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

TL;DR

This paper derives a formula for the one-parameter Nielsen number of homotopies on an n-torus using induced homomorphisms, linking it to the Lefschetz class in algebraic topology.

Contribution

It provides a novel explicit formula for the one-parameter Nielsen number on the n-torus based on the induced homomorphism of the homotopy.

Findings

Derived a formula for N(F) in terms of induced homomorphism

Connected N(F) to the one-parameter Lefschetz class L(F)

Established a relationship between N(F) and algebraic topology invariants

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 $\alpha \in H_{1}(\pi_{1}(T^{n}),\mathbb{Z}).$