Vol. 63, No. 2, 1976

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Mappings of polyhedra with prescribed fixed points and fixed point indices

Helga Schirmer

Vol. 63 (1976), No. 2, 521–530

The following problem is studied: If points ck of a polyhedron and integers ik are given, when does there exist a selfmap within a given homotopy class which has the ck as its fixed points and the ik as its fixed point indices? Necessary and sufficient conditions for the existence of such selfmaps are established if the selfmap is a deformation and the polyhedron is of type W, and if the selfmap is arbitrary and the polyhedron is of type S. It is further shown that there always exists a selfmap of an n-sphere (n 2) which has arbitrarily prescribed locations and indices of its fixed points. The proofs are based on Shi Gen-Hua’s construction of selfmaps with a minimum number of fixed points.

Mathematical Subject Classification
Primary: 55C20, 55C20
Secondary: 57C05
Received: 20 September 1974
Revised: 22 January 1975
Published: 1 April 1976
Helga Schirmer