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The Fixed-Point Property for Simply Connected Plane Continua

Charles L. Hagopian
Transactions of the American Mathematical Society
Vol. 348, No. 11 (Nov., 1996), pp. 4525-4548
Stable URL: http://www.jstor.org/stable/2155431
Page Count: 24
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Fixed-Point Property for Simply Connected Plane Continua
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Abstract

We answer a question of R. Mańka 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, D is a decomposition of M, and each element of D is simply connected, then every map of M that sends each element of D into itself has a fixed point.

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