Equivariant preimage theory for G-maps

Monis, Thaís F M; Wong, Peter

doi:10.2140/agt.2026.26.1529

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ISSN (electronic): 1472-2739

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Equivariant preimage theory for $G$-maps

Thaís F M Monis and Peter Wong

Algebraic & Geometric Topology 26 (2026) 1529–1548

DOI: 10.2140/agt.2026.26.1529

Abstract

Let $X$ and $Y$ be closed $G$ -manifolds and $B \subset Y$ a closed invariant nonempty subset where $G$ is a finite group. For any $G$ -map $f : X \to Y$ and for every subgroup $H \leq G$ , we introduce a Nielsen type number $N (f^{H}, B^{H})$ which is a lower bound for the number of connected components of $W H$ -orbits of ${(f^{H})}^{- 1} (B^{H})$ . This theory generalizes existing Nielsen type numbers for various $G$ and $B$ with an application to the Nielsen Borsuk–Ulam theory for the minimal number of coincidences of $f (x) = f τ (x)$ where $f : X \to Y$ and $τ$ a free involution on $X$ .

Keywords

Nielsen theory, Borsuk–Ulam type theorem, preimage theory

Mathematical Subject Classification

Primary: 55M20

Secondary: 57S99

References

Publication

Received: 1 June 2024

Revised: 22 April 2025

Accepted: 4 May 2025

Published: 25 April 2026

Authors

Thaís F M Monis
Instituto de Geociências e Ciências Exatas Universidade Estadual Paulista Rio Claro Brazil

Peter Wong
Department of Mathematics Bates College Lewiston, ME United States

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Volume 26, issue 4 (2026)