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Unpublished results of S Straus and W Browder state that two notions ofhomotopy equivalence for manifolds with smooth group actions - isovariant andequivariant - often coincide under a condition called the Gap Hypothesis; theproofs use deep results in geometric topology. This paper analyzes thedifference between the two types of maps from a homotopy theoretic viewpointmore 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.

Isovariant mappings of degree 1 and the Gap Hypothesis