Diagrams for relative trisections

Nickolas A. Castro, David T. Gay and Juanita Pinzón-Caicedo

Vol. 294 (2018), No. 2, 275–305

DOI: 10.2140/pjm.2018.294.275

Abstract

We establish a correspondence between trisections of smooth, compact, oriented $4$-manifolds with connected boundary and diagrams describing these trisected $4$-manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise condition of being standard. This should be thought of as the $4$-dimensional analog of a sutured Heegaard diagram for a sutured $3$-manifold. We also give many foundational examples.

Keywords

trisection, open book, Heegaard diagram, monodromy, 3-manifold, 4-manifold

Mathematical Subject Classification 2010

Primary: 57M99

Milestones

Received: 18 January 2017

Revised: 16 October 2017

Accepted: 30 October 2017

Published: 20 February 2018