Trisecting 4-manifolds
type: Item
instance of: scholarly article
title: Trisecting 4-manifolds
zbMATH Open document ID: 1372.57033
author: David Gay
author: Robion C. Kirby
published in: Geometry \& Topology
publication date: 2017-02-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.1565
review text: This paper introduces trisections, a new decomposition of smooth 4-manifolds analogous to Heegaard splittings of 3-manifolds. The paper establishes fundamental properties and examples of trisections, with a particular emphasis on placing them within a larger framework. A \textit{\((g,k)\)-trisection} of a smooth 4-manifold \(X\) is a decomposition \(X=X_1\cup X_2\cup X_3\) into three genus \(k\) 1-handlebodies (i.e., \(X_i\cong \natural^k S^1\times D^3\)), glued along their boundaries which are each equipped with a fixed genus \(g\geq k\) Heegaard splitting. Namely, the triple intersection \(X_1\cap X_2\cap X_3\) is a closed orientable surface of genus \(g\) (the \textit{trisection surface}) that separates \(\partial X_i\) into two 3-dimensional 1-handlebodies, and these are precisely the pairwise intersections \(X_i\cap X_{i+1}\) and \(X_i\cap X_{i-1}\) (indices modulo 3). A trisection is shown to arise from a smooth generic map \(X\to {\mathbb{R}}^2\) (a \textit{Morse 2-function}) and a trisection of \({\mathbb{R}}^2\), in much the same way as a Heegaard splitting of a 3-manifold \(Y\) arises from a Morse function \(Y\to {\mathbb{R}}\) and a bisection of \({\mathbb{R}}\). The main result of the paper is that every smooth, closed, orientable, connected 4-manifold admits a \((g,k)\)-trisection for some \(g,k\) (existence), and any two trisections for the same 4-manifold become isotopic after some iterations of a certain stabilization operation (uniqueness), akin to the Reidemeister-Singer theorem in dimension three. Extending the analogy, a trisection is represented by a \textit{trisection diagram}, a particular triple of closed curves on a closed orientable surface, and the existence and uniqueness statements are formulated in terms of trisection diagrams and analogues of the usual operations on Heegaard diagrams. (In particular, trisection stabilization means replacing a trisection diagram by its connected sum with the first non-trivial trisection diagram of \(S^4…
reviewed by: Ash Lightfoot
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 6687794
zbMATH Keywords: 4-manifold
zbMATH Keywords: trisection
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: Heegaard triple
zbMATH Keywords: Morse 2-function
MaRDI profile type: Publication
OpenAlex ID: W3103271661
arXiv ID: 1205.1565
DOI: 10.2140/GT.2016.20.3097
description: scientific article