Fixed Point Theory and Applications
Volume 2006 (2006), Issue 2, Pages Article 46052, 20 p.
doi:10.1155/FPTA/2006/46052

Fixed point sets of maps homotopic to a given map

Abstract

Let f:X→X be a self-map of a compact, connected polyhedron and Φ⊆X a closed subset. We examine necessary and sufficient conditions for realizing Φ as the fixed point set of a map homotopic to f. For the case where Φ is a subpolyhedron, two necessary conditions were presented by Schirmer in 1990 and were proven sufficient under appropriate additional hypotheses. We will show that the same conditions remain sufficient when Φ is only assumed to be a locally contractible subset of X. The relative form of the realization problem has also been solved for Φ a subpolyhedron of X. We also extend these results to the case where Φ is a locally contractible subset.