The equivalence of two methods: finding representatives of non-empty Nielsen classes (Q1704387)

A central question in fixed point theory is the computation of the Nielsen number. In [\textit{J. Wagner}, Trans. Am. Math. Soc. 351, No. 1, 41--62 (1999; Zbl 0910.55001)], an algorithm was given for calculating the Nielsen number of self-maps of hyperbolic surfaces with boundary where it is only needed to know the induced map on the fundamental group level. In [\textit{E. Fadell} and \textit{S. Husseini}, Contemp. Math. 21, 59--98 (1983; Zbl 0563.55001)], a procedure for calculating the Nielsen number by using Fox calculus was presented. In this note, the authors prove an equivalence among these techniques.