Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces

TL;DR

This paper develops a method to compute the mod p Quillen homology of spectral Lie algebras, revealing non-formality of the spectral Lie operad and providing bounds and explicit calculations for the mod p homology of labeled configuration spaces.

Contribution

It introduces a general computational approach for mod p Quillen homology of spectral Lie algebras and applies it to analyze the homology of labeled configuration spaces, highlighting new structural insights.

Findings

The spectral Lie operad is not formal over _p.

Provides upper bounds for the mod p homology of configuration spaces.

Explicit calculations for small k and dependence on cohomology ring when p is odd.

Abstract

We provide a general method computing the mod $p$ Quillen homology of algebras over a monad that parametrizes the structure of mod $p$ homology of spectral Lie algebras. This is the $E^2$-page of the bar spectral sequence converging to the mod $p$ topological Quillen homology of spectral Lie algebras. The computation of the Quillen homology of the trivial algebra allows us to deduce that the $\mathbb F_p$-linear spectral Lie operad is not formal. As an application, we study the mod $p$ homology of the labeled configuration space $B_k(M;X)$ of $k$ points in a manifold $M$ with labels in a spectrum $X$, which is the mod $p$ topological Quillen homology of a certain spectral Lie algebra by a result of Knudsen. We obtain general upper bounds for the mod $p$ homology of $B_k(M;X)$, as well as explicit computations for small $k$. When $p$ is odd, we observe that the mod $p$ homology of $B_k(M^n;S^r)$ for small $k$ depends on and only on the cohomology ring of the one-point compactification of $M$ when $n+r$ is even. This supplements and contrasts with the result of B\"{o}digheimer-Cohen-Taylor when $n+r$ is odd.