Duan's fixed point theorem: Proof and generalization

Abstract

Let X be an H-space of the homotopy type of a connected, finite CW-complex, f:X→X any map and pk:X→X the kth power map. Duan proved that pkf:X→X has a fixed point if k≥2. We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces X 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 θ-structure μθ:X→X as defined by Hemmi-Morisugi-Ooshima. The conclusion is that μθf and fμθ each has a fixed point.