A Nielsen type periodic number for maps over $B$

TL;DR

This paper introduces a Nielsen-type periodic number for fiber maps over a base space, extending classical Nielsen periodic number concepts to a fibered setting with both geometric and algebraic considerations.

Contribution

It defines a new invariant, $N_B P_n(f)$, for fiber maps over a base space, generalizing Nielsen periodic numbers to fibered spaces.

Findings

$N_B P_n(f)$ coincides with classical Nielsen periodic number when $B$ is a point.

The work provides a framework for analyzing periodic points in fibered spaces.

It explores geometric and algebraic Reidemeister classes of iterates of fiber maps.

Abstract

Let $Y \to E \stackrel{p}{\to} B$ be a fibration and let $f: E \to E$ be a fiber map over $B$. In this work, we study the geometric and algebraic Reidemeister classes of the iterates of $f$ and introduce a Nielsen-type periodic number over $B$, denoted by $N_B P_n(f)$. When $B$ is a point, then $N_B P_n(f)$ coincides with the classical Nielsen periodic number.