Absolute Torsion

In this paper we use the results of our previous work [FT] in order to compute the phase of the torsion of an Euler structure ξ in terms of the characteristic class c(ξ). Also, we introduce here a new notion of an absolute torsion, which does not require a choice of any additional topological information (like an Euler structure). We prove that in the case of closed 3-manifolds obtained by 0-surgery on a knot in S the absolute torsion is equivalent to the Conway polynomial. Hence the absolute torsion can be viewed as a high-dimensional generalization of the Conway polynomial. §1. Euler structures and Poincaré-Reidemeister metric In this section we give a brief review of the main results of [FT], which we will use in the sequel. The proofs of all theorems, appearing in this section, can be found in [FT]. 1.1. Determinant lines. We shall denote by k a fixed ground field of characteristic zero. The most important special cases are k = R and k = C. If V is a finite dimensional vector space over k, the determinant line of V is denoted by detV and is defined as the top exterior power of V , i.e., ΛV , where n = dimV . The dual line Homk(V,k) is denoted by (detV ) . For a finite dimensional graded vector space V = V0 ⊕V1 ⊕· · ·⊕Vm, its determinant line detV is defined as the tensor product detV = detV0 ⊗ (detV1) −1 ⊗ detV2 ⊗ · · · ⊗ (detVm) (−1) . Let C be a finite dimensional chain complex over k. In the theory of torsion a crucial role is played by a canonical isomorphism φC : detC → detH∗(C), (1-1) where both C and H∗(C) are considered as graded vector spaces. The definition of the mapping φC is as follows. Choose for each q = 0, ..., m non-zero elements cq ∈ detCq and hq ∈ detHq(C). Set c = c0 ⊗ c −1 1 ⊗ c2 ⊗ · · · ⊗ c (−1) m ∈ detC and h = h0 ⊗ h −1 1 ⊗ h2 ⊗ · · · ⊗ h (−1) m ∈ detH∗(C), where −1 in the exponent denotes the dual functional. *Partially supported by a grant from the US Israel Binational Science Foundation and by the Herman Minkowski Center for Geometry **Partially supported by the EC TMR network ”Algebraic Lie Representations”, EC-contract no ERB FMRX-CT97-0100 Typeset by AMS-TEX 1 2 M. FARBER AND V. TURAEV We define φC by φC(c) = (−1) N(C) [c : h] h, where N(C) is a residue modulo 2 defined below and [c : h] is a nonzero element of k, defined by