Poincaré duality, Hilbert complexes and geometric applications

Let (M,g)(M,g) be an open, oriented and incomplete riemannian manifold. The aim of this paper is to study the following two sequences of L2L2-cohomology groups:1.H2,m→Mi(M,g) defined as the image (H2,mini(M,g)→H2,maxi(M,g))2.H¯2,m→Mi(M,g) defined as the image (H¯2,mini(M,g)→H¯2,maxi(M,g)). We show, under suitable hypothesis, that the first sequence is the cohomology of a Hilbert complex which contains the minimal one and is contained in the maximal one. In particular this leads us to prove a Hodge theorem for these groups. We also show that when the second sequence is finite dimensional then Poincaré duality holds and that, with the same assumptions, when dim(M)=4ndim(M)=4n then we can employ H¯2,m→M2n(M,g) in order to define an L2L2-signature on M  . We prove several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension ΔiF of ΔiΔi.