Fixed points and homotopy fixed points

2020

Smith theory says that the fixed point of a semi-free action of a group G on a contractible space is Z p-acyclic for any prime factor p of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain K-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the K-theoretical obstruction.

arXiv (Cornell University), 2020

arXiv (Cornell University), 2020

Proceedings of the National Academy of Sciences of the United States of America, 1959

Annales Henri Lebesgue, 2019

The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of topological groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts. Résumé.-La correspondance de Kechris-Pestov-Todorcevic établit une relation entre la moyennabilité extrême des groupes topologiques et les propriétés de type Ramsey de certaines classes de structures finies. Le but de cet article est de la resituer comme une instance particulière d'une construction plus générale, permettant ainsi de montrer que des énoncés de type Ramsey apparaissent en fait comme l'expression combinatoire naturelle de l'existence de points fixes dans certaines compactifications de groupes, et que des correspondances similaires sont en réalité présentes dans toute une variété de contextes dynamiques.

arXiv: Algebraic Topology, 2020

Smith theory says that the fixed point of a semi-free action of a group G on a contractible space is Z_p-acyclic for any prime factor p of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain K-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the K-theoretical obstruction.

arXiv (Cornell University), 2021

Let H be a group acting on a simply-connected diagrammatically reducible combinatorial 2-complex X with fine 1skeleton. If the fixed point set X H is non-empty, then it is contractible. Having fine 1-skeleton is a weaker version of being locally finite.

数理解析研究所講究録, 2014

K-Theory, 2000

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each K ___ G, yK is dim(X x)-connected and simple. G acts on the function complex F(X, Y) by conjugation of maps. We give a complete analysis of the homotopy fixed point set of the space 92~E~ Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on f~E~AX, where AX denotes the free loop space of X, and X is a simply connected finite complex.

Fundamenta Mathematicae, 2007

Let G be a compact group and X a G-ANR. Then X is a G-AR iff the H-fixed point set X H is homotopy trivial for each closed subgroup H ⊂ G.