Computing the one-parameter Nielsen number for homotopies on the n-torus

Abstract

Let $F: T^{n} \times I \to T^{n}$ be a homotopy on a n-dimensional torus. The main purpose of this paper is to present a formula for the one-parameter Nielsen number $N(F)$ of $F$ in terms of its induced homomorphism. If $L(F)$ is the one-parameter Lefschetz class of $F$ then $L(F)$ is given by $L(F) = \ N(F)\alpha,$ for some $\alpha \in H_{1}(\pi_{1}(T^{n}),\mathbb{Z}).$