2 SEBASTIAN GOETTE In [ BL ]

We compare the higher analytic torsion T of Bismut and Lott of a fibre bundle p:M → B equipped with a flat vector bundle F → M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F , thereby extending previous joint work with Bismut. Under additional conditions on F , the torsion T is related to Igusa’s higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles Mj → B with diffeomorphic fibres that are homeomorphic but not diffeomorphic as bundles. This is the first of two papers devoted to a comparison of Igusa’s and Klein’s higher FranzReidemeister torsion τ with Bismut’s and Lott’s higher analytic torsion T . In this paper, we evaluate the Bismut-Lott torsion form T for families that carry a fibre-wise Morse function, thus extending earlier work with Bismut ([BG1]). In the second part of the series, we relate τ and T for families with fibre-wise Morse functions in those situations where both invariants are defined. Let us recall the development of higher torsion invariants. Franz and Reidemeister constructed a numerical invariant τFR of chain complexes in [R], [F], and used it to detect homeomorphism types of homotopy-equivalent Lense-spaces. The higher FranzReidemeister torsion τ is an extension of τFR to families of manifolds p:M → B, which takes values in the cohomology of B. It was first constructed by John Klein using a variation of Waldhausen’s A-theory. Other describtions of τ were later given by Igusa and Klein, see [I4] for more detailled references. In [IK], they computed τ in the special case of circle bundles. We will constantly refer to the construction of τ in [I4], see also [I2], [I3]. Given a family p:M → B of smooth manifolds, one first finds a function h:M → R that has only Morse-type and cubical singularities along each fibre of p, such that the unstable manifolds of the fibre-wise singularities are trivialised in a compatible way. By [I1], such a “framed” function always exists if dimM > 2 dimB. If dimM ≤ 2 dimB, one may replace M by M × RP 2N for some sufficiently large N . Let F → M be a unitarily flat complex vector bundle that is fibre-wise acyclic, then F and h give rise to a functor from the category of generic small simplices on B to the simplicial Whitehead category Wh(M(C), U) associated to the infinite matrix ringM(C) = lim→ Mn(C) and the infinite unitary group U = lim→ U(n). Now, the k-th higher torsion τk(M/B;F ) is defined as the pull-back of a certain cohomology class D2k ∈ H ( Wh(M(C), U),R ) . On the other hand, Ray and Singer defined an analytic torsion TRS of unitarily flat complex vector bundles on compact manifolds in [RS] and conjectured that TRS = τFR. This conjecture was established independendly by Cheeger ([C]) and Müller ([M1]), and further generalised by Müller ([M2]) to unimodular flat bundles, and by Bismut and Zhang ([BZ1]) to arbitrary flat complex vector bundles. 2000 Mathematics Subject Classification. Primary 58J52; Secondary 57R22, 57Q10.