Isovariant homotopy equivalences of manifolds with group actions (Q2790212)

Let \(M\) and \(N\) be smooth semi-free \(G\)-manifolds, where \(G\) is a finite group, and let \(f : M \to N\) be an equivariant homotopy equivalence. Then \(f\) is called \textit{G-isovariant} if \(f\) preserves the isotropy groups, i.e. \(G_x = G_{f(x)}\) for all \(x \in M\). We say that a \(G\)-manifold \(M\) satisfies a Codimension \(\geq 3\) Gap Hypothesis if it satisfies the condition: \(\dim(C)\leq \dim M-3\) for all components of \(M^G\). The main theorem states that if the Codimension \(\geq 3\) Gap Hypothesis is satisfied, then \(f\) becomes a homotopy equivalence in the category of isovariant mappings. The proof of this theorem greatly depends on results by \textit{G. Dula} and the author [Mem. Am. Math. Soc. 527, 82 p. (1994; Zbl 0815.55002)]. Let \(N_\alpha\) be a component of \(M^G\), then we set \(M_\alpha=f^{-1}(N_\alpha)\cap M^G\). By \(E_M\) and \(E_N\) we denote the unions of closed tubular neighborhoods of \(M_\alpha\) and \(N_\alpha\), and also write \(C_M\) and \(C_N\) for the closures of the complements of \(E_M\) and \(E_N\), respectively. The proof consists of two steps. In the first step it is proved that \(f\) is isovariantly homotopic to a map of triads \((M; E_M, C_M) \to (N; E_N, C_N)\) and in the second step it is proved that the induced map \(C_M \to C_N\) is a homotopy equivalence. Combining these two facts we obtain the desired result. Moreover, four examples are given of how the conditions do contribute to the result that \(f\) is an isovariant homotopy equivalence. In particular, the last example illustrates that the result given here is the best possible in the sense that it does not necessarily hold true when one replaces the inequality in the Gap Hypothesis above by \(\dim(C)\leq \dim M-2\).