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Mappings of polyhedra with prescribed fixed points and fixed point indices
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.
... 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. ...
The Bass trace conjectures are placed in the setting of homotopy idempotent selfmaps of manifolds. For the strong conjecture, this is achieved via a formulation of Geoghegan. The weaker form of the conjecture is reformulated as a comparison of ordinary and L^2-Lefschetz numbers.
... 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 ...
In recent years it has been shown that many spaces have the so-called complete invariance property, i.e. that every closed and nonempty subset of them can be realized as the fixed point set of a continuous selfmap. Here a related result is obtained for homotopies H: X × I → X rather than self maps of a space X. The theorem proved here states that if P is a compact and connected polyhedron without local cut points and K ⊂ P × I a closed set which contains a continuum intersecting both X × 0 and X × 1, then there exists a homotopy H: P × I → P with fixed point set K.
This paper deals with some questions concerning the existence of a fixed point, minimal fixed point sets, and prescribed fixed point sets on A, X − A and Cℓ (X − A) for a given selfmap f: (X,A) → (X,A) of a pair of spaces (X,A) and maps homotopic to f. Some new answers are presented, and existing results as well as open questions are discussed. The main tool is the relative Nielsen number, which was recently introduced for selfmaps of a pair of compact metrizable ANR’s, and which is here generalized to “admissible” selfmaps of pairs of arbitrary metrizable ANR’s.
For a symmetric monoidal closed category B satisfying certain completeness conditions, consider a B-category u, a 5u, a subcategory ∈ of u which admits a B-calculus of left fractions, and a B-monad I = (T, η, μ) on u. Suppose I is compatible with ∈ so that a B-monad I' is induced on u[∈-1] and the canonical projection B-functor ɸ: u →[∈-1] induces a B-functor L:uT' on the B-categories of Eilenberg- Moore algebras. Suppose that ∈ is conice and uT has coequalizers. We prove that, if L preserves coequalizers (which is true in the case where T preserves coequalizers), then L is the canonical projection for the B-localization of a subcategory of BT which admits a B-calculus of left fractions.
In this paper, we introduce a Nielsen type number N(ƒ, p) for a fibre preserving map ƒ of a fibration p; we show that it is a lower bound for the least number of fixed points within the fibre homotopy class of ƒ. The number N(ƒ, p), which can be thought of as the dual of the relative Nielsen number due to Schirmer, is often much bigger than the ordinary Nielsen number, N(ƒ), of ƒ. It shares with N(ƒ) such properties as homotopy invariance and commutativity. The definition of N(ƒ, p) is reminiscent of the so-called naïve product formula due to Brown.In this paper, we also exhibit and exploit a connection between the relative Nielsen number and N(ƒ, p); we compare N(ƒ, p) and N(ƒ); give necessary and sufficient conditions for N(ƒ, p) and N(ƒ) to coincide, and show, under fairly mild conditions, that our lower bound is sharp. Some corollaries concerning minimum fixed point sets for ordinary Nielsen numbers of a fibre map are given.
Let ⨍:X→X be a self-map of a compact ANR and A ⊂ X a closed subset. Two conditions are stated which are necessary for the realization of A as the fixed point set of a map g in the homotopy class of f, and it is shown that these conditions are also sufficient in many cases if X is a compact connected polyhedron and if A can be by-passed. The proofs use methods from relative Nielsen fixed point theory.
This paper gives a partial answer to the problem of establishing conditions for the existence of selfmaps of one-dimensional spaces with prescribed fixed points and fixed point indices. Two types of isolated fixed points on dendrites are defined, and called effluent and n on effluent fixed points. They correspond on polyhedral trees to fixed points of minimal or maximal algebraic index, but are characterized by separation properties. Necessary and sufficient conditions are given for the existence of a selfmap of a dendrite which has a prescribed set of effluent and noneffluent fixed points.
A relative Nielsen number N(f; X, A) for a self map f: (X, A) → (X, A) of a pair of spaces is introduced which shares such properties with the Nielsen number N(f) as homotopy invariance and homotopy type invariance. As N(f; X, A) ≥ N(f) = N(f; X, Ø), the relative Nielsen number is in the case A ≠ Ø a better lower bound than N(f) for the minimum number MF[f; X, A] of fixed points of all maps in the homotopy class of f. Conditions for a compact polyhedral pair (X, A) are given which ensure that the relative Nielsen number is in fact the best possible lower bound, i.e. that N(f; X, A) = MF[f; X, A].
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