Duan's fixed point theorem: Proof and generalization

Let "Equation missing" be an H-space of the homotopy type of a connected, finite CW-complex, "Equation missing" any map and "Equation missing" the "Equation missing" th power map. Duan proved that "Equation missing" has a fixed point if "Equation missing" . We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces "Equation missing" 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 "Equation missing" -structure "Equation missing" as defined by Hemmi-Morisugi-Ooshima. The conclusion is that "Equation missing" and "Equation missing" each has a fixed point.