Abstract

Over a field of characteristic zero, the homology groups of configuration spaces of points on a manifold are by now well understood and are known to be determined by the rational homotopy type of the manifold. Over a field of positive characteristic, the situation is more complicated.

In this talk, based on joint work with Victor Roca i Lucio, I will explain how recent operadic constructions can be used to attack this question. Starting from a theorem of Knudsen describing the stable homotopy type of configuration spaces in terms of free spectral Lie algebras, we obtain explicit chain complexes (“point-set models”) computing the homology of possibly labeled configuration spaces of parallelizable manifolds over an arbitrary field. These models combine functorial cofibrant resolutions from operadic bar-cobar constructions, classical models for the E-infinity operad such as the Barratt—Eccles and surjection operads, and the recently developed theory of quasi-planar cooperads. As a bonus, they are concrete enough to be implemented on a computer, and we have developed a SageMath library that carries out the relevant calculations. I will also outline how this framework allows us to formulate precise conjectures about the homotopy invariance of configuration spaces in positive characteristic, in terms of an equivalence of twisted E-infinity-coalgebra structures whose characteristic-zero version recovers the invariance results of myself and Campos—Willwacher.