Some examples in fixed point theory

Abstract

It is known that the fixed point property (f.p.p.) is not invariant under suspension and join in the category of simply connected polyhedra. In this paper we exhibit examples to show that f .p.p. is not invariant under suspension and join in the category of simply connected polyhedra satisfying the Shi condition and more strongly, in the category of simply connected compact manifolds. We also exhibit a simply connected polyhedron X such that the smash product X A X fails to have f.p.p. if one choice of base point is used to form X A X, while X A X has f .p.p. using another choice of base point. In the last section we prove that f .p.p. is invariant under Cartesian products in very special circumstances.