On the $L^2-$Poincar\'e duality for incomplete riemannian manifolds: a general construction with applications

Let $(M,g)$ be an open, oriented and incomplete riemannian manifold of dimension $m$. Under some general conditions we show that it is possible to build a Hilbert complex $(L^2{\mathrm{\Omega }}^i(M,g),d_{\mathfrak{M},i})$ such that its cohomology groups, labeled with $H_{2,\mathfrak{M}}^i(M,g)$, satisfy the following properties: \begin{itemize} \item $H_{2,\mathfrak{M}}^i(M,g)=ker(d_{max,i})\mathrm{/}\text{im}(d_{min,i})$ \item $H_{2,\mathfrak{M}}^i(M,g)\cong H_{2,\mathfrak{M}}^{m-i}(M,g)$ (Poincar'e duality holds) \end{itemize} Finally in the rest of the paper we study some properties of this complex with particular attention to the sufficient conditions which make it a Fredholm complex.