A method of combining fixed points

Abstract:

It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points $x$ and $y$ of a self-map of a polyhedron $K$ can be combined in a certain way if a certain criterion is satisfied by the $f$-image of a path from $x$ to $y$. Several corollaries follow, one of which is that if $K$ is a finite simply connected $2$-polyhedron with no local separating points, ${H_2}(K) \ne 0$, and $K$ has a $2$-simplex $\sigma$ such that ${\pi _1}(K - \operatorname {Int} \sigma ,z)$ is cyclic, then $K$ fails to have the fixed point property. This eliminates many $2$-dimensional polyhedra from consideration as examples.