A homotopical description of Deligne-Mumford compactifications

TL;DR

This paper provides a homotopical framework for understanding Deligne-Mumford compactifications by trivializing S1 families of annuli in Riemann surface properads, connecting to symplectic cohomology operations.

Contribution

It introduces a homotopical description of Deligne-Mumford properads via trivialization of annuli, extending prior results to a new setting with applications to symplectic cohomology.

Findings

Homotopical trivialization of S1 families of annuli in Riemann surface properads.

A new partial compactification of Riemann surfaces relevant to symplectic cohomology.

Extension of Drummond-Cole and Oancea--Vaintrob results to properads.

Abstract

We give a description of the Deligne-Mumford properad expressing it as the result of homotopically trivializing S1 families of annuli (with appropriate compatibility conditions) in the properad of smooth Riemann surfaces with parameterized boundaries. This gives an analog of the results of Drummond-Cole and Oancea--Vaintrob in the setting of properads. We also discuss a variation of this trivialization which gives rise to a new partial compactification of Riemann surfaces relevant to the study of operations on symplectic cohomology.