Homology of configuration spaces in positive characteristic via point-set constructions

Abstract

The first goal of this paper is to provide concrete chain complexes computing the homology of (unordered) configuration spaces of manifolds in positive characteristic, lifting a theorem by Knudsen to the model category level. We make them fully explicit and provide a computer program to compute their homology. Our methods also allow us to construct several new spectral sequences converging to these homology groups. Finally, we conjecture that this equivalence of chain complexes can be promoted to an equivalence of \emph{twisted} $\mathcal{E}_\infty$-coalgebras in right $\mathcal{E}_d$-modules, and we explain how this conjecture would imply the homotopy invariance of the $\mathcal{E}_d$-homotopy type of configuration spaces in positive characteristic via new “twist” and “detwist” functors.