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Non-additivity of the fixed point property for tree-like continua

Volume 231 / 2015

C. L. Hagopian, M. M. Marsh Fundamenta Mathematicae 231 (2015), 113-137 MSC: Primary 54F15, 54H25; Secondary 54G20. DOI: 10.4064/fm231-2-2

Abstract

We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge $\hat X$ of tree-like continua, each having the fpp, and $\hat X$ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua $H$ and $K$ such that $H$, $K$, and $H\cap K$ have the fixed point property, but $H\cup K$ admits a fixed-point-free homeomorphism. In an appendix we verify some of the properties of Bellamy's continuum.

Authors

  • C. L. HagopianDepartment of Mathematics & Statistics
    California State University, Sacramento
    Sacramento, CA 95819-6051, U.S.A.
    e-mail
  • M. M. MarshDepartment of Mathematics & Statistics
    California State University, Sacramento
    Sacramento, CA 95819-6051, U.S.A.
    e-mail

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