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Transactions of the American Mathematical Society

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

 
 

 

The fixed-point property
for simply connected plane continua


Author: Charles L. Hagopian
Journal: Trans. Amer. Math. Soc. 348 (1996), 4525-4548
MSC (1991): Primary 54F15, 54H25; Secondary 55M20, 57N05
DOI: https://doi.org/10.1090/S0002-9947-96-01582-6
MathSciNet review: 1344207
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Abstract: We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let $M$ be a plane continuum with the property that every simple closed curve in $M$ bounds a disk in $M$. Then every map of $M$ that sends each arc component into itself has a fixed point. Hence every deformation of $M$ has a fixed point. These results are corollaries to the following general theorem. If $M$ is a plane continuum, $\mathcal {D}$ is a decomposition of $M$, and each element of $\mathcal {D}$ is simply connected, then every map of $M$ that sends each element of $\mathcal {D}$ into itself has a fixed point.


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Additional Information

Charles L. Hagopian
Affiliation: Department of Mathematics, California State University, Sacramento, California 95819-6051
Email: hagopian@csus.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01582-6
Keywords: Fixed-point property, plane continuum, simply-connected set, arcwise connectivity, fundamental group, interior domain, deformation, decomposition, ray, dog-chases-rabbit principle
Additional Notes: The CSUS Research, Scholarship, and Creative Activities Program supported this work. Piotr Minc proofread the original manuscript and made several comments that led to its improvement.
Article copyright: © Copyright 1996 American Mathematical Society

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