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

Zhang, Adela YiYu¹

¹Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 1945-1997

Résumé

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 E2-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 Bk(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 Bk(M;X), as well as explicit computations for k ≤ 3. When p is odd, we observe that the mod p homology of Bk(Mn;Sr) 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.

Détail

DOI : 10.2140/agt.2025.25.1945

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

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