Parametrised moduli spaces of surfaces as infinite loop spaces

Abstract

We study the $E_2$-algebra $\Lambda {\mathfrak{M}}_{\ast ,1}={\coprod }_{g⩾0}\Lambda {\mathfrak{M}}_{g,1}$ 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 $\Omega B\Lambda {\mathfrak{M}}_{\ast ,1}$: 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 $g⩾0$ and $n⩾1$.

59 pages, 11 figures; accepted version (Forum of Mathematics, Sigma)

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