Fixed point sets of fiber-preserving maps (Q2461361)

\textit{H. Schirmer} [Topology Appl. 37, 153--162 (1990; Zbl 0717.55001)] studied the question of when a map \(f:X\to X\) of a finite connected polyhedron admits a homotopic map \(g\) which has a prescribed subset \(A\) of \(X\) as its set of fixed points. The present authors consider the analogous question for fiber-preserving maps: Let \(\mathcal{F}=(E,p,B;Y)\) be a fiber bundle where all spaces are connected finite polyhedra and let \(f:E\to E\) be a fiber-preserving map and let \(A\subset E\). If there exists a map \(g\) which is fiber homotopic to \(f\) such that \(A\) is the fixed point set of \(g\) then the following conditions hold: (C1) There is a fiber-preserving homotopy \(H:A\times I\to E\) from \(f| \,A\) to the inclusion \(i:A\to E\). (C2) For every essential fixed point class \(\mathbb{F}\) of \(f\) there exists a path \(\alpha:I\to E\) starting in \(\mathbb{F}\) and ending in \(A\) such that \(\alpha\) is homotopic (with fixed endpoints) to the path product of \(f\circ\alpha\) and \(H(\alpha(1),\cdot)\). For the converse, call \(A\subset E\) a bundle subset of \(\mathcal{F}\) if for each component \(C\) of \(p(A)\) the pair \((p^{-1}(C),A\cap p^{-1}(C))\) is a bundle pair with respect to \(\mathcal{F}| C\). Moreover, call a topological pair \((X,A)\) suitable if \(X\) is a finite polyhedron with no local cut points and \(A\) is a closed locally contractible subspace of \(X\) such that \(X\setminus A\) is not a 2-manifold and \(A\) can be by-passed in \(X\) [cf., H. Schirmer (op. cit.)]. Let \(\mathcal{F}\) and \(f\) be as above and assume that \(A\) is closed and locally contractible, that all components of \(p(A)\) are contractible and that \((B,p(A))\) and \((Y,W)\) are suitable pairs whenever \(W\) is a subbundle fiber of \(A\). Assume that (C1) and \((C2)\) hold and that \(A\) intersects every essential fixed point class of \(f_b:p^{-1}(\{b\})\to p^{-1}(\{b\})\) for at least one \(b\) in each component of \(p(A)\). If then \(Z\) is a closed bundle subset of \(A\) that intersects every component of \(A\) then there exists a map \(g:E\to E\) that is fiber-homotopic to \(f\) which has \(Z\) as its fixed point set.