Trisections of 3-Manifolds

TL;DR

This paper introduces the concept of trisections for 3-manifolds, explores their properties, and establishes a stable equivalence theorem, providing a new framework analogous to Heegaard splittings.

Contribution

It defines trisections of 3-manifolds, introduces the trisection genus, and proves a stable equivalence theorem similar to Reidemeister-Singer for these structures.

Findings

Trisection genus $t(M)$ is related to Heegaard genus $g(M)$ by $t(M) \\le g(M) \\le 2t(M)$.

All trisections of a 3-manifold are stably equivalent.

Existence of complicated trisections of $S^3$.

Abstract

We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus $t(M)$ of a 3-manifold, and relate it to the Heegaard genus $g(M)$, showing that $t(M) \le g(M) \le 2t(M)$. We show moreover that the bound $g(M) \le 2t(M)$ is tight. We define stabilizations of trisections and show that all trisections of a 3-manifold are stably equivalent, providing an analogue of the Reidemeister-Singer theorem for trisections. We conclude by showing that there exist complicated trisections of $S^3$.