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 $N\left(F\right)$ of F in terms of its induced homomorphism. If $L\left(F\right)$ is the one-parameter Lefschetz class of F then $L\left(F\right)$ is given by $L\left(F\right)=N\left(F\right)\alpha$, for some $\alpha \in H_1\left({\pi }_1\right(T^n),Z)$.