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

C. L. Hagopian ¹   ; M. M. Marsh ¹

¹ Department of Mathematics & Statistics California State University, Sacramento Sacramento, CA 95819-6051, U.S.A.

Fundamenta Mathematicae, Tome 231 (2015) no. 2, pp. 113-137

Résumé

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.

Détail

DOI : 10.4064/fm231-2-2

Keywords: investigate fixed point property tree like continua unions tree like continua obtain positive result finitely many tree like continua fixed point property have dendrites pairwise intersections using bellamys seminal example define countable wedge hat tree like continua each having fpp hat admitting fixed point free homeomorphism tree like continua cap have fixed point property cup admits fixed point free homeomorphism appendix verify properties bellamys continuum