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In this paper we study the minimal and maximal $L^{2}$-cohomology oforiented, possibly not complete, Riemannian manifolds. Our focus will be onboth the reduced and the unreduced $L^{2}$-cohomology groups. In particular wewill prove that these groups are invariant under uniform homotopy equivalencequasi-isometric on the unbounded ends. A uniform map is a uniformly continuousmap such that the diameter of the preimage of a subset is bounded in terms ofthe 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 withboundary, $E_X$ is an open of $X$ and the restriction of $f$ to $E_X$ is aquasi-isometry. Finally some consequences are shown: the main ones aredefinition of a mapping cone for $L^2$-cohomology and the invariance of the$L^2$-signature.

$L^2$-cohomology and quasi-isometries on the ends of unbounded geometry