  In this paper we study the reduced and unreduced $L^2$-cohomology groups of
manifolds of bounded geometry and their behavior under uniformly proper
lipschitz maps. A \textit{uniformly proper map} is a map such that the diameter
of the preimage of a subset depends only on the diameter of the subset. In
general, the pullback map along a uniformly proper lipschitz map doesn't induce
a morphism in, reduced or not, $L^2$-cohomology. Then, our goal is to introduce
two contravariant functors between the category of manifolds of bounded
geometry and uniformly proper lipschitz maps and the category of complex vector
spaces and linear maps. As consequence we obtain that the, reduced or not,
$L^2$-cohomology is a lipschitz-homotopy invariant. Moreover these functors
coincide with the pullback, when the pullback does induce a map between the
(un)-reduced $L^2$-cohomologies.
