Path and quasi-homotopy for Sobolev maps between manifolds

摘要

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic - and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere. (C) 2021 Elsevier Inc. All rights reserved.