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Higher Franz-Reidemeister Torsion: Low Dimensional

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Higher Franz-Reidemeister Torsion: Low Dimensional Contemp orary Mathematics Volume 00 Higher FranzReidemeister torsion low dimensional applications JOHN R KLEIN Abstract In this exp ository article I will discuss the theory of higher FranzReidemeister torsion and its application to the construction of non trivial classes in the algebraic K theory of elds I will also explain how these ideas may b e used to construct classes in the cohomology of the Torelli group Intro duction In this note I shall discuss the higher FranzReidemeister torsion invariants of K and IK What I shall b e rep orting on is joint work with Kiyoshi Igusa The rst sections will set up the relevent formalism and state the general result I then provide the results of the known calculations of the invariants in the sp ecial case of circle bundles This leads to a geometric interpretation of the algebraic K theory of cyclotomic elds Finally in the case of surface bundles I shall show how torsion can b e used to construct classes in the rational homology of the Torelli group and I end by asking whether there is any connection b etween them and the MillerMoritaMumford classes I Preliminaries In what follows X will denote a xed top ological space M will b e a compact smo oth manifold and M X will b e a xed map We let DiM denote the top ological group of dieomorphisms of M and MapsM X the space of maps from M to X with basep oint Then MapsM X has a left DiM action on it dened by f f Consider the orbit map of given by DiM MapsM X f f Mathematics Subject Classication Primary J Secondary D R The author wishes to acknowledge the supp ort of the Alexander von Humb oldt Foundation This pap er is in nal form and no version of it will b e submitted for publication elsewhere c American Mathematical So ciety p er page JOHN R KLEIN and let Di M denote the homotopy bre This has a delo oping B Di M which can b e dened as the balanced pro duct E DiM MapsM X DiM We will b e mainly interested in the homotopy and homology typ es of B Di M Note that B Di M classies smo oth M bundles p E K over a p ointed nite complex together with a map E X extending M X we are xing once and for all an identication of M with the bre over the basep oint of K This information is summarized in the diagram M X E p K We will concentrate on two particular cases Example In this situation we take X M and M M the identity Then B Di M classies smo oth M bundles which are trivialized as homotopy brations Furthermore B Di M GM DiM breB DiM BGM where GM is the top ological monoid of self homotopy equivalences of M Example M B is a map to a K space Let M b e the covering space of M dened by Then B Di M classies bundles with bre M the covering space of M dened by with structure group the equivariant dieomorphisms of M Remark Regarding these examples it should b e noted that the rst ts naturally within the context of the theory of higher Whitehead torsion where we take p E K to b e a bundle of hcob ordisms which is trivial along one b oundary comp onent The second example is connected with higher Franz Reidemeister torsion as we shall see b elow We single out a particular case of which is relevent to mapping class groups Example Let M b e a closed orientable surface of genus g H M and M BH M the classifying map for the universal ab elian cover of M Then Di M consists of dieomorphisms of M which induce the identity on onedimensional homology The set of path comp onents of this is HIGHER FRANZREIDEMEISTER TORSION by denition the Torelli group T Moreover the homomorphism of top ological g groups 0 Di M T g is a homotopy equivalence ie the homotopy groups of Di M in p ositive dimensions vanish Consequently the study of B Di M is the same as the study of BT g I I K Theory Spaces over X Following Waldhausen Wal let R X denote the f category of retractive relative nite CWcomplexes over X An ob ject of R X f is a space Z endowed with a pair of maps i X Z r Z X resp ectively called inclusion and retraction which satisfy r i id X the inclusion i is such that Z is obtained from X by the inductive at tachment of a nite numb er of cellsthe attaching order given by the dimension of such cells A morphism Y Z of R X is a map of underlying spaces which commutes f with inclusion and retraction This is a category with cobrations in the sense of Waldhausen A cobration Y Z is a morphism whose underlying map of spaces is a cobration in the usual sense of the term Chain complexes Fix an asso ciative ring R with unit Let C R f b e the category of nite free chain complexes over R An ob ject of C R f is a nite dimensional