Analytic torsion for group actions

Abstract

The Reidemeister torsion is a classical topological invariant for nonsimply-connected manifolds [14]. Let M be a closed oriented smooth manifold with the fundamental group πλ(M) nontrivial. Let p:πx{M) -• O(N) be an orthogonal representation of nx(M). If the twisted real cohomology H*(M, Ep) vanishes, then one can define the Reidemeister torsion τp GR , which is a homeomorphism invariant of M. The original interest of τp was that it is not a homotopy invariant, and so can distinguish spaces which are homotopy equivalent but are not homeomorphic. Ray and Singer asked whether, as for many other topological quantities, one can compute τp by analytic methods [17], Given a Riemannian metric g on M, they defined an analytic torsion T' e R as a certain combination of the eigenvalues of the Laplacian acting on twisted differential forms. They showed that under the above acyclicity condition, T is independent of the metric g, and they conjectured that the analytic expression T equals the combinatorial expression τ . This conjecture was proven to be true independently by Cheeger [4] and Mϋller [15]. One can look at the above situation in the following way. The group nχ{M) acts freely on the universal cover M, and so one has an invariant for free group actions. A natural question is whether the Reidemeister torsion can be extended to an invariant for more general group actions. For a finite group acting (not necessarily freely) on a closed oriented PL manifold X, a Reidemeister torsion was defined algebraically by Rothenberg [19]. One can then ask whether there is a corresponding analytic torsion when X is smooth, and whether the analytic torsion equals the combinatorial torsion. In §11 we define the analytic torsion Tp for a finite group action and show that if the relevant cohomology groups vanish, then it is independent of the G-invariant metric used in its definition. (This was shown previously in unpublished work by Cheeger [5].) The analysis involved to