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 $\alpha \in H_1({\pi }_1(T^n),\mathbb{Z}).$