We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.
Geometry & Topology
This paper owes an obvious intellectual debt to the illuminating treatments of factorization homology by J. Francis, D. Gaitsgory, and J. Lurie in [GL,G1, FG]. The author would like to thank B. Farb and J. Wolfson for bringing the question of explaining coincidences in homological densities to his attention. Moreover, the author thanks J. Wolfson for many helpful conversations on the subject, O. Randal-Williams for many comments which greatly help improve the exposition, and G. C. Drummond-Cole for many useful conversations on L∞-algebras. Finally, the author is grateful to the anonymous referee for carefully reading the manuscript and for providing numerous comments which greatly helped improve the clarity and precision of the exposition. This work is supported by the Advanced Grant “Arithmetic and Physics of Higgs moduli spaces” No. 320593 of the European Research Council and the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization Homology,” Austrian Science Fund (FWF): M 2751.
Ho QP. Homological stability and densities of generalized configuration spaces. Geometry & Topology. 2021;25(2):813-912. doi:10.2140/gt.2021.25.813
Ho, Q. P. (2021). Homological stability and densities of generalized configuration spaces. Geometry & Topology. Mathematical Sciences Publishers. https://doi.org/10.2140/gt.2021.25.813
Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration Spaces.” Geometry & Topology. Mathematical Sciences Publishers, 2021. https://doi.org/10.2140/gt.2021.25.813.
Q. P. Ho, “Homological stability and densities of generalized configuration spaces,” Geometry & Topology, vol. 25, no. 2. Mathematical Sciences Publishers, pp. 813–912, 2021.
Ho QP. 2021. Homological stability and densities of generalized configuration spaces. Geometry & Topology. 25(2), 813–912.
Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration Spaces.” Geometry & Topology, vol. 25, no. 2, Mathematical Sciences Publishers, 2021, pp. 813–912, doi:10.2140/gt.2021.25.813.