  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 $(L^2\Omega^i(M,g),d_{\mathfrak{M},i})$ such that its
cohomology groups, labeled with $H^i_{2,\mathfrak{M}}(M,g)$, satisfy the
following properties: \begin{itemize} \item
$H^i_{2,\mathfrak{M}}(M,g)=ker(d_{max,i})/\im(d_{min,i})$ \item
$H^i_{2,\mathfrak{M}}(M,g)\cong H^{m-i}_{2,\mathfrak{M}}(M,g)$ (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.
