One-parameter Lefschetz class of homotopies on torus

The main result this paper states that if $F: T \times I \to T$ is a homotopy on torus then the one-parameter Lefschetz class $L(F)$ of $F$ is given by $L(F) = \pm N(F)\alpha$, where $N(F)$ is the one-parameter Nielsen number of $F$ and $\alpha$ is one of the two generators in $H_{1}(\pi_{1}(T),\mathbb{Z})$.