Equivariant Cohomology of Configuration Spaces mod 2: The State of the Art

TL;DR

This paper reviews the history and recent developments in the mod 2 equivariant cohomology of configuration spaces, providing a new proof of a key result that impacts lower bounds for regular embeddings.

Contribution

It offers a new, detailed proof of Hung's main result on mod 2 equivariant cohomology, correcting previous arguments and refining bounds for regular embeddings.

Findings

Validated Hung's main result with a new proof.

Corrected previous errors in the literature.

Derived new lower bounds for regular embeddings.

Abstract

The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojevi\'c, L\"uck \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.