  Unpublished results of S Straus and W Browder state that two notions of
homotopy equivalence for manifolds with smooth group actions - isovariant and
equivariant - often coincide under a condition called the Gap Hypothesis; the
proofs use deep results in geometric topology. This paper analyzes the
difference between the two types of maps from a homotopy theoretic viewpoint
more generally for degree one maps if the manifolds satisfy the Gap Hypothesis,
and it gives a more homotopy theoretic proof of the Straus-Browder result.
