let ∂ v E be the subspace B. BADZIOCH, W. DORABIA(cid:32)LA, AND B. WILLIAMS of E consisting of all boundary points of the fibers of p . If F is a fiber of p then p | ∂ v E : ∂ v E → B is a subbundle of p with fiber ∂F .For b ∈ B let F b denote the fiber of p over b . We can find an open neighborhood U ⊆ E in such way that for all b ∈ B the intersection U ∩ F b is a collar neighborhoodof ∂F b in F b [1, p.590]. For a partition ( M, s ) ∈ P ( M ) let Q ( p ! )( M ) ⊆ E × I begiven by Q ( p ! )( M ) := U × [0 ,
13 ] ∪ (( p × id) − ( M ) ∩ (( E \ U ) × I ))We need then to modify the vector field ( p × id) ∗ s so that it is normal to Q ( p ! )( M ).This can be done in a way similar to the one we used to define stabilization ofpartitions.In a similar way given a bundle p : E → B we can construct maps A ( p ! ) : A ( B ) → A ( E ) and A p ( p ! ) : A p ( B ) → A p ( E ). Each of these maps is induced by a functor ofcategories of retractive spaces which assigns to a retractive space over B its pullbackalong p (in order to make this compatible with the construction of Q ( p ! ) we needto modify these pullback slightly in a neighborhood of the boundaries of fibers of p ). These three maps induce in turn a map of homotopy pullbacks˜ Q ( p ! ) : ˜ Q ( B + ) → ˜ Q ( E + )The map p ! : B → ˜ Q ( E + ) will be obtained as the composition of ˜ Q ( p ! ) with acoaugmentation map η : B → ˜ Q ( B + ) which we describe below. For simplicity wewill assume first that B is a closed manifold.Let S ( B ) denote the simplicial set of singular simplices of B . It will be conve-nient to consider it as a simplicial category with identity morphisms only on eachsimplicial level. We have a weak equivalence B (cid:39) | S ( B ) | , so it will suffice to con-struct a map | S ( B ) | → ˜ Q ( B + ). Recall that ˜ Q ( B + ) was defined as the homotopypullback of the diagram (1). Notice also that we have a commutative diagram | T R fd ( B ) | (cid:47) (cid:47) (cid:15) (cid:15) | T R fd • ( B ) | (cid:15) (cid:15) | T P • ( B ) | (cid:111) (cid:111) (cid:15) (cid:15) A ( B ) (cid:47) (cid:47) A p ( B ) Q ( B + ) (cid:111) (cid:111) where the vertical maps are obtained as in Remark 2.5. It will then suffice todefine a map from | S ( B ) | to the homotopy limit of the upper row of this diagram.This, in turn, can be accomplished by specifying functors S ( B ) → T R fd ( B ) and S ( B ) → T P • ( B ) and a zigzag of natural transformations joining these functors in T R fd • ( B ). A minor inconvenience here is the fact that T P • ( B ), T R fd • ( B ) and S ( B ) are simplicial categories while R fd ( B ) is not, but we can think about R fd ( B )as of a constant simplicial object in the category of small categories.We will start then with a diagram of categories T R fd ( B ) i (cid:47) (cid:47) T R fd • ( B ) T P • ( B ) a (cid:111) (cid:111) MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 9 and we will extend it to a diagram S ( B ) η R (cid:120) (cid:120) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113) η p R (cid:15) (cid:15) η P (cid:38) (cid:38) (cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76) T R fd ( B ) i (cid:47) (cid:47) T R fd • ( B ) T P • ( B ) a (cid:111) (cid:111) such that the two triangles of functors commute up to natural transformations.The functor η R : S ( B ) → T R fd ( B ) is defined as follows: given a singular simplex σ : ∆ k → B consider the retractive space B (cid:116) ∆ k over B . We set η R ( σ ) to be thecofibration of retractive spaces B (cid:44) → B (cid:116) ∆ k .In order to define the functor η P : S ( B ) → T P • ( B ) fix a Riemannian metricon the tangent bundle T B . Choose (cid:15) >
