Bonn Topology Group - Abstracts

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Talk

November 23, 2021
Florian Kranhold (Bonn University): Parametrised moduli spaces of surfaces as infinite loop spaces

Abstract

The collection M of moduli spaces of Riemann surfaces with one parametrised boundary component is equivalent to the infinite loop space assigned to the affine Thom spectrum MTSO(2). We address the following generalisation: for each space X we consider the mapping space M^X, the space of surface bundles over X, and show that its group completion splits as the product of the infinite loop space assigned to MTSO(2) and a certain free infinite loop space. The proof of this result combines two seemingly unrelated ingredients: on the one hand, we show a structure theorem for centralisers of mapping classes in mapping class groups of surfaces, and on the other hand, we develop some operadic machinery which enables us to understand group completions of relatively free algebras over coloured operads with homological stability.
This is joint work with Andrea Bianchi (Copenhagen) and Jens Reinhold (Münster)

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