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The Lubin-Tate Theory of Configuration Spaces: I

Lukas Brantner, Jeremy Hahn, Ben Knudsen

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.

The Lubin-Tate Theory of Configuration Spaces: I

Abstract

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

Paper Structure

This paper contains 37 sections, 72 theorems, 194 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Future directions
  4. Linear outline
  5. Acknowledgements
  6. Preliminaries
  7. Weighted graded $R$-modules
  8. The derived category of $\mathop{\mathrm{Mod}}\nolimits_R^\mathbb{N}$.
  9. Symmetric powers
  10. Divided Powers
  11. Lie algebras and their homology
  12. Simplicial Lie algebras over a graded ring
  13. The universal enveloping algebra
  14. Lie algebra homology
  15. Derivations and the proof of Proposition \ref{['prop:lie homology is tor']}
  16. ...and 22 more sections

Key Result

Theorem 1

(Hecke spectral sequence) Let $M$ be a framed $n$-manifold and $X$ a spectrum, and suppose that the Hecke Lie algebra $\mathfrak{g}(M;X):=E_*^{\wedge} (\mathop{\mathrm{Free}}\nolimits_{\mathop{\mathrm{Lie}}\nolimits}(\Sigma^{n-1}X)^{M^+} )$ is a finite and free $E_\ast$-module in each weight. There

Figures (2)

  • Figure 1: The Hecke Lie algebra homology $H^{\mathop{\mathrm{Lie}}\nolimits^{\mathcal{H}_u}}( \mathfrak{g}(\mathbb{R}^n, S^k))(p)$.
  • Figure :

Theorems & Definitions (171)

  • Theorem 1: \ref{['thm:main']}
  • Remark 1.1
  • Theorem 2: \ref{['prop:general calculations']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 3: \ref{['thm:open surfaces']}
  • Remark 1.7
  • ...and 161 more