Geometry & Topology

Subflexible symplectic manifolds

Emmy Murphy and Kyler Siegel

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Abstract

We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.

Article information

Source
Geom. Topol., Volume 22, Number 4 (2018), 2367-2401.

Dates
Received: 8 December 2016
Revised: 16 July 2017
Accepted: 17 August 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584825

Digital Object Identifier
doi:10.2140/gt.2018.22.2367

Mathematical Reviews number (MathSciNet)
MR3784524

Zentralblatt MATH identifier
06864340

Subjects
Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 32E20: Polynomial convexity

Keywords
symplectic geometry Weinstein manifolds h principles symplectic cohomology polynomial convexity exotic symplectic structures

Citation

Murphy, Emmy; Siegel, Kyler. Subflexible symplectic manifolds. Geom. Topol. 22 (2018), no. 4, 2367--2401. doi:10.2140/gt.2018.22.2367. https://projecteuclid.org/euclid.gt/1523584825


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References

  • M Abouzaid, Symplectic cohomology and Viterbo's theorem, from “Free loop spaces in geometry and topology” (J Latschev, A Oancea, editors), IRMA Lect. Math. Theor. Phys. 24, Eur. Math. Soc., Zürich (2015) 271–485
  • M Abouzaid, P Seidel, Altering symplectic manifolds by homologous recombination, preprint (2010)
  • F Bourgeois, T Ekholm, Y Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012) 301–389
  • R Casals, E Murphy, F Presas, Geometric criteria for overtwistedness, preprint (2015)
  • K Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. 4 (2002) 115–142
  • K Cieliebak, Y Eliashberg, From Stein to Weinstein and back: symplectic geometry of affine complex manifolds, Amer. Math. Soc. Colloq. Publ. 59, Amer. Math. Soc., Providence, RI (2012)
  • K Cieliebak, Y Eliashberg, The topology of rationally and polynomially convex domains, Invent. Math. 199 (2015) 215–238
  • T Ekholm, Y Eliashberg, E Murphy, I Smith, Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal. 23 (2013) 1772–1803
  • Y Eliashberg, E Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013) 1483–1514
  • E Giroux, J Pardon, Existence of Lefschetz fibrations on Stein and Weinstein domains, Geom. Topol. 21 (2017) 963–997
  • M Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. 9, Springer (1986)
  • R M Harris, Distinguishing between exotic symplectic structures, J. Topol. 6 (2013) 1–29
  • O van Koert, Lecture notes on stabilization of contact open books, preprint (2010)
  • M Maydanskiy, Exotic symplectic manifolds from Lefschetz fibrations, PhD thesis, Massachusetts Institute of Technology (2009) Available at \setbox0\makeatletter\@url https://dspace.mit.edu/bitstream/handle/1721.1/50268/465218504-MIT.pdf {\unhbox0
  • M Maydanskiy, P Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 3 (2010) 157–180
  • M Maydanskiy, P Seidel, Corrigendum: Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 8 (2015) 884–886
  • M McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009) 1877–1944
  • E Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, preprint (2012)
  • A Oancea, The Künneth formula in Floer homology for manifolds with restricted contact type boundary, Math. Ann. 334 (2006) 65–89
  • A F Ritter, Deformations of symplectic cohomology and exact Lagrangians in ALE spaces, Geom. Funct. Anal. 20 (2010) 779–816
  • A F Ritter, Topological quantum field theory structure on symplectic cohomology, J. Topol. 6 (2013) 391–489
  • P Seidel, Floer homology and the symplectic isotopy problem, DPhil thesis, University of Oxford (1997) Available at \setbox0\makeatletter\@url http://www-math.mit.edu/~seidel/thesis.pdf {\unhbox0
  • P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc., Zürich (2008)
  • P Seidel, Lectures on four-dimensional Dehn twists, from “Symplectic $4$–manifolds and algebraic surfaces” (F Catanese, G Tian, editors), Lecture Notes in Math. 1938, Springer (2008) 231–267
  • K Siegel, Squared Dehn twists and deformed symplectic invariants, preprint (2016)
  • E L Stout, Polynomial convexity, Progress in Math. 261, Birkhäuser, Boston (2007)
  • C Viterbo, Functors and computations in Floer homology with applications, I, Geom. Funct. Anal. 9 (1999) 985–1033
  • A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241–251
  • T Yoshiyasu, On Lagrangian embeddings into the complex projective spaces, Internat. J. Math. 27 (2016) 1650044, 12