Journal of Symplectic Geometry

Volume 20 (2022)

Number 3

Contact categories of disks

Pages: 665 – 759



Ko Honda (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Yin Tian (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


In the first part of the paper we associate a pre-additive category $\mathcal{C}(\Sigma)$ to a closed oriented surface $\Sigma$, called the contact category and constructed from contact structures on $\Sigma \times [0, 1]$. There are also $\mathcal{C}(\Sigma, F)$, where $\Sigma$ is a compact oriented surface with boundary and $F \subset \partial\Sigma$ is a finite oriented set of points which bounds a submanifold of $\partial\Sigma$, and universal covers $\widetilde{\mathcal{C}}(\Sigma)$ and $\widetilde{\mathcal{C}}(\Sigma, F)$ of $\mathcal{C}(\Sigma)$ and $\mathcal{C}(\Sigma, F)$. In the second part of the paper we prove that the universal cover of the contact category of a disk admits an embedding into its “triangulated envelope.”

K.H. is supported by NSF Grants DMS-0805352, DMS-1105432, DMS-1406564, and DMS-154914.

Y.T. is supported by NSFC 11601256 and 11971256.

Received 6 June 2017

Accepted 19 April 2021

Published 28 February 2023