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Proceedings of the American Mathematical Society

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



A method of combining fixed points

Author: Roger Waggoner
Journal: Proc. Amer. Math. Soc. 51 (1975), 191-197
MSC: Primary 54H25; Secondary 57C05
MathSciNet review: 0402713
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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.

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Keywords: Fixed point property, Lefschetz number, fixed point index
Article copyright: © Copyright 1975 American Mathematical Society