T B (cid:15) consisting of vectors of
T M of length ≤ (cid:15) has the property that the exponential mapexp : T B (cid:15) → M restricted to each fiber of T B (cid:15) is a diffeomorphism onto its image.Given a singular simplex σ : ∆ k → B consider the induced disc bundle σ ∗ T B (cid:15) over∆ k . The exponential map gives a map of bundles σ ∗ T B (cid:15) exp (cid:47) (cid:47) (cid:35) (cid:35) (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71) B × ∆ k (cid:123) (cid:123) (cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118) ∆ k which is a fiberwise embedding. Fix numbers a, b such that < a < b <
1. Wehave a fiberwise embedding of fiber bundles over ∆ k :( σ ∗ T B (cid:15) × [ a, b ]) ∪ ( B × [0 , ] × ∆ k ) (cid:47) (cid:47) (cid:41) (cid:41) (cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84)(cid:84) B × I × ∆ k (cid:121) (cid:121) (cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116) ∆ k This fiberwise embedding defines a family of partitions parametrized by ∆ k . Weset η P ( σ ) to be the inclusion of families of partitions B × [0 ,
13 ] × ∆ k (cid:44) → ( σ ∗ T B (cid:15) × [ a, b ]) ∪ ( B × [0 ,
13 ] × ∆ k )Next, we need to define the functor η p R : S ( B ) → T R fd • ( B ). For a singularsimplex σ : ∆ k → B we have the mapid ∆ k (cid:116) pr ∆ k : ∆ k (cid:116) B × ∆ k → ∆ k For each t ∈ ∆ k the fiber of this map over t is in a natural way a retractive spaceover B , so we can think of ∆ k (cid:116) B × ∆ k as of a family of retractive spaces over B parametrized by ∆ k . The functor η p R is given by the assignment η p R ( σ ) := ( B × ∆ k (cid:44) → ∆ k (cid:116) B × ∆ k )In order to describe a natural transformation from η p R to iη R notice that for σ : ∆ k → B we obtain iη R ( σ ) by taking the retractive space η R ( σ ) = ∆ k (cid:116) B andmultiplying it by ∆ k which makes it into a retractive space over B parametrizedby ∆ k . The natural transformation is then defined by the maps η p R ( σ ) = ∆ k (cid:116) B × ∆ k → (∆ k × ∆ k ) (cid:116) ( B × ∆ k ) = (∆ k (cid:116) B ) × ∆ k = iη R ( σ ) which restrict to the identity map on B × ∆ k and which send x ∈ ∆ k to ( x, x ) ∈ ∆ k × ∆ k . The natural transformation from aη P to η p R is easy to define.If B is a manifold with a boundary we need to modify this construction somewhatso that the values of the functor η P • are still partitions. This can be done bychoosing an open collar neighborhood U of the boundary of B . On B \ U the map η can be now defined exactly as before. We then extend it to B by composing itwith the retraction B → B \ U .4.1. Proposition. If p : E → B is a smooth fibration then the map p ! := ˜ Q ( p ! ) ◦ η : B → Q ( E + ) is the Becker-Gottlieb transfer of p .Proof. The composition ˜ a ◦ p ! is homotopic to χ h ( p ) - the homotopy Euler charac-teristic of the bundle p as defined in [4]. By [4, Thm. 3.12] we obtain that tr ◦ χ h ( p )is the Becker-Gottlieb transfer where tr : A ( E ) → ˜ Q ( E + ) is the Waldhausen’s tracemap [11]. By [11] we have tr ◦ ˜ a ∼ id ˜ Q ( E + ) , sotr ◦ χ h ( p ) ∼ tr ◦ ˜ a ◦ p ! ∼ p ! (cid:3) Linearization
