0 M ay 2 00 3 MORSE THEORY AND HIGHER TORSION INVARIANTS II

Let p:M → B be a family of compact manifolds equipped with a unitarily flat vector bundle F → M . We generalize Igusa’s higher Franz-Reidemeister torsion τ(M/B;F ) to the case that the fibre-wise cohomologyH(M/B;F ) → B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the difference of τ(M/B;F ) and the higher analytic torsion T (M/B;F ). We also generalise the examples given in [G]. This is the second 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 the first part [G], we evaluated the Bismut-Lott torsion form T for families that carry a fibre-wise Morse function, thus extending earlier work with Bismut ([BG]). Here, we relate τ and T for families with fibre-wise Morse functions in those situations where both invariants are defined. We start by recalling some notation and results of [G]. Let p:M → B be a family of compact manifolds, let F → M be a flat vector bundle, and let h:M → R be a fibre-wise Morse function. Then the fibre-wise critical points of h form a covering p̂:C → B. Let o(T X) → C denote the orientation bundle of the unstable vertical tangent bundle at C, and construct a vector bundle V = p̂∗ ( F |C ⊗ o(T X) ) . Then V carries a flat connection ∇V induced by ∇F . In [G], we defined a family Thom-Smale complex by constructing a flat superconnection A′ on V . We also constructed a generalised “integration over the fibre”, i.e., an Ω∗(B)-linear cochain map I: (Ω∗(M ;F ),∇F ) → (Ω∗(B;V ), A′). This cochain map naturally identifies the fibre-wise cohomology H = H∗(M/B;F ) → B, equipped with the flat Gauß-Manin connection ∇H , with the bundle H∗(V, a0) → B as follows. Both Ω∗(M ;F ) and Ω∗(B;V ) can be filtered by horizontal degree. The cochain map I respects these filtrations and induces isomorphisms of the Ep-terms of the associated spectral sequences for p ≥ 1. The E1-term of both spectral sequences is isomorphic to (Ω∗(B;H),∇H). A metric g on F induces a metric g on V , and the Morse function h induces an endomorphism h of V . Using these data, we defined an analytic torsion form T (A′, g , h ) ∈ Ω∗(B) . The metric g induces a metric g V on H by finite-dimensional Hodge theory. Let ch (∇F , g ) denote the characteristic form of F → M defined in [BL], written in the normalisation of [BG]. Then dT (A′, g , h ) = ch(∇ , g )− ch(∇ , g V ). 2000 Mathematics Subject Classification. Primary 58J52; Secondary 57R22, 57Q10.