Parametrised moduli spaces of surfaces as infinite loop spaces

Abstract

We study the $E_2$ -algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ 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 $n\geqslant 1$ .

Vertical Configuration Spaces and Their Homology

“…and we are broadly interested in understanding, in concrete examples, whether this map is an equivalence, or rather how much this map is not an equivalence. In [BKR22], we determined, in a slightly different setting, the homotopy type of the group-completion of Map(S 1 , A), where A is the E 1 -algebra given by the disjoint union of classifying spaces of either of the following three sequences of groups: mapping class groups of oriented surfaces of genus g ⩾ 0 with one boundary curve, Artin braid groups, and symmetric groups. In all these cases, the above map is far from being an equivalence.…”