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

2021/10/16 by Zhang, Adela YiYu · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.08428

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 E²-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(Mⁿ;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 Bödigheimer-Cohen-Taylor when n+r is odd.

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• Odd-primary torsion in the homology of unordered configurations on the torus

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