A Cohomological Criterion for Density of Smooth Maps in Sobolev Spaces Between Two Manifolds

Abstract

We consider in this paper two compact Riemannian manifolds M and N, and a map f in W ^{1, p}(M, N), where 1 ≤ p > m = dim M. We assume that N is ([p]-1)-connected and that H _[p](N, Q) ≃ Π_[p](N) where [p] is the largest integer less or equal to p. We prove that f can be approximated by smooth maps between M and N if and only if the pullback by f of any closed [p]-form on N is closed.