Open Access

Fixed point sets of maps homotopic to a given map

Fixed Point Theory and Applications20062006:46052

DOI: 10.1155/FPTA/2006/46052

Received: 3 December 2004

Accepted: 24 July 2005

Published: 20 April 2006

Abstract

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.

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Authors’ Affiliations

(1)
Department of Mathematics, California Lutheran University

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Copyright

© Christina L. Soderlund. 2006

This article is published under license to BioMed Central Ltd. 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.