The Lubin-Tate Theory of Configuration Spaces: I

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.

Final version to appear in the Journal of Topology. 61 pages, 1 figure