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The genus filtration on the modular surface operad

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Jan Steinebrunner || GER
University of Cambridge
Wed, 29/06/2022 - 11:45 - 12:15
The moduli spaces of surfaces assemble into a modular $\infty$-operad, closely related to the $2$-dimensional bordism category.
I will establish an obstruction theory for algebras over this surface modular operad where the obstruction for extending from genus $g-1$ to $g$ is controlled by the curve complex $C(\Sigma_g)$.
For invertible algebras this yields a spectral sequence with $E^1$-page given by unstable homology of mapping class groups, which converges to the spectrum homology of $\mathit{MTSO}_2$.
The required cancellations in this spectral sequence imply for instance that the top-dimensional homology group $H_{14}(\mathcal{M}_5)$ has at most rank $2$.


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