Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED POINT SETS OF MAPS HOMOTOPIC TO A GIVEN MAP | FIXED POINT SETS OF MAPS HOMOTOPIC TO A GIVEN MAP CHRISTINA L. SODERLUND Received 3 December 2004 Revised 20 April 2005 Accepted 24 July 2005 Let f X - X be a self-map of a compact connected polyhedron and o Q X a closed subset. We examine necessary and sufficient conditions for realizing o as the fixed point set of a map homotopic to f. For the case where o 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 o is only assumed to be a locally contractible subset of X. The relative form of the realization problem has also been solved for o a subpolyhedron of X. We also extend these results to the case where o is a locally contractible subset. 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. 1. Introduction Let f X - X be a self-map of a compact connected polyhedron. For any map g denote the fixed point set of g as Fixg x e X I g x x . In this paper we are concerned with the realization of an arbitrary closed subset o Q X as the fixed point set of a map g homotopic to f . Several necessary conditions for this problem are well known. If o Fixg for some map g homotopic to f it is clear that o must be closed. Further by the definition of a fixed point class cf. 1 page 86 7 page 5 all points in a given component of o must lie in the same fixed point class. Thus as the Nielsen number cf. 1 page 87 7 page 17 of any map cannot exceed the number of fixed point classes and as the Nielsen number is also a homotopy invariant the set o must have at least N f components. In particular if N f 0 then o must be nonempty. It is also necessary that f Io the restriction of f to the set o must be homotopic to the inclusion map i o X. In
