Fixed point sets of fiber-preserving maps

Abstract

Let \(\mathfrak{F} = (E, p, B; Y)\) be a fiber bundle where E, B and Y are connected finite polyhedra. Let \(f : E \longrightarrow E\) be a fiber-preserving map and \(A \subseteq E\) a closed, locally contractible subset. We present necessary and sufficient conditions for A and its subsets to be the fixed point sets of maps fiber homotopic to f. The necessary conditions correspond to those introduced by Schirmer in 1990 but, in the fiber-preserving setting, homotopies are fiberpreserving. Those conditions are shown to be sufficient in the presence of additional hypotheses on the bundle and on the map f. The hypotheses can be weakened in the case that f is fiber homotopic to the identity.