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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fixed point sets of deformations of polyhedra with local cut points
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by Peter Wolfenden PDF
Trans. Amer. Math. Soc. 350 (1998), 2457-2471 Request permission

Abstract:

A locally finite simplicial complex $X$ is said to be 2-dimensionally connected if $X - \{\text {local cut points of } X \}$ 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.
References
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  • 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
  • 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
  • Received by editor(s): June 20, 1995
  • Received by editor(s) in revised form: August 12, 1996
  • © Copyright 1998 American Mathematical Society
  • 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