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

Zhang, Adela YiYu

doi:10.2140/agt.2025.25.1945

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

Adela YiYu Zhang

Algebraic & Geometric Topology 25 (2025) 1945–1997

DOI: 10.2140/agt.2025.25.1945

Abstract

We provide a general method computing the mod $p$ Quillen homology of algebras over the monad that parametrizes the structure of mod $p$ homology of spectral Lie algebras, including the construction of a May-type spectral sequence when $p = 2$ . This is the $E^{2}$ -page of the bar spectral sequence converging to the mod $p$ topological Quillen homology of spectral Lie algebras. 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 $k \leq 3$ . When $p$ is odd, we observe that the mod $p$ homology of $B_{k} (M^{n}; S^{r})$ for small $k$ depends 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 Bödigheimer, Cohen and Taylor when $n + r$ is odd.

Keywords

spectral Lie algebra, Quillen homology, Lie algebra homology, configuration spaces

Mathematical Subject Classification

Primary: 17B56, 55N20, 55R80

References

Publication

Received: 23 December 2021

Revised: 1 February 2024

Accepted: 7 April 2024

Published: 11 August 2025

Authors

Adela YiYu Zhang
Department of Mathematical Sciences University of Copenhagen Copenhagen Denmark

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Volume 25, issue 4 (2025)