Title: Analytic torsion and š¯‘…-torsion for unimodular representations
Abstract:Let M be a closed C?-manifold of dimension n. Both R-torsion and analytic torsion are smooth invariants of acyclic orthogonal (or unitary) representations p of the fundamental group 7I (M) . The Reide...Let M be a closed C?-manifold of dimension n. Both R-torsion and analytic torsion are smooth invariants of acyclic orthogonal (or unitary) representations p of the fundamental group 7I (M) . The Reidemeister-Franz torsion (or R-torsion) Tm(p) of p is defined in terms of the combinatorial structure of M given by its smooth triangulations. The analytic torsion TM(p) was introduced by Ray and Singer [RS] as an analytic counterpart of R-torsion. In order to define the analytic torsion one has to choose a Riemannian metric on M. Then TM(p) is a certain weighted alternating product of regularized determinants of the Laplacians on differential qforms of M with values in the flat bundle E, defined by p. It was conjectured by Ray and Singer [RS] that Tm (p) = Trm(p) for all acyclic orthogonal (or unitary) representations p. This conjecture was proved independently by Cheeger [C] and the author [Mu]. The restriction to orthogonal (or unitary) representations is certainly a limitation of the applicability of this result if 7I (M) is infinite because an infinite discrete group will have, in general, many nonorthogonal finite-dimensional representations. It is the purpose of the present paper to remove this limitation. We call a representation p: r, (M) -* GL(E) on a finite-dimensional real or complex vector space E unimodular if I det p(y)I = 1 for all y E I(M) . Then we define R-torsion and analytic torsion for unimodular representations, and the main result is that for odd-dimensional manifolds M the equality of the two torsions extends to all unimodular representations.Read More