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# Mappings of polyhedra with prescribed fixed points and fixed point indices

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## Abstract

The following problem is studied: If points ck of a polyhedron and integers ik are given, when does there exist a self map 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.

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... First, various results in the literature (see in particular Wecken [31], Brown [4], and Shi [27] for the PL case; Jiang [16] for the smooth case) show that every selfmap of M with Nielsen number N is homotopic to a map with exactly N fixed points. By a result of Schirmer [26,Lemma 2], these fixed points may be chosen arbitrarily. ...
... Second, recall from [26] that every fixed-point-free selfmap of a connected compact PL manifold of dimension at least 3 is homotopic to a selfmap having an arbitrary unique fixed point. The argument there can be adapted as follows. ...
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... is fixed point free, it follows from the homotopy property of the index that ind(/£+i> cl °( s ') X (/ + 1)/2 W+1 ) = 0. So we can use the proof of Lemma 1 in [6] to split the fixed point of i^' + j into two points, one of them of index x(^) an< 3 the other of index — x(^)> an d then use [2], Ch. VIII C, Lemmas 2 and 3, to move the fixed point of index -x(^) to y x and the one of index x(^) to JV Thus we change i^' +1 to a map F n+x \c\O{S') X (/+ l)/2* +1 which satisfies (F n+l ,l) to (F n + l9 4) and ...
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It is shown that every closed nonempty subset of a polyhedron can be the fixed point set of a suitable self-map if the polyhedron satisfies a certain connectedness condition. Hence the same is true for all compact triangulable manifolds with or without boundary. The proof uses existing results on deformations of polyhedra with a minimum number of fixed points if the dimension of the polyhedron is at least two, and on self-maps of dendrites with given fixed point sets if the dimension of the polyhedron is one.
On the least number of fixed points and Nielsen numbers This research was partially supported by the National Research Council of Canada (Grant A- 7579)
• Shi Gen
Shi Gen-Hua, On the least number of fixed points and Nielsen numbers, Chinese Math., 8 (1966), 234-243. Received September 20, 1974 and in revised form January 22, 1975. This research was partially supported by the National Research Council of Canada (Grant A- 7579). CARLETON UNIVERSITY, OTTAWA, CANADA