The Lubin–Tate theory of configuration spaces: I

Abstract

We construct a spectral sequence converging to the Lubin–Tate theory, that is, 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 (parameterizing the weight $p$ operations on $E_n$-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 $F_p$-homology of the space of unordered configurations of $p$ particles on a punctured surface.