Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Nathaniel Bottman

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.

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

Abstract

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

Paper Structure

This paper contains 14 sections, 4 theorems, 25 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Getzler--Jones' attempted decomposition
  3. Tamarkin's counterexample
  4. An overview of our construction
  5. The relationship between our construction and $\mathsf{Symp}$
  6. Future directions
  7. Acknowledgments
  8. A "W-version" of the 2-associahedra
  9. A warm-up: $K^W$, i.e. $W({\operatorname{Ass}})$, i.e. a W-version of the associahedra
  10. Metric tree-pairs and the definition of $W_\mathbf{n}^W$
  11. The construction of ${\operatorname{FM}}_2^W$
  12. Quotient maps on 2-associahedra
  13. The construction of ${\operatorname{FM}}_2^W(k)$
  14. The operad structure on ${\operatorname{FM}}_2^W$

Key Result

Proposition \oldthetheorem

The spaces $\bigl(K_r^W\bigr)_{r\geq 1}$ form a non-$\Sigma$ operad of CW complexes, and the composition maps defined in Def. def:K_r_composition are cellular.

Figures (9)

  • Figure 1:
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 4 more figures

Theorems & Definitions (39)

  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem: Def. 2.2, b:2-associahedra
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Definition 2.4, b:2-associahedra
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • ...and 29 more