The equivalence of two methods: finding representatives of non--empty Nielsen classes

Abstract

Let $f:X\to X$ be a self--map with $X$ a wedge of circles or a compact surface with boundary, so that the fundamental group of $X$ is finitely generated and free. In [3], Wagner presents an algorithm for extracting information from the homomorphism induced by $f$ on the fundamental group. This information involves the fixed point index of $f$ and the Nielsen classes of fixed points of $f$. The step in which the representatives of Nielsen classes, Wagner tails, are calculated is equivalent to a step in the method presented by Fadell and Husseini in [1]. The Fadell--Husseini method was designed for closed two dimensional CW--complexes, but the step in which they use the Fox calculus, to determine terms in the unreduced Reidemeister trace, produces the Wagner tails and their indices. The equivalence of these steps was stated in [2] without proof. Further developments in this area have caused continued interest in the techniques, and a clarification of the equivalence is needed. Here we provide the proof and an example.