Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Y \rightarrow E {\mathop {\rightarrow }\limits ^{p}} B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>E</mi> <mover> <mo stretchy="false">→</mo> <mi>p</mi> </mover> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> be a fibration and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f: E \rightarrow E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N_B P_n(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>B</mi> </msub> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. When B is a point, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_B P_n(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>B</mi> </msub> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> coincides with the classical Nielsen periodic number.

A Nielsen type periodic number for fiber maps over the base

期刊论文
发表日期：

2025年9月20日

查看全文

Weslem Liberato Silva,
Rafael Moreira de Souza

摘要

Let $Y \rightarrow E {\mathop {\rightarrow }\limits ^{p}} B$ $Y \to E \overset{p}{\to} B$ be a fibration and let $f: E \rightarrow E$ $f : E \to E$ 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 $N_B P_n(f)$ $N_{B} P_{n} (f)$ . When B is a point, then $N_B P_n(f)$ $N_{B} P_{n} (f)$ coincides with the classical Nielsen periodic number.