The surface category and tropical curves

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• Jan Steinebrunner, University of Cambridge
• Wednesday 02 February 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Henry Wilton.

This talk will discuss the (truncated) surface category Cob_2 in which morphisms are diffeomorphism classes of surfaces and composition is defined by gluing along boundaries. This is a simplified version of the topological bordism categories that have become a key tool for studying the algebraic topology of diffeomorphism groups.

What makes Cob_2 simpler is that it does not have moduli spaces built in: it only knows about the combinatorics of how surfaces glue up to diffeomorphism. Nevertheless, I will show that the classifying space of a certain subcategory of Cob_2 has a surprisingly rich structure: it is splits off free infinite loop spaces on the moduli stacks of tropical curves of arbitrary genus. The splitting map admits a nice pictorial description.

If time permits, I will also speculate on how one might use Cob_2 to study the unstable homology of M_g.

This talk is part of the Differential Geometry and Topology Seminar series.

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