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Algebraic and Giroux torsion in higher-dimensional contact manifolds

Agustin Moreno

Abstract

We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic $1$-torsion in any odd dimension, which proves a conjecture by Massot-Niederkrueger-Wendl. These results are part of the author's PhD thesis.

Algebraic and Giroux torsion in higher-dimensional contact manifolds

Abstract

We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic -torsion in any odd dimension, which proves a conjecture by Massot-Niederkrueger-Wendl. These results are part of the author's PhD thesis.

Paper Structure

This paper contains 16 sections, 16 theorems, 146 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Algebraic torsion computations
  3. Construction of the model contact manifolds
  4. Compatible almost complex structure
  5. Finite energy foliation
  6. Index computations
  7. Fredholm regularity
  8. Regularity for genus zero page-like curves in the Morse--Bott case
  9. From Morse--Bott to Morse
  10. Uniqueness of the curves in the Morse/Morse--Bott case
  11. From the SHS to a sufficiently non-degenerate contact structure
  12. Proof of Theorem \ref{['thm5d']}
  13. Giroux torsion implies algebraic 1-torsion in higher dimensions
  14. Giroux SOBDs
  15. Prequantization SOBDs (non-trivial case)
  16. ...and 1 more sections

Key Result

Theorem 1.3

For any $k\geq 1$, and $g\geq k$, the $(2n+1)$-dimensional contact manifolds $(M_g=Y\times \Sigma,\xi_k)$ satisfy $AT(M_g,\xi_k)\leq k-1$. Moreover, if $(Y,\alpha_\pm)$ are hypertight, and $k\geq 2$, the corresponding contact manifold $(M_g,\xi_k)$ is also hypertight. In particular, $AT(M_g,\xi_k)>0

Figures (9)

  • Figure 1: The SOBD structure in $M$.
  • Figure 2: The qualitative behaviour of the flow of the Liouville vector field $V$ on any cylindrical Liouville semi-filling, for which the central slice ($r=0$) is invariant. One may informally think of such a Liouville domain as being obtained by gluing two negative symplectizations along a "non-contact hypersurface".
  • Figure 3: The paths $\rho \mapsto (F_\pm(\rho),G_\pm(\rho))$.
  • Figure 4: The double completion, and a Morse function $H$ along the spine.
  • Figure 5: The double completion $E^{\infty,\infty}$, containing $M$ and its perturbed version $M^\epsilon$ as contact type hypersurfaces. The foliation by holomorphic curves is shown in green (for the non-trivial curves) and blue (for the trivial cylinders).
  • ...and 4 more figures

Theorems & Definitions (34)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 24 more