Homotopies and the Universal Fixed Point Property
A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.
KeywordsFixed point property Homotopy Kolmogorov space Finite poset
Unable to display preview. Download preview PDF.
- 3.Birkhoff, G.: Lattice theory. 3rd. edition. American Mathematical Society, Providence, R.I., 1967 (1967)Google Scholar
- 5.Carlsson, G.: Segal’s Burnside ring conjecture and related problems in topology. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley CA 1986) 574–579. Amer. Math. Soc., Providence, RI (1987)Google Scholar
- 10.Miller, H.R.: The Sullivan conjecture and homotopical representation theory. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley CA 1986) 580–589. Amer. Math. Soc., Providence, RI (1987)Google Scholar