Trisecting 4-manifolds

Abstract

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of ${♮}^k(S^1\times B^3)$, 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 $G:X\to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3--manifold $Y$ arises from a Morse function $g:Y\to B^1$ and the obvious bisection of $B^1$.

38 pages, 29 figures; minor improvements to exposition, more examples, and more discussion

Cited by in corpus (40)

• Bridge trisections of knotted surfaces in $S^4$
• Bridge trisections of knotted surfaces in 4--manifolds
• Genus two trisections are standard
• Trisecting non-Lagrangian theories
• Trisections of 4-manifolds with Boundary
• Calculating the homology and intersection form of a 4-manifold from a trisection diagram
• Isotopies of surfaces in 4-manifolds via banded unlink diagrams
• Simplified broken Lefschetz fibrations and trisections of 4-manifolds
• Trisections of surface complements and the Price twist
• Bridge trisections in ${\mathbb{CP}}^2$ and the Thom conjecture (with Corrigendum)
• Doubly pointed trisection diagrams and surgery on 2-knots
• Computing trisections of 4-manifolds
• The relative $\mathcal{L}$-invariant of a compact $4$-manifold
• Characterizing Dehn surgeries on links via trisections
• Torsions and intersection forms of 4-manifolds from trisection diagrams
• Trisections of Lefschetz Pencils
• Trisections of 3-manifold bundles over $S^1$
• Kirby-Thompson distance for trisections of knotted surfaces
• Group trisections and smooth 4-manifolds
• Symplectic 4-manifolds admit Weinstein trisections
• An Algorithm taking Kirby diagrams to Trisection diagrams
• Trisections, intersection forms and the Torelli group
• Bi-filtrations and persistence paths for 2-Morse functions
• Filling braided links with trisected surfaces
• The Dihedral Genus of a Knot
• Trisecting a 4-dimensional book into three chapters
• Bridge trisections and classical knotted surface theory
• Pseudo-trisections of four-manifolds with boundary
• 4-Manifold Invariants From Hopf Algebras
• Bounding the Kirby-Thompson invariant of spun knots
• Comparing 4-Manifolds in the Pants Complex via Trisections
• Infinitely many standard trisection diagrams for Gluck twisting
• Shadows of 2-knots and complexity
• Bridge trisections and Seifert solids
• Trisections of PL 4-manifolds arising from colored triangulations
• Thin Position through the lens of trisections of 4-manifolds
• Nonsmooth manifold decompositions
• Efficient multisections of odd-dimensional tori
• Band diagrams of immersed surfaces in 4-manifolds
• The algebraic topology of 4-manifolds multisections