Duan's fixed point theorem: Proof and generalization

Martin Arkowitz

doi:10.1155/fpta/2006/17563

Abstract

Let be an H-space of the homotopy type of a connected, finite CW-complex, any map and the th power map. Duan proved that has a fixed point if . We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a -structure as defined by Hemmi-Morisugi-Ooshima. The conclusion is that and each has a fixed point.

Highlights

Let G be a topological group and f : G → G a map
We begin by briefly discussing the Lefschetz number and H-spaces
We examine the matrix of pk∗n and f ∗n with respect to this basis

Summary

Introduction

Let G be a topological group and f : G → G a map (i.e., a continuous function). If G is a compact, connected topological group and f : G → G is a map, for any k ≥ 2, the map pk f : G → G has a fixed point. This theorem was proved more generally for homotopy-associative H-spaces having the homotopy type of a finite, connected CW-complex (Theorem 2.2). In 1996, Lupton and Oprea [2] gave a new proof of Duan’s theorem using rational homotopy theory.

Duan’s fixed point theorem: proof and generalization

Duan’s theorem

Fixed points and eigenvalues

Theta spaces