For a ring R let C h ( R ) denote the category of finitely homotopy dominatedchain complexes of projective R -modules. The category Ch ( R ) can be equippedwith a Waldhausen model category structure with degreewise monomorphisms ascofibrations and quasi isomorphisms as weak equivalences. Applying Waldhausen’s S • -construction we obtain a simplicial category S • Ch ( R ). The associated space K ( R ) = Ω( | S • Ch ( R ) | ) is homotopy equivalent to the infinite loop space underlyingthe K -theory spectrum of the ring R .Let X be a space and let ρ : M → X be a locally constant sheaf of finitelygenerated projective R -modules. As in [5, p.40] we notice that we have a functor λ R ρ : R fd ( X ) → Ch ( R ) which assigns to every retractive space Y ∈ R fd ( X ) therelative singular chain complex of C ∗ ( Y, X, ρ ) with coefficients in ρ . This func-tor induces a map λ R ρ : A ( X ) → K ( R ). Recall that for a smooth manifold X we constructed the assembly map ˜ a : ˜ Q ( X + ) → A ( X ). By the linearization map λ ρ : ˜ Q ( X ) → K ( R ) we will understand the composition λ R ρ ◦ ˜ a .6. Smooth torsion
We are now in position to define smooth Reidemeister torsion of a bundle ofmanifolds. We will do it under two different sets of assumptions, one replicatingthe conditions of [5], and the other conforming to the axiomatic setup of Igusa[7]. We note, however, that the idea underlying both constructions is essentiallythe same: given a bundle p : E → B and a sheaf of R -modules ρ : M → E wehave constructed maps p ! : B → ˜ Q ( E + ) and λ ρ : ˜ Q ( E + ) → K ( R ). Consider thecomposition λ ρ ◦ p ! : | S ( B ) | → K ( R )Under certain conditions on the bundle p and the sheaf ρ this map is homotopic toa constant map via a preferred homotopy. As a consequence we obtain a lift of p !MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 11 to a map τ sρ ( p ) : B → hofib( ˜ Q ( E + ) → K ( R )). This lift is the smooth torsion of thebundle p .6.1 . Acyclicity. Assume that we are given a sheaf ρ such that for any b ∈ B wehave H ∗ ( F b ; ρ ) = 0 where F b is the fiber of p over b . Notice that the map λ ρ ◦ p ! comes from a functor S ( B ) → Ch ( R ) which assigns to each simplex σ : ∆ k → B the relative chain complex C ∗ ( σ ∗ E (cid:116) E, E ; ρ ) where σ ∗ E denotes the pullback ofthe diagram ∆ k σ (cid:47) (cid:47) B E p (cid:111) (cid:111) Vanishing of homology groups of the fibers of p implies that all these chain com-plexes are acyclic, and thus the maps C ∗ ( σ ∗ E (cid:116) E, E ; ρ ) → S ( B ) to the zero chain complex. On the level of spaces this natural trans-formation defines a homotopy ω ρ : | S ( B ) | × I → K ( R ). Denote by Wh ρ ( E ) thehomotopy fiber of the linearization map λ ρ taken over the basepoint of K ( R ) rep-resented by the zero chain complex. The smooth torsion of the bundle p is the map τ sρ ( p ) : | S ( B ) | → Wh ρ ( E ) determined by the transfer p ! together with the homotopy ω ρ .6.2 . Unipotent bundles. Let F be a field and let p : E → B be a bundle such that B is a connected manifold with a basepoint b ∈ B . Assume that the fundamentalgroup π ( B, b ) acts trivially on the homology H ∗ ( F b ; F ) of the fiber over b .Consider the map λ ρ : ˜ Q ( E + ) → K ( F ) associated with the constant sheaf of 1-dimensional vector spaces over F . In this case the composition λ ρ ◦ p ! assigns toa simplex σ the chain complex C ∗ ( σ ∗ E (cid:116) E, E ; F ). We will construct a sequenceof homotopies joining this map with the constant function which maps the wholespace | S ( B ) | to H ∗ ( F b ; F ), which we will consider as a chain complex with thetrivial differentials.Let wCh ( F ) denote the subcategory of Ch ( F ) with the same objects as Ch ( F ),but with quasi-isomorphisms as morphisms. We have the canonical map k : | wCh ( F ) | → K ( F )Notice that the map λ ρ ◦ p ! admits a factorization | S ( B ) | λ ρ ◦ p ! (cid:47) (cid:47) (cid:37) (cid:37) (cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75) K ( F ) | wCh ( F ) | k (cid:57) (cid:57) (cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116) We have6.3.
