Twisted Lefschetz number and Nielsen number

TL;DR

This paper explores the twisted Lefschetz number, providing a homology-based formula, and introduces a method to estimate the Nielsen number, including an algorithm for abelian fundamental groups and a proof related to Bass' conjecture.

Contribution

It presents a new formula for the twisted Lefschetz number and a novel approach to estimate the Nielsen number, with applications to algorithms and group theory conjectures.

Findings

Derived a homology formula for the twisted Lefschetz number

Proposed a method to estimate the Nielsen number

Developed an algorithm for spaces with abelian fundamental groups and proved Bass' conjecture for certain groups

Abstract

We study the twisted Lefschetz number, which is a generalization of classical Lefschetz number. A formula in terms of homology with local coefficients is given. We then propose a method to estimate the Nielsen number. As applications, an algorithm for computing the Nielsen number of a self map of a space with abelian fundamental group is discussed; and Bass' conjecture for a class of groups is proved.