Symplectic forms on trisected 4-manifolds
Abstract: Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the converse direction and give explicit criteria on a trisection of a closed, smooth 4-manifold $X$ that allows one to construct a symplectic structure on $X$. Combined, these give a new characterization of 4-manifolds that admit symplectic structures. This construction motivates several problems on taut foliations, the Thurston norm and contact geometry in 3-dimensions by connecting them to questions about the existence, classification and uniqueness of symplectic structures on 4-manifolds.
- Denis Auroux. Symplectic 4-manifolds as branched coverings of ℂℙ2ℂsuperscriptℙ2\mathbb{C}\mathbb{P}^{2}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Invent. Math., 139(3):551–602, 2000.
- Eugenio Calabi. An intrinsic characterization of harmonic one-forms. In Global Analysis (Papers in Honor of K. Kodaira), pages 101–117. Univ. Tokyo Press, Tokyo, 1969.
- Danny Calegari. Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007.
- A transfer principle: from periods to isoperiodic foliations, 2015.
- From Stein to Weinstein and back, volume 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds.
- Morse theory of harmonic forms. Topology, 37(3):469–483, 1998.
- Calculating the homology and intersection form of a 4-manifold from a trisection diagram. Proc. Natl. Acad. Sci. USA, 115(43):10869–10874, 2018.
- Torsions and intersection forms of 4-manifolds from trisection diagrams. Canad. J. Math., 74(2):527–549, 2022.
- Trisecting 4-manifolds. Geom. Topol., 20(6):3097–3132, 2016.
- Trisections of surface complements and the Price twist. Algebr. Geom. Topol., 20(1):343–373, 2020.
- Peter Lambert-Cole. Stein trisections and homotopy 4-balls, 2021.
- Peter Lambert-Cole. Lecture notes on trisections and cohomology. volume 5 of Open Book Series, pages 245–264, 2022.
- Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022.
- Symplectic 4-manifolds admit Weinstein trisections. J. Topol., 14(2):641–673, 2021.
- A note on 4444-dimensional handlebodies. Bull. Soc. Math. France, 100:337–344, 1972.
- Introduction to symplectic topology. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, third edition, 2017.
- Dietmar Salamon. Uniqueness of symplectic structures. Acta Math. Vietnam., 38(1):123–144, 2013.
- Martin Scharlemann. Sutured manifolds and generalized Thurston norms. J. Differential Geom., 29(3):557–614, 1989.
- William P. Thurston. A norm for the homology of 3333-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.
- Alexander Zupan. Hexagonal lattice diagrams for complex curves in ℂℙ2ℂsuperscriptℙ2\mathbb{CP}^{2}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, 2022.
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