  In this paper we study the minimal and maximal $L^{2}$-cohomology of
oriented, possibly not complete, Riemannian manifolds. Our focus will be on
both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we
will prove that these groups are invariant under uniform homotopy equivalence
quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous
map such that the diameter of the preimage of a subset is bounded in terms of
the diameter of the subset itself. A map $f$ between two Riemannian manifolds
$(X,g)$ and $(Y,h)$ is \textit{quasi-isometric on the unbounded ends} if $X = M
\cup E_X$ where $M$ is the interior of a manifold of bounded geometry with
boundary, $E_X$ is an open of $X$ and the restriction of $f$ to $E_X$ is a
quasi-isometry. Finally some consequences are shown: the main ones are
definition of a mapping cone for $L^2$-cohomology and the invariance of the
$L^2$-signature.
