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Contact big fiber theorems

Yuhan Sun, Igor Uljarevic, Umut Varolgunes

TL;DR

The paper develops a contact-geometry analogue of Entov–Polterovich’s big fiber theorem by introducing and employing symplectic cohomology with support, together with Rabinowitz Floer theory, to detect non-displaceable fibers in contact involutive maps under Liouville-fillable, nonzero symplectic cohomology hypotheses. It proves a general contact big fiber theorem (and a prequantization circle-bundle variant) and demonstrates broad rigidity phenomena, including contact non-squeezing, through descent arguments and a Mayer–Vietoris framework. The results are illustrated with concrete applications to Ustilovsky spheres and quadrics, establishing non-displaceable fibers and non-squeezing phenomena in several non-toric, non-fillable contexts where Floer theory can still be employed. Overall, the work provides a flexible, non-toric, Floer-theoretic approach that yields robust rigidity results in contact topology with potential for further extensions and applications.

Abstract

We prove contact big fiber theorems, analogous to the symplectic big fiber theorem by Entov and Polterovich, using symplectic cohomology with support. Unlike in the symplectic case, the validity of the statements requires conditions on the closed contact manifold. One such condition is to admit a Liouville filling with non-zero symplectic cohomology. In the case of Boothby-Wang contact manifolds, we prove the result under the condition that the Euler class of the circle bundle, which is the negative of an integral lift of the symplectic class, is not an invertible element in the quantum cohomology of the base symplectic manifold. As applications, we obtain new examples of rigidity of intersections in contact manifolds and also of contact non-squeezing.

Contact big fiber theorems

TL;DR

The paper develops a contact-geometry analogue of Entov–Polterovich’s big fiber theorem by introducing and employing symplectic cohomology with support, together with Rabinowitz Floer theory, to detect non-displaceable fibers in contact involutive maps under Liouville-fillable, nonzero symplectic cohomology hypotheses. It proves a general contact big fiber theorem (and a prequantization circle-bundle variant) and demonstrates broad rigidity phenomena, including contact non-squeezing, through descent arguments and a Mayer–Vietoris framework. The results are illustrated with concrete applications to Ustilovsky spheres and quadrics, establishing non-displaceable fibers and non-squeezing phenomena in several non-toric, non-fillable contexts where Floer theory can still be employed. Overall, the work provides a flexible, non-toric, Floer-theoretic approach that yields robust rigidity results in contact topology with potential for further extensions and applications.

Abstract

We prove contact big fiber theorems, analogous to the symplectic big fiber theorem by Entov and Polterovich, using symplectic cohomology with support. Unlike in the symplectic case, the validity of the statements requires conditions on the closed contact manifold. One such condition is to admit a Liouville filling with non-zero symplectic cohomology. In the case of Boothby-Wang contact manifolds, we prove the result under the condition that the Euler class of the circle bundle, which is the negative of an integral lift of the symplectic class, is not an invertible element in the quantum cohomology of the base symplectic manifold. As applications, we obtain new examples of rigidity of intersections in contact manifolds and also of contact non-squeezing.

Paper Structure

This paper contains 18 sections, 34 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Symplectic cohomology with support
  3. Symplectic and contact big fiber theorems
  4. Big fiber theorem for prequantization circle bundles
  5. Applications
  6. Ustilovsky spheres
  7. Quadrics
  8. Non-compact skeleta
  9. Contact non-squeezing
  10. Proofs
  11. The definition of symplectic cohomology with support
  12. Proof of Theorem \ref{['t:conformal']}
  13. Proof of Theorem \ref{['thm-rfh-sh-vanish']}
  14. Proof of Theorem \ref{['thm-prequantum']}
  15. Comparison results
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let ${K}_1, \cdots,{K}_N$ be a Poisson commuting collection of compact subsets of $M$. Let $K:=K_1\cup\ldots\cup K_N.$ If $SH^*_M(K)\neq 0,$ then at least one of ${K}_1, \cdots,{K}_N$ is not displaceable from itself by a Hamiltonian diffeomorphism.

Figures (5)

  • Figure 1: Two moment polytopes for $Q_2$. On the left, the dot represents the barycenter. On the right, the line segment is contained in the median and the interior dot is at the midpoint of this median.
  • Figure 2: Moment polytope for $Q_3$ and a probe in $T^*S^3$. The vertices of the polytope on the left are $(0,0,0),(0,0,3),(\pm 3,3,3).$
  • Figure 3: Hamiltonian functions in the cylindrical coordinate.
  • Figure 4: Hamiltonian functions for restriction maps.
  • Figure 5: Hamiltonian functions for a filled cobordism.

Theorems & Definitions (88)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • Definition 1.7
  • Definition 1.8
  • ...and 78 more