Contact big fiber theorems

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.

💡 Research Summary

The paper “Contact big fiber theorems” establishes contact‑geometric analogues of the symplectic big‑fiber theorem of Entov and Polterovich, using a version of symplectic cohomology with support. The authors first develop the theory of symplectic cohomology with support on a Liouville manifold (M) and a compact (or suitably invariant) subset (K\subset M). They prove three basic properties: (i) if (K) is Hamiltonian‑displaceable then (SH^(M;K)=0); (ii) when (K) is a compact Liouville subdomain, (SH^(M;K)) coincides with the Viterbo symplectic cohomology of the completion of (K); (iii) for a field coefficient, (SH^*(M;\partial K)) is isomorphic to Rabinowitz Floer homology of (\partial K).

A key new notion is a “Poisson‑commuting collection” of compact subsets, defined via families of functions whose Poisson brackets vanish. Using a Mayer–Vietoris exact sequence for supported symplectic cohomology, they prove Theorem 1.3: if a Poisson‑commuting cover ({K_i}) satisfies (SH^*(M;\bigcup K_i)=0), then at least one (K_i) cannot be Hamiltonian‑displaceable. This is the contact‑geometric counterpart of the Entov‑Polterovich result, but now the non‑vanishing of symplectic cohomology of a filling becomes a necessary hypothesis.

The authors then apply this machinery in two settings.

1. Strong symplectic fillings. Let ((C,\xi)) be a closed contact manifold admitting a strong symplectic filling (\bar M) with completion (M). If for some sequence (r_i\to\infty) the supported cohomology (SH^*(M;{r_i}\times\partial\bar M)=0), then any contact‑involutive map (\pi:C\to\mathbb R^N) (i.e. a map whose components Poisson‑commute with respect to the Jacobi bracket of a contact form (\alpha)) possesses a fiber that is not contact‑displaceable. Moreover, there is a point in that fiber whose image under any contactomorphism isotopic to the identity has conformal factor 1 with respect to (\alpha). This is Theorem 1.9, and Corollary 1.10 deduces the same conclusion when the filling is a Liouville domain with non‑zero symplectic cohomology.

2. Prequantization circle bundles. For a closed symplectic manifold ((D,\omega_D)) with integral lift (\sigma\in H^2(D;\mathbb Z)), consider the principal (S^1)‑bundle (p:C\to D) with Euler class (-\sigma) and a connection form (\alpha) satisfying (d\alpha=p^*\omega_D). The bundle is a contact manifold (the prequantization). The authors prove that if the quantum multiplication operator \