Infinity (modular) operads and PROPs for surfaces

Explicit fat graph models of homotopy-coherent (modular) operads for moduli spaces of curves.

Together with Luciana Bonatto, Saffiah Chettih, Abigail Linton, Marcy Robertson and Nathalie Wahl

Moduli spaces of algebraic curves may be composed by gluing surfaces along boundary components. At the level of homology, this composition admits a combinatorial model in terms of a PROP of (metric) fat graphs (or ribbon graphs). But, Daniela Egas noticed that this metric fat graph composition is not strictly associative, and therefore does not describe a PROP in the category of spaces.

As part of Women in Topology, 2019, we considered explicit topological models of composition of moduli spaces, up to all coherent homotopies.

Our first paper focused on the genus 0 case.

An infinity operad of normalized cacti, to appear in special issue of Topology and its Applications.

We show that normalized cacti form an ∞-operad in the form of a dendroidal space satisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketed trees and an enrichment of the dendroidal category.

Extending this result to a modular operad of fat graphs to model higher genus curves presents a number of interesting challenges. This is the subject of current work.

(More details to come.)