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On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

Samuel Lisi, Jeremy Van Horn-Morris, Chris Wendl

Abstract

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to deformation equivalence in terms of diffeomorphism classes of Lefschetz fibrations. This extends previous results of the third author to a much wider class of contact manifolds, which we illustrate here by classifying the strong and Stein fillings of all oriented circle bundles with non-tangential $S^1$-invariant contact structures. Further results include new vanishing criteria for the ECH contact invariant and algebraic torsion in SFT, classification of fillings for certain non-orientable circle bundles, and a general "symplectic quasiflexibility" result about deformation classes of Stein structures in real dimension four.

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

Abstract

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to deformation equivalence in terms of diffeomorphism classes of Lefschetz fibrations. This extends previous results of the third author to a much wider class of contact manifolds, which we illustrate here by classifying the strong and Stein fillings of all oriented circle bundles with non-tangential -invariant contact structures. Further results include new vanishing criteria for the ECH contact invariant and algebraic torsion in SFT, classification of fillings for certain non-orientable circle bundles, and a general "symplectic quasiflexibility" result about deformation classes of Stein structures in real dimension four.

Paper Structure

This paper contains 46 sections, 87 theorems, 392 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Some remarks on the context
  3. Classification of fillings
  4. Weak fillings deform to strong fillings
  5. Symplectic deformation implies Stein deformation
  6. Filling obstructions and contact invariants
  7. Fillings of circle and torus bundles
  8. Classification of fillings
  9. Vanishing results for contact invariants
  10. Parabolic torus bundles
  11. The non-amenable case and exotic fibers
  12. Generalities on punctured holomorphic curves
  13. Stable Hamiltonian structures and symplectization ends
  14. Finite energy holomorphic curves
  15. Intersection theory
  16. ...and 31 more sections

Key Result

Theorem 1.5

Suppose $(M,\xi)$ is a closed contact $3$-manifold that is strongly fillable and contains a compact domain $M_0 \subset M$, possibly with boundary, on which $\xi$ is supported by a partially planar spinal open book $\boldsymbol{\pi}$. Then $M = M_0$ and $\boldsymbol{\pi}$ is uniform. Moreover, if $\

Figures (10)

  • Figure 1: The picture at the left shows a holomorphic building with arithmetic genus two, which is broken up at the right into three maximal non-nodal subbuildings, one with arithmetic genus $1$ and two with arithmetic genus $0$.
  • Figure 2: The darkly shaded region is the model collar neighborhood $E$ (with boundary $\partial E = \partial_v E \cup \partial_h E$ and corner $\partial_v E \cap \partial_h E$), together with the smooth hypersurfaces $M^0,M^- \subset E$ defined in §\ref{['sec:collars']}. The lightly shaded region represents the rest of the double completion $\widehat{E} \supset E$ as defined in §\ref{['sec:doubleCompletion']}.
  • Figure 3: The decomposition $M^0 = \widecheck{M}^0_P \cup \widecheck{M}^0_{\mathcal{I}} \cup \widecheck{M}^0_\Sigma$.
  • Figure 4: The stabilizing vector field $Z$ transverse to $M^0 \subset \widehat{E}$.
  • Figure 5: The perturbed stable hypersurface $M^+ \subset \widehat{E}$.
  • ...and 5 more figures

Theorems & Definitions (175)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 165 more