Borsuk--Ulam property for graphs II: The $\mathbb{Z}_n$-action

TL;DR

This paper characterizes when the Borsuk--Ulam property holds for maps between spaces with group actions, using geometric and algebraic conditions, especially focusing on graphs and cyclic groups with applications to configuration spaces.

Contribution

It provides a new geometric and algebraic framework to determine the Borsuk--Ulam property for spaces with group actions, extending previous results to graphs and cyclic groups.

Findings

Geometric condition for Borsuk--Ulam property involving equivariant maps

Algebraic condition equivalent to geometric one for aspherical spaces

Effective control of graph-braid groups via discrete Morse theory

Abstract

For a finite group $H$ and connected topological spaces $X$ and $Y$ such that $X$ is endowed with a free left $H$-action $\tau$, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the triple $(X,Y;\tau)$) to decide whether the Borsuk--Ulam property holds for based homotopy classes $\alpha\in[X,Y]_0$, as well as for free homotopy classes $\alpha\in[X,Y]$. Here, a homotopy class $\alpha$ is said to satisfy the Borsuk--Ulam property if, for each of its representatives $f\in\alpha$, there exists an $H$-orbit where $f$ fails to be injective. Our geometric characterization is attained by constructing an $H$-equivariant map from $X$ to the classical configuration space $F_{|H|}(Y)$. We derive an algebraic condition from the geometric characterisation, and show that the former one is in fact equivalent to the latter one when $X$ and $Y$ are aspherical. We then specialize to the 1-dimensional case, i.e., when $X$ is an arbitrary connected graph, $H$ is cyclic, and $Y$ is either an interval, a circle, or their wedge sum. The graph-braid-group ingredient in our characterizations is then effectively controlled through the use of discrete Morse theory.