Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 46052, 20 pages

Fixed point sets of maps homotopic to a given map

Christina L. Soderlund

Department of Mathematics, California Lutheran University, 60 West Olsen Road 3750, Thousand Oaks 91360-2700, CA, USA

Received 3 December 2004; Revised 20 April 2005; Accepted 24 July 2005

Copyright © 2006 Christina L. Soderlund. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let f:XX be a self-map of a compact, connected polyhedron and ΦX 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.