Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 17563, 10 pages
doi:10.1155/FPTA/2006/17563
Duan's fixed point theorem: Proof and generalization
Department of Mathematics, Dartmouth College, Hanover 03755, NH, USA
Received 25 July 2004; Revised 6 January 2005; Accepted 21 July 2005
Copyright © 2006 Martin Arkowitz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.