Isovariant mappings of degree 1 and the Gap Hypothesis

TL;DR

This paper explores the relationship between isovariant and equivariant homotopy equivalences for manifolds with group actions under the Gap Hypothesis, providing a homotopy theoretic perspective and proof of a known result.

Contribution

It offers a homotopy theoretic analysis of the difference between isovariant and equivariant maps for degree one maps under the Gap Hypothesis, extending prior geometric topology results.

Findings

Isovariant and equivariant maps often coincide under the Gap Hypothesis.

Provides a homotopy theoretic proof of Straus and Browder's result.

Clarifies the distinction between map types in a broader setting.

Abstract

Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions - isovariant and equivariant - often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.