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Transactions of the American Mathematical Society

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



Fixed points of arc-component-preserving maps

Author: Charles L. Hagopian
Journal: Trans. Amer. Math. Soc. 306 (1988), 411-420
MSC: Primary 54F20; Secondary 54H25
MathSciNet review: 927698
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Abstract: The following classical problem remains unsolved: If $M$ is a plane continuum that does not separate the plane and $f$ is a map of $M$ into $M$, must $f$ have a fixed point? We prove that the answer is yes if $f$ maps each arc-component of $M$ into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author’s theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.

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Keywords: Fixed-point property, deformation, arc-component-preserving map, nonseparating plane continua, indecomposable continua, internal composant, tree-like continua, uncountably many disjoint triods, free chain, Borsuk ray, dog-chases-rabbit principle
Article copyright: © Copyright 1988 American Mathematical Society