Lemma ([5, Prop. 6.6]) . Let H : | wCh ( F ) | → K ( F ) be the map which assigns toeach chain complex C its homology complex H ∗ ( C ) . There is a preferred homotopy k (cid:39) H Proof.
For a chain complex C = ( . . . (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C (cid:47) (cid:47) P q C denote the complex such that ( P q C ) i = 0 for i > q + 1, ( P q C ) q +1 = ∂ ( C q +1 ), and ( P q C ) i = C i for i ≤ q . Let Q q C be the kernel of the map P q C → P q +1 C . We obtain cofibration sequences Q q C → P q C → P k − C functorial in C . Notice that the complex Q q C is quasi-isomorphic to its homologycomplex H ∗ ( Q q C ) and this last complex has only one non-zero module H q ( C )in the degree q . By Waldhausen’s additivity theorem we obtain that the map P q : | wCh ( F ) | → K ( F ) which assigns to C ∈ wCh ( F ) the chain complex P q C ishomotopic to the map H q : | wCh ( F ) | → K ( F ) sending C to P q − C ⊕ H ∗ ( Q q C ).Iterating this argument we see that for each q the map P q is homotopic to the mapwhich assigns to a complex C the chain complex (cid:76) qi =0 H i ( C ). Since C = lim q P q C we obtain the statement of the lemma. (cid:3) As a consequence of the lemma we get a homotopy between λ ρ ◦ p ! and the mapwhich assigns to a simplex σ the homology chain complex H ∗ ( σ ∗ E (cid:116) E, E ; F ). Sincethis last chain complex is isomorphic to the chain complex H ∗ ( σ ∗ E ; F ) we obtaina homotopy from λ ρ ◦ p ! to the map represented by a functor v : S ( B ) → Ch ( R )given by v ( σ ) = H ∗ ( σ ∗ E ; F ).Next, let v : S ( B ) → Ch ( R ) denote the functor this assigns to each singularsimplex σ the complex H ∗ ( F σ (0) ; F ) where F σ (0) is the fiber of the bundle p takenover the zero vertex of σ . The isomorphisms H ∗ ( F σ (0) ; F ) → H ∗ ( σ ∗ E ; F ) form anatural transformation of functors v and v . Finally, given any point b ∈ B choose apath joining this point to the basepoint b . Lifting it to the space E we can producea homotopy equivalence F b → F b . The map which it induces on the homologygroups will not depend on the choice of the path by our assumption the π ( B ) actstrivially on the homology of the fibers. The maps H ∗ ( F σ (0); F ) → H ∗ ( F b ; F ) yieldthe natural transformation from v to the constant functor.On the level of spaces the natural transformations of functors we described abovedefine a homotopy from the map λ ρ ◦ p ! : | S ( B ) | → K ( F ) to the constant map whichmaps | S ( B ) | to the point of K ( F ) represented by the chain complex H ∗ ( F b ; F ).This homotopy taken together with the transfer map p ! : | S ( B ) | → ˜ Q ( E + ) defines amap ˜ τ F ( p ) : | S ( B ) | → hofib( ˜ Q ( E + ) → K ( F )) H ∗ ( F b , F ) . We will call this element theunreduced Reidemeister torsion of the bundle p . The obvious inconvenience of thisdefinition is that changing a basepoint in B changes the target of the map ˜ τ F ( p ).This can be fixed by shifting this map so it takes values in the space Wh F ( E ) :=hofib( ˜ Q ( E + → K ( F )) – the homotopy fiber taken over the zero chain complex.Since both Q ( E + ) and K ( F ) are infinite loop spaces this shift can be accomplishedby subtracting the element p ! ( b ) from the map p ! and subtracting H ∗ ( F b ; F ) fromthe contracting homotopy | S ( B ) | × I → K ( F ). One could make it more explicit byconstructing models for inverses of elements in ˜ Q ( E + ) and K ( F ). This is not hardto do. The new map τ F ( p ) : | S ( B ) | → Wh F ( E ) is the (reduced) torsion of p .The above construction can be also carried out under more general conditionswhich conform to the