Some spaces that do not have the common fixed point property

Abstract

For what topological spaces X do every pair of self maps of X which commute under composition have a common fixed point? No nontrivial examples of such spaces are known. Since every self map commutes with itself, X does not have this property if X does not have the fixed point property. It is shown that every completely regular Hausdorff space containing an arc does not have this property. In general, the self maps for these spaces are not surjective. The image is the arc. For surjective self maps it is shown that every topological manifold with nonnegative Euler characteristic does not have this property. An earlier counterexample for the closed interval is used in all proofs. This counterexample is due to Huneke.