Fixed point sets of deformations of polyhedra with local cut points

Author:
Peter Wolfenden

Journal:
Trans. Amer. Math. Soc. **350** (1998), 2457-2471

MSC (1991):
Primary 54C99; Secondary 05C90

DOI:
https://doi.org/10.1090/S0002-9947-98-01941-2

MathSciNet review:
1422912

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Abstract: A locally finite simplicial complex is said to be 2-dimensionally connected if is connected. Such spaces exhibit ``classical'' behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi's results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall's theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz's result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.

**1.**P. Hall,*On representatives of subsets*, J. London Math Soc.**10**(1935), 26-30.**2.**Kiang Tsai-han,*The theory of fixed point classes*, Translated from the second Chinese edition, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1989. MR**1002187****3.**Oystein Ore,*Theory of graphs*, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962. MR**0150753****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. MR**852598****5.**Gen Hua Shi,*On the least number of fixed points for infinite complexes*, Pacific J. Math.**103**(1982), no. 2, 377–387. MR**705237**

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Additional Information

**Peter Wolfenden**

Email:
wolfen@bbnplanet.com

DOI:
https://doi.org/10.1090/S0002-9947-98-01941-2

Keywords:
Deformation,
fixed point,
local cut point,
part,
welding vertex,
marriage theorem

Received by editor(s):
June 20, 1995

Received by editor(s) in revised form:
August 12, 1996

Article copyright:
© Copyright 1998
American Mathematical Society