Topology of Sobolev mappings III

We establish a necessary and sufficient topological condition for maps that are in W^1,p(M, N) to be connected by continuous paths in W^1,p(M, N) to maps in W^1,q(M, N), q > q ≥ 1. We also obtain a necessary and sufficient topological condition under which W^1,q(M, N) is strongly dense in W^1,p(M, N). Several results concerning sequential weak density of smooth (or W^1,q(M, N) maps in W^1,p(M, N) are also proven. © 2003 Wiley Periodicals, Inc.