Celebratio Mathematica

D. T. Gay and R. Kirby: “Constructing symplectic forms on 4-manifolds which vanish on circles,” Geom. Topol. 8 (2004), pp. 743–777. MR 2057780 Zbl 1054.57027

D. T. Gay and R. Kirby: “Constructing Lefschetz-type fibrations on four-manifolds,” Geom. Topol. 11 (2007), pp. 2075–2115. MR 2350472 Zbl 1135.57009

D. T. Gay and R. Kirby: “Indefinite Morse 2-functions: Broken fibrations and generalizations,” Geom. Topol. 19 : 5 (2015), pp. 2465–2534. MR 3416108 Zbl 1328.57019 article

D. Gay and R. Kirby: “Trisecting 4-manifolds,” Geom. Topol. 20 : 6 (2016), pp. 3097–3132. MR 3590351 Zbl 1372.57033 article

We show that any smooth, closed, oriented, connected 4-manifold can be trisected into three copies of \( \#^k(\mathbb{S}^1\times \mathbb{B}^3) \), intersecting pairwise in 3-dimensional handlebodies, with triple intersection a closed 2-dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3-manifolds. A trisection of a 4-manifold \( X \) arises from a Morse 2-function \( G:X\to \mathbb{B}^2 \) and the obvious trisection of \( \mathbb{B}^2 \), in much the same way that a Heegaard splitting of a 3-manifold \( Y \) arises from a Morse function \( g:Y\to \mathbb{B}^1 \) and the obvious bisection of \( \mathbb{B}^1 \).

A. Abrams, D. Gay, and R. Kirby: “Group trisections and smooth 4-manifolds,” Geom. Topol. 22 : 3 (2018), pp. 1537–1545. MR 3780440 ArXiv 1605.06731 article

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby’s existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group-theoretic. For example, the smooth 4-dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.

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