The Lubin-Tate Theory of Configuration Spaces: I

TL;DR

This paper develops a spectral sequence approach to compute Morava $E$-theory of configuration spaces and related iterated loop spaces, connecting algebraic structures with topological invariants and confirming conjectures about Morava $K$-theory groups.

Contribution

It introduces a spectral sequence linking Morava $E$-theory of configuration spaces to Chevalley-Eilenberg complexes for Hecke Lie algebras, enabling explicit computations.

Findings

Computed Morava $E$-theory of weight $p$ summands of iterated loop spaces.

Determined Morava $K$-theory groups related to Ravenel's conjecture.

Calculated $F_p$-homology of configuration spaces on punctured surfaces.

Abstract

We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $E$-theory of the weight $p$ summands of iterated loop spaces of spheres (parametrising the weight $p$ operations on $\mathbb{E}_n$-algebras), as well as the $E$-theory of the configuration spaces of $p$ points on a punctured surface. We read off the corresponding Morava $K$-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the $\mathbb{F}_p$-homology of the space of unordered configurations of $p$ particles on a punctured surface.