Analytic and topological torsion for manifolds with boundary and symmetry

LetG be a finite group acting on a Riemannian manifoldM by isometries. We introduce analytic torsion ρan(M,M1;V ) ∈ R⊗Z RepR(G) PL-torsion ρpl(M,M1;V ) ∈ K1(RG) Poincare torsion ρpd(M,M1;V ) ∈ K1(RG) and Euler characteristic χ(M,M1;V ) ∈ RepR(G) for ∂M the disjoint union of M1 and M2 and V an equivariant coefficient system. The analytic torsionis defined in terms of the spectrum of the Laplace operator, the PL-torsionis based on the cellular chain complex and Poincare torsionmeasures the failure of equivariant Poincare duality in the PL-setting, which does hold in the analytic context. Denote by RepR(G) the subgroup of RepR(G) generated by the irreducible representations of real or complex type. We define an isomorphism