Inductive detection for homotopy equivalences of manifolds
Let \(f:M^n \to N^n\) be a homotopy equivalence of CAT manifolds (CAT=PL, TOP, or DIFF) with finite fundamental groups. Let \(C\) be a class of subgroups \(H\) of \(\pi_1 (M)\) and \(f_H\) a lift of \(f\) to the corresponding covering spaces. The authors consider the following problem: Suppose that for each \(H\) in \(C\), \(f_H\) is homotopic to a CAT isomorphism, is \(f\) itself homotopic to a CAT isomorphism? A typical result is Theorem A: The question has an affirmative answer in the smooth category, if \(n\geq 5\), \(f\) is a simple homotopy equivalence, the Sylow 2-subgroups of \(\pi_1 (M)\) are normal, and the class \(C\) consists of the Sylow or 2-hyperelementary subgroups. Examples are given where the conclusion is that \(f \times \text{id}: M \times\mathbb{R} \to N\times \mathbb{R}\) is properly homotopic to an isomorphism but \(f\) itself is not homotopic to a homeomorphism.
scientific article; zbMATH DE number 1036285