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× S 1 and RP 4 .

Note on three-fold branched covers of S 4

“…-The trisection parameters (g; k 1 , k 2 , k 3 ) are additive under connected sum of trisected 4-manifolds [9]. The 4-manifolds S 4 , ±CP 2 , S 2 × S 2 and K3 admit (g, 0)-trisections with g = 0, 1, 2 and 22, respectively [9,20,31]. As before, we denote the g = 0 trisection of S 4 by T 0 .…”

“…Consider the collection M of 4-manifolds that contains CP 2 , S 2 × S 2 , and K3 and is closed under the operations of taking connected sums and reversing orientations. Each member of M admits a (g, 0)-trisection [20,31]. Every smooth, simply-connected 4-manifold that satisfies the 11/8-Conjecture is homeomorphic to a member of M [8,7]; for details, see [12,Section 1.2].…”