Lie algebras and the (co)homology of configuration spaces

TL;DR

This paper surveys various approaches to understanding the (co)homology of configuration spaces through Lie algebra (co)homology, highlighting conceptual differences and open problems.

Contribution

It provides a comprehensive overview of multiple perspectives on the connection between configuration space (co)homology and Lie algebra (co)homology, emphasizing conceptual insights.

Findings

Different routes to the proto-theorem offer unique explanations.

The survey clarifies the conceptual core of each approach.

Lists open problems in the field.

Abstract

We survey decades of research identifying the (co)homology of configuration spaces with Lie algebra (co)homology. The different routes to this one proto-theorem offer genuinely different explanations of its truth, and we attempt to convey some sense of the conceptual core of each perspective. We close with a list of problems.