setting of [7]. We will say that a smooth bundle p : E → B is unipotent if the homology groups H ∗ ( F b ; F ) admit a filtration0 = V ( F b ) ⊆ V ( F b ) ⊆ . . . V k ( F b ) = H ∗ ( F b ; F )such that π ( B ) acts trivially on the quotients V i /V i − . In this case consider thefunctor v : S ( B ) → Ch ( F ) defined above. Waldhausen’s additivity theorem impliesthat the map v : | S ( B ) | → K ( R ) is canonically homotopic to the map which assignsto a simplex σ the direct sum (cid:76) V i ( F σ (0) ) /V i − ( F σ (0) ). Triviality of the action of MOOTH PARAMETRIZED TORSION – A MANIFOLD APPROACH 13 the fundamental group of B on the quotients V i ( F σ (0) ) /V i − ( F σ (0) ) implies that wecan construct a map τ F : | S ( B ) | → Wh F ( E ) similarly as before.7. Characteristic classes
As we mentioned at the beginning of this paper the torsion invariants of smoothbundles constructed by Igusa-Klein and Bismut-Lott are constructed as certaincohomology classes associated to the bundle. More precisely, for a bundle p : E → B its torsion in both of these settings is an element of (cid:76) k> H k ( B ; R ). Our finalgoal in this note is to show that the construction of torsion described above alsogives rise to an element of (cid:76) k> H k ( B ; R ) which brings it on a common groundwith the other notions of torsion.Let then p : E → B be a bundle with a unipotent action of π ( B ) on the homologyof the fiber H ∗ ( F b ; R ), so that the torsion τ R ( p ) : | S ( B ) | → Wh R ( E ) is defined.Consider an embedding i : E → D N where D N is a closed disc in R M for some large M >
0. Let
N E be a closed tubular neighborhood of E in D N . Considering N E as a disc bundle over E we obtain a transfer map ˜ Q ( E + ) → ˜ Q ( N E + ). Also, sincethe construction of ˜ Q is functorial with respect to codimension 0 embeddings ofmanifolds the inclusion N E (cid:44) → D M induces a map ˜ Q ( N E ) → ˜ Q ( D M ). Composingit with the transfer of the bundle N E → E we obtain a map ˜ Q ( E + ) → ˜ Q ( D M + ).Consider the diagram Wh R ( E ) (cid:15) (cid:15) (cid:47) (cid:47) (cid:95)(cid:95)(cid:95) Wh R ( D M ) (cid:15) (cid:15) B τ R ( p ) (cid:60) (cid:60) (cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120) p ! (cid:47) (cid:47) ˜ Q ( E + ) λ R (cid:15) (cid:15) (cid:47) (cid:47) ˜ Q ( D M + ) λ R (cid:120) (cid:120) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113) K ( R )One can check that the lower triangle commutes up to a preferred choice of homo-topy, so that we obtain a map of homotopy fibers Wh R ( E ) → Wh R ( D M ).Now, consider the fibration sequenceΩ K ( R ) → Wh R ( D M ) → ˜ Q ( D M + ) → K ( R )Since this is a fibration of infinite loop spaces after applying the rationalizationfunctor we obtain a new fibration sequence.Ω K ( R ) Q → Wh R ( D M ) Q → ˜ Q ( D M + ) Q → K ( R ) Q Since the homotopy groups of π i ˜ Q ( D M + ) ∼ = π i Q ( S ) are torsion for i > Q ( D M + ) is homotopically discrete. It follows that every connected componentof Wh R ( D M ) Q is weakly equivalent to the space Ω K ( R ) Q . On the other hand by[3] we have a weak equivalence K ( R ) Q (cid:39) Z × (cid:89) k> K ( R , k + 1)Let Wh R ( D M ) B Q denote the connected component of Wh R ( D M ) Q which is a targetof the map B → Wh R ( D M ) Q . By the observation above we have Wh R ( D M ) B Q (cid:39) (cid:81) k> K ( R , k ), and so the homotopy class of the map B → Wh R ( D M ) Q determinesan element in (cid:76) k> H k ( B ; R ). References [1] James C. Becker and Reinhard E. Schultz. Axioms for bundle transfers and traces.
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