Adela Zhang: Mod p homology of unordered configuration spaces of surfaces.

Abstract: We give a complete calculation of the homology groups of unordered configuration spaces of all surfaces with coefficients in a field \(k\). We use factorization homology to construct a small simplicial commutative ring model of unordered configurations spaces. Then we leverage the theory of \(\mathrm{SL}_2\) and \(\mathrm{SpO}(2|2)\) representations, in particular the cohomology of Tilting modules, to compute the homology of these simplicial commutative ring models for surfaces. Our method recovers the results of Drummond-Cole-Knudsen over \(\mathbb{Q}\), Bödigheimer-Cohen-Talyor and Milgram-Löffler over \(\mathbb{F}_2\), as well as the odd primary results of Bianchi-Stavrou for once-punctured orientable surfaces. This is joint work with Robert Burklund.