Fixed point sets of maps homotopic to a given map (Q2491520)

Let \(X\) be a compact, connected polyhedron and let \(f:X \to X\) be a map. The author studies necessary and sufficient conditions for a subset \(\Phi\) of \(X\) to be the fixed point set of a map \(g\) homotopic to \(f\). In 1986 \textit{H. Schirmer} studied this problem in the case when \(\Phi\) is a closed subpolyhedron of \(X\). She invoked the notion of by-passing and stated certain conditions, called conditions (C1) and (C2), to answer the problem. The author of the paper under review extends Schirmer's result from polyhedra to locally contractible subsets \(\Phi\) as follows. Theorem. Let \(f: X \to X\) be a map of a compact, connected polyhedron \(X\) into itself. Let \(\Phi\) be a closed, locally contractible subset of \(X\) such that: (1) \(X - \Phi\) is not a 2-manifold, (2) \(f\) satisfies (C1) and (C2), (3) \(\Phi\) can be bypassed. Then for every closed subset \(\Gamma\) of \(\Phi\) that has nonempty intersection with every component of \(\Phi\), there is a map \(g\) homotopic to \(f\) with Fix \(g = \Gamma\). In particular, if \(\Phi\) is connected, then every closed subset of \(\Phi\), including \(\Phi\) itself, is the fixed point set of a map homotopic to \(f\).