Parametrised moduli spaces of surfaces as infinite loop spaces

Abstract We study the $E_2$ -algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda {M}_{g,1}$ consisting of free loop spaces moduli Riemann surfaces with one parametrised boundary component, and compute homotopy type group completion $\Omega B\Lambda {M}_{*,1}$ : it is product ^{\infty }\mathbf {MTSO}(2)$ a certain }$ -space depending on family all boundary-irreducible mapping classes in class groups $\Gamma _{g,n}$ $g\geqslant 0$ $n\geqslant 1$ .