Duan's Fixed Point Theorem: Proof and Generalization

Authors

Martin Arkowitz, Dartmouth College

Document Type

Article

Publication Date

2-26-2006

Publication Title

Fixed Point Theory and Applications

Department

Department of Mathematics

Abstract

Let X be an H-space of the homotopy type of a connected, finite CW-complex, f : X→X any map and p_k : X→X the k^th power map. Duan proved that p_kf : 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 μ_θ : X → X as defined by Hemmi-Morisugi-Ooshima. The conclusion is that μ_θf and f_μθ each has a fixed point.

DOI

10.1155/FPTA/2006/17563

Original Citation

Arkowitz, M. Duan's fixed point theorem: Proof and generalization. Fixed Point Theory Appl 2006, 17563 (2006). https://doi.org/10.1155/FPTA/2006/17563

Dartmouth Digital Commons Citation

Arkowitz, Martin, "Duan's Fixed Point Theorem: Proof and Generalization" (2006). Dartmouth Scholarship. 2419.
https://digitalcommons.dartmouth.edu/facoa/2419