Pullback functors for reduced and unreduced $$L^{q,p}$$-cohomology

Abstract

In this paper we study the reduced and unreduced $L^{q,p}$-cohomology groups of manifolds of bounded geometry and their behavior under uniform maps. A \textit{uniform map} is a uniformly continuous map such that the diameter of the preimage is bounded in terms of 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, $L^{q,p}$-cohomology. Then, our goal is to introduce some contravariant functors between the category of manifolds of bounded geometry and uniform maps and the category of complex vector spaces and linear maps. As consequence we obtain that the, reduced or not, $L^{q,p}$-cohomology is a uniform homotopy invariant. Moreover these functors coincide with the pullback, when the pullback does induce a map between the reduced and unreduced $L^{q,p}$-cohomologies.