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 ∧ X fails to have f.p.p. if one choice of base point is used to form
X ∧X, while X ∧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.
