Parametrised moduli spaces of surfaces as infinite loop spaces

TL;DR

This paper investigates the homotopy type of a certain $E_2$-algebra related to moduli spaces of Riemann surfaces with boundary, revealing its structure as a product of infinite loop spaces and connecting it to mapping class groups.

Contribution

It computes the homotopy type of the group completion of an $E_2$-algebra of free loop spaces of moduli spaces, linking it to infinite loop spaces and mapping class groups.

Findings

Homotopy type of the group completion is a product of $oldsymbol{ ext{Ω}^oldsymbol{ ext{∞}} extbf{MTSO}(2)}$ and a free $oldsymbol{ ext{Ω}^ extbf{∞}}$-space.

Explicit description involving boundary-irreducible mapping classes across all mapping class groups.

Connection established between parametrized moduli spaces and infinite loop space structures.

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 $n\geqslant 1$.