Filtering the modular surface operad
Moduli and Friends seminar
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Monday 21 March 2022

Jan Steinebrunner (University of Cambridge)Filtering the modular surface operad

I will explain how the moduli spaces of surfaces assemble into a graded modular operad whose envelope is the surface category studied by Galatius-Madsen-Tillmann-Weiss. For any graded modular operad there is a filtration by genus, which in this case yields a filtration of MTSO2. The associated graded can be described in terms of the curve complex.
This leads to a spectral sequence with E1-page the (dualised) unstable homology of mapping class groups and which converges to the spectrum homology of MTSO2. As a computational consequence, we will see that the top-dimensional homology group H14(𝔐5), which was shown to be non-zero by Chan-Galatius-Payne, has at most rank 2.

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