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

Abstract

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 improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4–manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the simplicial structure of triangulations using the Budney-Burton census of closed triangulated 4-manifolds. These experiments include the construction of minimal genus trisections of the non-orientable 4-manifolds $S^3\tilde{\times }{S}^1$ and $\mathbb{R}{P}^4.$

Keywords:

• combinatorial topology
• triangulated manifolds
• simply connected 4-manifolds
• K3 surface
• trisections of 4-manifolds
• handlebodies
• algorithms
• experiments

MSC2010:

• 57Q15
• 57N13
• 14J28
• 57R65

Acknowledgements

The authors thank the anonymous referee for useful comments. The second author thanks the DFG Collaborative Center SFB/TRR 109 at TU Berlin, where parts of this work have been carried out, for its hospitality.

Notes

1 A d-dimensional handlebody (or, more precisely, 1-handlebody) is the regular neighborhood of a graph embedded into Euclidean d-space.

2 A bistellar 2-4-move replaces a pair of pentachora glued along a common tetrahedron by a collection of four pentachora glued around a common edge. Accordingly, each 2-4-move increases the number of pentachora in the triangulation by two.

3 A stabilization of an orientable trisection with central surface of genus g is obtained by attaching a 1-handle to a 4-dimensional handlebody along a properly embedded boundary parallel arc in the 3-dimensional handlebody that is the intersection of the other two 4-dimensional handlebodies of the trisection. The result is a new trisection with central surface of genus g + 1, and exactly one of the 4-dimensional handlebodies has its genus increased by one.

4 Note that, on the one hand there exist simply connected topological 4-manifolds with an infinite number of PL structures, and on the other hand, every topological type of simply connected PL 4-manifold can only admit a finite number of simple crystallisations.

5 Given a triangulation, its isomorphism signature is a canonical representative invariant under combinatorial isomorphisms (relabelings) of the simplices of the triangulation. See [4, Section 3.2] for a detailed description.

6 Property c) in the definition of a simple crystallisation states that $\mathcal{T}$ must have 5 vertices and 10 edges. Since all ten edges of $\mathcal{T}$ are contained in a single pentachoron, $M{\cong }_{PL}\left|\mathcal{T}\right|$ must be simply connected and thus its Euler characteristic is given by $\chi (M)=2+{\beta }_2(M,\mathbb{Z}).$ The other entries of the face-vector are then determined by the Dehn–Sommerville equations for 4-manifolds.

Additional information

Funding

Research of the first author was supported by the Einstein Foundation (project “Einstein Visiting Fellow Santos”). Research of the second author was supported in part under the Australian Research Council’s Discovery funding scheme (project number DP160104502).