#### Volume 19, issue 2 (2019)

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$2$–associahedra

### Nathaniel Bottman

Algebraic & Geometric Topology 19 (2019) 743–806
##### Abstract

For any $r\ge 1$ and $n\in {ℤ}_{\ge 0}^{r}\phantom{\rule{0.3em}{0ex}}\setminus \left\{0\right\}$ we construct a poset ${W}_{n}$ called a $2$–associahedron. The $2$–associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. We prove that the completion ${\stackrel{̂}{W}}_{n}$ is an abstract polytope of dimension $|n|+r-3$. There are forgetful maps ${W}_{n}\to {K}_{r}$, where ${K}_{r}$ is the $\left(r-2\right)$–dimensional associahedron, and the $2$–associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, we work out the $2$– and $3$–dimensional $2$–associahedra in detail.

##### Keywords
polytopes, associahedra, Fukaya category, pseudoholomorphic quilts
Primary: 53D37
##### Publication
Received: 3 September 2017
Revised: 19 June 2018
Accepted: 20 August 2018
Published: 12 March 2019
##### Authors
 Nathaniel Bottman School of Mathematics Institute for Advanced Study Princeton, NJ United States