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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