Signatures of Witt spaces with boundary.

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 : sign dR (M ‾ , ∂ M ‾) − sign dR Γ ( M ‾ Γ , ∂ M ‾ Γ) = ρ Γ (∂ M ‾ Γ). We end the paper with two geometric applications of our results. [ABSTRACT FROM AUTHOR]