General Bourgin-Yang theorems

TL;DR

This paper presents a unified method for estimating the dimension of inverse images under equivariant maps, extending Bourgin-Yang theorems to various group actions based on connectivity and dimension considerations.

Contribution

It introduces a general framework for Bourgin-Yang type theorems applicable to cyclic groups, p-tori, and tori, unifying previous results in equivariant topology.

Findings

Provides bounds on the dimension of inverse images for equivariant maps

Extends Bourgin-Yang theorems to new classes of groups

Unifies various existing results under a common approach

Abstract

We describe a unified approach to estimating the dimension of $f^{-1}(A)$ for any $G$-equivariant map $f \colon X \to Y$ and any closed $G$-invariant subset $A\subseteq Y$ in terms of connectivity of $X$ and dimension of $Y$, where $G$ is either a cyclic group of order $p^k$, a $p$-torus ($p$ a prime), or a torus.