# Computing trisections of 4-manifolds

@article{Bell2018ComputingTO, title={Computing trisections of 4-manifolds}, author={Mark C. Bell and Joel Hass and Joachim Hyam Rubinstein and Stephan Tillmann}, journal={Proceedings of the National Academy of Sciences}, year={2018}, volume={115}, pages={10901 - 10907} }

Significance Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a four-dimensional manifold as a union of three four-dimensional 1-handlebodies. The complexity of the 4-manifold is captured in a collection of curves on a surface, which guide the gluing of the 1…

## 8 Citations

Generalized trisections in all dimensions

- MathematicsProceedings of the National Academy of Sciences
- 2018

This paper constructs multisections, which describe an n-dimensional manifold as a union of k-dimensional handlebodies, where n=2k or 2k+1, and describes a generalization of Heegaard splittings of 3-manifolds and trisection of 4- manifolds to all dimensions, using triangulations as a key tool.

The Trisection Genus of Standard Simply Connected PL 4-Manifolds

- MathematicsSoCG
- 2018

It is shown that the K3 surface has trisection genus 22, which implies that the trisected genus of all standard simply connected PL 4-manifolds is known.

Trisections of 4-manifolds

- MathematicsProceedings of the National Academy of Sciences
- 2018

The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic, so what is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions.

Determining the Trisection Genus of Orientable and Non-Orientable PL 4-Manifolds through Triangulations

- MathematicsExperimental Mathematics
- 2020

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper…

ON THE ADDITIVITY OF TRISECTION GENUS

- Mathematics
- 2021

In this paper, we explain what a trisection of a 4-manifold is and show how to construct one. A trisection naturally gives rise to a quadruple of non-negative integers (g; g0, g1, g2), encoding the…

Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants

- Mathematics
- 2020

The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {\it semi-simple} and…

Trisections in colored tensor models

- Mathematics
- 2021

We give a procedure to construct (quasi-)trisection diagrams for closed (pseudo-)manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation,…

Gem-induced trisections of compact PL $4$-manifolds

- Mathematics
- 2019

The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann,…

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It is shown that the K3 surface has trisection genus 22, which implies that the trisected genus of all standard simply connected PL 4-manifolds is known.

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