  We study the relationship between quasihomotopy and path homotopy for Sobolev
maps between manifolds. We employ singular integrals on manifolds to show that,
in the critical exponent case, path homotopy implies quasihomotopy - and
observe the rather surprising fact that $n$-quasihomotopic maps need not be
path homotopic. We also study the case where the target is an aspherical
manifold, e.g. a manifold with nonpositive sectional curvature, and the
contrasting case of the target being a sphere.
