One-parameter Lefschetz class of homotopies on torus

TL;DR

This paper establishes a formula linking the one-parameter Lefschetz class of a homotopy on a torus to its Nielsen number and fundamental group generators, providing a clear algebraic-topological relationship.

Contribution

It proves that the one-parameter Lefschetz class for homotopies on a torus can be explicitly expressed in terms of the Nielsen number and fundamental group generators.

Findings

L(F) = ± N(F)α, relating Lefschetz class to Nielsen number and generators

Provides an explicit algebraic formula for homotopies on the torus

Connects topological invariants with algebraic structures in homotopy theory

Abstract

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 $H_{1}(\pi_{1}(T),\mathbb{Z})$.