Analytic and topological torsion for manifolds with boundary and symmetry
Let \(G\) be a finite group, \(M\) a Riemannian \(G\)-manifold, i.e. \(M\) is a compact, smooth manifold with differentiable \(G\)-action and invariant Riemannian metric and \(V\) an equivariant coefficient system which is coherent to a \(G\)-representation. If the boundary \(\partial M\) is the disjoint union of \(M_ 1\) and \(M_ 2\), the author defines the analytic torsion \(\rho^ G_{\text{an}}(M,M_ 1;V)\) of \((M,M_ 1)\). Poincaré duality as well as double and product formulae for analytic torsion are established. The PL-torsion \(\rho^ G_{p\ell}(M,M_ 1;V)\) and the Poincaré torsion \(\rho^ G_{\text{pd}}(M,M_ 1;V)\) are also defined in the case when \((M;M_ 1,M_ 2)\) is a Riemannian \(G\)-manifold triad, i.e. \(M\) is a Riemannian \(G\)-manifold \(M\) together with \(G\)-invariant codimension zero submanifolds \(M_ 1\) and \(M_ 2\) of the boundary \(\partial M\) satisfying \(\partial M = M_ 1 \cup M_ 2\) and \(\partial M_ 1 = M_ 1 \cap M_ 2 = \partial M_ 2\). Poincaré duality, double product and sum formulae are deduced for Poincaré and PL- torsion. The main result of this paper is a torsion formula relating \(\rho^ G_{\text{an}}(M,M_ 1;V)\), \(\rho^ G_{p\ell}(M,M_ 1;V)\), \(\rho^ G_{\text{pd}}(M,M_ 1;V)\) and \(\chi^ G(\partial M;V)\) (\(\chi^ G\) denotes the equivariant Euler characteristic) in the case when the metric is a product near the boundary \(\partial M = M_ 1 \coprod M_ 2\). The proof is based on the work of \textit{J. Lott} and \textit{M. Rothenberg} [J. Differ. Geom. 34, No. 2, 431-481 (1991; Zbl 0744.57021)]. For a manifold with boundary without group action, \textit{J. Cheeger} [Ann. Math., II. Ser. 109, 259-322 (1979; Zbl 0412.58026)] has given without proof a formula relating analytic and PL-torsions which is not as precise as the author's one. Some special cases are investigated. For example, a sum formula for the analytic torsion is derived from the sum formula in the PL-case.
scientific article; zbMATH DE number 444707