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Fixed point sets of deformations of polyhedra with local cut points. (English) Zbl 0897.54015

Given a polyhedron \(X\) (i.e. a locally finite connected simplicial complex), a deformation of \(X\) is a map homotopic to the identity map. Polyhedra that are 2-dimensionally connected, i.e. those \(X\) for which \(X-\){local cut points of \(X\}\) is connected, admit deformations with at most one fixed point. The study of polyhedra which are not 2-dimensionally connected began with establishing an equivalence between fixed point sets of deformations and fixed point sets of suitable combinatorial maps called good displacements [G.-H. Shi, Pac. J. Math. 103, No. 2, 377-387 (1982; Zbl 0522.55003)].
The main result of this paper is a realization theorem based on an extension of Hall’s theorem. The realization theorem gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. From the theorem, both Scholz’s characterization of all finite polyhedra which admit fixed point free deformation and a characterization of deformations with only one fixed point follow.
Reviewer: M.Demlová (Praha)

MSC:

54C99 Maps and general types of topological spaces defined by maps
05C90 Applications of graph theory
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0522.55003
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References:

[1] P. Hall, On representatives of subsets, J. London Math Soc. 10 (1935), 26-30. · Zbl 0010.34503
[2] Kiang Tsai-han, The theory of fixed point classes, Translated from the second Chinese edition, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1989. · Zbl 0676.55001
[3] Oystein Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962.
[4] U. Kurt Scholz, Fixed point deformations on compact polyhedra, Nonlinear functional analysis and its applications (Maratea, 1985) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 173, Reidel, Dordrecht, 1986, pp. 387 – 392. · Zbl 0647.55001
[5] Gen Hua Shi, On the least number of fixed points for infinite complexes, Pacific J. Math. 103 (1982), no. 2, 377 – 387. · Zbl 0522.55003
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