A simplicial version of the 2-dimensional Fulton-MacPherson operad

TL;DR

This paper introduces a new simplicial operad in topology, called $ ext{FM}_2^W$, with cellular compositions and CW decompositions, aiming to relate it to the classical 2D Fulton-MacPherson operad and its applications in symplectic geometry.

Contribution

The paper constructs a simplicial, cellular operad $ ext{FM}_2^W$ in Top, providing a new perspective on the 2D Fulton-MacPherson operad and its potential applications in symplectic cohomology.

Findings

$ ext{FM}_2^W$ has CW decompositions with cellular operad compositions.

Each space in $ ext{FM}_2^W$ is a realization of a simplicial set.

The construction suggests a link to symplectic cochain complexes and homotopy BV algebras.

Abstract

We define an operad in Top, called $\text{FM}_2^W$. The spaces in $\text{FM}_2^W$ come with CW decompositions, such that the operad compositions are cellular. In fact, each space in $\text{FM}_2^W$ is the realization of a simplicial set. We expect, but do not prove here, that $\text{FM}_2^W$ is isomorphic to the 2-dimensional Fulton-MacPherson operad $\text{FM}_2$. Our construction is connected to the author's work on the symplectic $(A_\infty,2)$-category, and suggests a strategy toward equipping the symplectic cochain complex with the structure of a homotopy Batalin-Vilkoviskiy algebra.