Fixed points of arc-component-preserving maps

Hagopian, Charles L.

doi:10.1090/S0002-9947-1988-0927698-2

Fixed points of arc-component-preserving maps
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Trans. Amer. Math. Soc. 306 (1988), 411-420 Request permission

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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