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. C. Kirby: “Fiber-connected, indefinite Morse 2-functions on connected \( n \)-manifolds,” Proc. Natl. Acad. Sci. USA 108 : 20 (2011), pp. 8122–8125. MR 2806648

D. Gay and R. Kirby: “Reconstructing 4-manifolds from Morse 2-functions,” pp. 103–114 in Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. Proceedings of a conference on low-dimensional manifolds and high-dimensional categories and a symposium dedicated to Mike Freedman on the occasion of his 60th birthday. MR 3084235 Zbl 1267.57020 ArXiv 1202.3487 incollection

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume
David T. Gay	Related

@incollection {key3084235m,
	AUTHOR = {Gay, David and Kirby, Robion},
	TITLE = {Reconstructing 4-manifolds from {M}orse
		 2-functions},
	BOOKTITLE = {Proceedings of the {F}reedman {F}est},
	EDITOR = {Kirby, Rob and Krushkal, Vyacheslav
		 and Wang, Zhenghan},
	SERIES = {Geometry \& Topology Monographs},
	NUMBER = {18},
	PUBLISHER = {Geometry \& Topology Publications},
	ADDRESS = {Coventry, UK},
	YEAR = {2012},
	PAGES = {103--114},
	DOI = {10.2140/gtm.2012.18.103},
	NOTE = {(Santa Barbara, CA, 15--17 April 2011
		 and Berkeley, CA, 6--10 June 2011).
		 Proceedings of a conference on low-dimensional
		 manifolds and high-dimensional categories
		 and a symposium dedicated to Mike Freedman
		 on the occasion of his 60th birthday.
		 ArXiv:1202.3487. MR:3084235. Zbl:1267.57020.},
	ISSN = {1464-8989},
}

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.

Aaron David Abrams	Related
David T. Gay	Related

year	title	people

			clear

Robion C. Kirby

Complete Bibliography

Works connected to David T. Gay

Filter the Bibliography List