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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A method of combining fixed points
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by Roger Waggoner PDF
Proc. Amer. Math. Soc. 51 (1975), 191-197 Request permission

Abstract:

It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points $x$ and $y$ of a self-map of a polyhedron $K$ can be combined in a certain way if a certain criterion is satisfied by the $f$-image of a path from $x$ to $y$. Several corollaries follow, one of which is that if $K$ is a finite simply connected $2$-polyhedron with no local separating points, ${H_2}(K) \ne 0$, and $K$ has a $2$-simplex $\sigma$ such that ${\pi _1}(K - \operatorname {Int} \sigma ,z)$ is cyclic, then $K$ fails to have the fixed point property. This eliminates many $2$-dimensional polyhedra from consideration as examples.
References
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 191-197
  • MSC: Primary 54H25; Secondary 57C05
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0402713-X
  • MathSciNet review: 0402713