Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 46052, 20 pages
Fixed point sets of maps homotopic to a given map
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 be a self-map of a compact, connected
polyhedron and a closed subset. We examine
necessary and sufficient conditions for realizing as the fixed point set of a map homotopic to . For the case where
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 is only assumed to be a locally
contractible subset of . The relative form of the realization
problem has also been solved for a subpolyhedron of . We also extend these results to the case where is a locally contractible subset.