A pullback functor for reduced and unreduced $L^{q,p}$-cohomology

TL;DR

This paper develops a functorial framework for $L^{q,p}$-cohomology on manifolds of bounded geometry, showing it is invariant under uniform homotopies and clarifying when pullback maps induce cohomology morphisms.

Contribution

It introduces contravariant functors for $L^{q,p}$-cohomology that are invariant under uniform homotopies and coincide with pullback maps when they exist.

Findings

$L^{q,p}$-cohomology is a uniform homotopy invariant.

Constructs functors that generalize pullback maps for $L^{q,p}$-cohomology.

Provides conditions under which pullback induces cohomology morphisms.

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.