Geodesic
Parametrised moduli spaces of surfaces as infinite loop spaces
Forum of Mathematics, Sigma, Tome 10 (2022)

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We study the $E_2$-algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda \mathfrak {M}_{*,1}$: it is the product of $\Omega ^{\infty }\mathbf {MTSO}(2)$ with a certain free $\Omega ^{\infty }$-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\Gamma _{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$. 
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Andrea Bianchi; Florian Kranhold; Jens Reinhold. Parametrised moduli spaces of surfaces as infinite loop spaces. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.29

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