Signatures of Witt spaces with boundary

Abstract

Let M‾ be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce the de Rham signature and the Hodge signature of M‾, and prove their equality. Next, building also on recent work of Albin and Gell-Redman, we extend the Atiyah-Patodi-Singer index theory established in our previous work under the hypothesis that M‾ has stratification depth 1 to the general case, establishing in particular a signature formula on Witt spaces with boundary.In a parallel way we also pass to the case of a Galois covering M‾Γ of M‾ with Galois group Γ. Employing von Neumann algebras we introduce the de Rham Γ-signature and the Hodge Γ-signature and prove their equality, thus extending to Witt spaces a result proved by Lück and Schick in the smooth case. Finally, extending work of Vaillant in the smooth case, we establish a formula for the Hodge Γ-signature. As a consequence we deduce the fundamental result that equates the Cheeger-Gromov rho-invariant of the boundary ∂M‾Γ with the difference of the signatures of and M‾ and M‾Γ:signdR(M‾,∂M‾)−signdRΓ(M‾Γ,∂M‾Γ)=ρΓ(∂M‾Γ). We end the paper with two geometric applications of our results.