A Method of Combining Fixed Points

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 /-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 AK) ?t 0, and K has a 2-simplex cr such that 17.(X — Inter, z) is cyclic, then K fails to have the fixed point property. This eliminates many 2-dimensional polyhedra from consideration as examples.