Isovariant homotopy equivalences of manifolds with group actions

Let f f be an equivariant homotopy equivalence f f of connected closed manifolds with smooth semifree actions of a finite group G G , and assume also that f f is isovariant. The main result states that f f is a homotopy equivalence in the category of isovariant mappings if the manifolds satisfy a Codimension ≥ 3 \geq 3 Gap Hypothesis; this is done by showing directly that f f satisfies the criteria in the Isovariant Whitehead Theorem of G. Dula and the author. Examples are given to show the need for the hypotheses in the main result.