Poincar\'e duality, Hilbert complexes and geometric applications

Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H_{2,m\to M}^i(M,g)$ defined as the image $\text{im}(H_{2,min}^i(M,g)\to H_{2,max}^i(M,g))$ and ${\bar{H}}_{2,m\to M}^i(M,g)$ defined as $\text{im}({\bar{H}}_{2,min}^i(M,g)\to {\bar{H}}_{2,max}^i(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence is finite dimensional then Poincar'e duality holds for it and that, in the same assumptions, when $dim(M)\equiv 0mod4$ we can use it to define a $L^2$ signature on $M$. Moreover we show several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension ${\mathrm{\Delta }}_i^{\mathcal{F}}$ of ${\mathrm{\Delta }}_i$.