In 1953, Barrett O’Neill stated
axioms for a “fixed point index” and obtained existence and uniqueness theorems for
the index on finite polyhedra. The proof of uniqueness consisted of showing that any
function which satisfied the axioms must agree with the index he had already defined.
This paper presents a proof of the uniqueness of the fixed point index on finite
polyhedra which depends only on the axioms and therefore is “elementary” in the
sense that it is independent of the existence of an index. The proof is “elementary”
also in that all the techniques used are taken from geometric topology or
calculus so that, in particular, no algebraic topology is required. An elementary
proof of the uniqueness of the fixed point index on compact metric absolute
neighborhood retracts is an immediate consequence of the material in this
paper.
