Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments. The classical Franz-Reidemeister torsion τFR is an invariant of manifolds with acyclic unitarily flat vector bundles [62], [33]. In contrast to most other algebraic-topological invariants known at that time, it is invariant under homeomorphisms and simple-homotopy equivalences, but not under general homotopy equivalences. In particular, it can distinguish homeomorphism types of homotopy-equivalent lens spaces. Hatcher and Wagoner suggested in [39] to extend τFR to families of manifolds p : E → B using pseudoisotopies and Morse theory. A construction of such a higher Franz-Reidemeister torsion τ was first proposed by John Klein in [48] using a variation of Waldhausen’s A-theory. Other descriptions of τ were later given by Igusa and Klein in [45], [46]. In this overview, we will refer to the construction in [42]. Let p : E → B be a family of smooth manifolds, and let F → E be a unitarily flat complex vector bundle of rank r such that the fibrewise cohomology with coefficients in F forms a unipotent bundle over B. Using a function h : E → R that has only Morse and birth-death singularities along each fibre of p, and with trivialised fibrewise unstable tangent bundle, one constructs a homotopy class of maps ξh(M/B;F ) fromB to a classifying spaceWh (Mr(C), U(r)). Now, the higher torsion τ(E/B;F ) ∈ H4•(B;R) is defined as the pull-back of a certain universal cohomology class τ ∈ H4• ( Wh(Mr(C), U(r));R ) . On the other hand, Ray and Singer defined an analytic torsion TRS of unitarily flat complex vector bundles on compact manifolds in [61] and conjectured that TRS = τFR. This conjecture was established independently by Cheeger [26] and Müller [59]. The most general comparison result was given by Bismut and Zhang in [17] and [18]. In [64], Wagoner predicted the existence of a “higher analytic torsion” that detects homotopy classes in the diffeomorphism groups of smooth closed manifolds. Such an invariant was defined later by Bismut and Lott in [15]. In [47], Kamber and Tondeur constructed characteristic classes ch(F ) ∈ Hodd(M ;R) of flat vector bundles F → M that provide obstructions towards finding a parallel metric. If p : E → B is a smooth bundle of compact manifolds and F → E is flat, Bismut and Lott proved a Grothendieck-Riemann-Roch theorem relating the characteristic classes of F to those of the fibrewise cohomology H(E/B;F ) → B. The higher analytic torsion form T (THE, gTX , gF ) 2000 Mathematics Subject Classification. 58J52 (57R22 55R40). Supported in part by DFG special programme “Global Differential Geometry”.