chain complex which has chains in dimensions and moreover which is nitely generated and free in each dimension as an Rmo dule A morphism of C R is a chain map A cobration is a chain map which is split f injective in each dimension A weak equivalence is a chain homotopy equivalence The sub category of weak equivalences will b e denoted by w C R f Proposition Waldhausen The Waldhausen K theory of C R is f natural ly homotopy equivalent to the representing space for Quil lens algebraic K theory of R ie jw S C Rj K R Z B GLR f jnRj where on the left w S is the S construction of lo c cit and on the right nR is the order of the class dened by R in the Grothendieck group K R and denotes Quil lens plus construction Q I I I Linearization Functors By a linearization functor we shall mean any functor R X C R f f JOHN R KLEIN satisfying the following is exact By this I mean that preserves the cobration prop erty and moreover if A Z y y Y P is a pushout diagram in R X with cobrations as horizontal arrows f then applied to the diagram is also a pushout diagram X where X is considered as an ob ject over itself via the identity map and denotes the trivial chain complex over R If Z is an ob ject of R X and i X Z is a homotopy equivalence f then Z is acyclic For a given linearization functor we obtain a sub category of weak equiv alences in R X by sp ecifying Y Z to b e a weak equivalence if and only f if Y Z is a weak equivalence in C R We let w R X denote the f f category of weak equivalences dened by Then the exactness prop erties of and show that Proposition Let A X jw S R X j be the Waldhausen K theory f of R X with respect to the weak equivalences dened by Then application f of denes a map A X K R which for short wil l also be denoted by Here are a few examples of linearization functors Example Let G b e the fundamental group of X and dene a lineariza tion functor R X C ZG by the rule f f cell G G Y C Y X G G G where X is the universal cover of X Y is the pullback of X X along cell the retraction r Y X and and C denotes the cellular chain complex considered as a free mo dule over ZG Note that the weak equivalences of R X with resp ect to this are the f Gequivariant relative homology equivalences or equivalently by Whiteheads theorem the relative homotopy equivalences Consequently A X is identical to Waldhausens AX and the map AX K ZG of is the usual linearization map into algebraic K theory HIGHER FRANZREIDEMEISTER TORSION Example Let F b e a eld with involution Supp ose we are provided with a unitary representation G U F where G is the fundamental group r of X Dene a linearization functor R X C F by the rule f f cell G G r Y C Y X F Z G r where on the right hand side the left factor is as in and the right factor F has the ZGmo dule structure dened by Prop osition then yields a map A X K F IV The Whitehead Space Let AX b e Waldhausens K theory of X dened as in using the sub cat egory hR X of R X of relative homotopy equivalences f f Let R X C R b e an arbitrary linearization functor By applica f f tion of axioms in the denition of a linearization functor it follows that hR X w R X We infer that there is a natural transformation f f N AX A X As is wellknown there is a natural map i Q X AX which is a split inclusion up to homotopy where Q denotes the unreduced stable homotopy functor Let W hX b e the homotopy bre of the comp os ite N i Q X AX A X K R There is then a homotopy bration W hX Q X K R Remarks The notation W hX for the bre Q X K R indicates that the bre delo ops This is indeed the case since the latter is an innite lo op map If is as in then the set of path comp onents of W hX is canonically isomorphic to the Whitehead group W h X If is as in the set of path comp onents is canonically isomorphic to the target group for the classical FranzReidemeister torsion see eg K JOHN R KLEIN V Higher Torsion We now return to the context of x Namely we consider classifying spaces asso ciated with data of the form M X E p K where p is a smo oth M bundle Fix once and for all a cell structure on M Form the following ob ject of R X f M M q X X with the obvious inclusion and where the retraction is r q id Similarly X let X b e the ob ject of R X given by X q X with retraction id q id and X f X X with inclusion X X q X given by the identity on the second comp onent Contractibility Hypothesis The map q id M X is a X X X weak equivalence in w R X f Acyclicity Hypothesis M obR X is weakly equivalent to X f the initial ob ject X in w R X ie the Rchain complex M is acyclic f X The main theorem on higher FranzReidemeisterWhitehead torsions may b e stated as follows Theorem IgusaKlein KIK If either the Contractibility Hy pothesis or the
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