Contact Big Fiber Theorem Overview

• Contact Big Fiber Theorem is a concept in contact geometry ensuring that every contact involutive map on a closed manifold possesses at least one non-displaceable fiber.
• It employs advanced tools like spectral invariants, Floer theory, quasi-states, and persistence modules to analyze rigidity and non-displaceability phenomena.
• The theorem generalizes symplectic results by extending rigidity, orderability, and non-squeezing principles to various classes of contact manifolds.

The Contact Big Fiber Theorem asserts that, under suitable geometric, topological, and (often Floer-theoretic) conditions on a closed contact manifold, every “contact involutive” map possesses at least one fiber that is contact non-displaceable—meaning no contactomorphism isotopic to the identity can move this fiber away from itself. This phenomenon generalizes the Entov–Polterovich big fiber theorem from symplectic topology and is foundational in the paper of rigidity in contact geometry. The recent literature has developed several approaches to establishing the theorem, leveraging spectral invariants from contact Hamiltonian Floer theory, quasi-states, quasi-measures, and new algebraic tools such as spectral selectors and gapped persistence modules.

1. Definitions and Foundational Framework

The Contact Big Fiber Theorem applies to closed contact manifolds (M, ξ) that admit a Liouville filling (W, λ) with non-vanishing and $\mathbb{Z}$-graded symplectic homology (Uljarević et al., 24 Mar 2025, Sun et al., 6 Mar 2025). A map $F : M \to \mathbb{R}^N$ is called contact involutive if its coordinate functions are Reeb-invariant and mutually commute under the contact Hamiltonian Poisson bracket; equivalently, they generate commuting flows that preserve the contact structure.

A fiber of F refers to $F^{-1}(x)$ for $x \in \mathbb{R}^N$. A fiber is said to be contact non-displaceable if for every contactomorphism $\phi$ isotopic to the identity, $\phi(F^{-1}(x)) \cap F^{-1}(x) \neq \emptyset$.

In Boothby–Wang manifolds (prequantization bundles over symplectic bases), existence of a non-displaceable fiber requires, in addition, that the Euler class of the S¹-bundle (the negative of an integral lift of the symplectic class) is not invertible in the quantum cohomology of the base (Sun et al., 6 Mar 2025).

2. Contact Spectral Invariants and Persistence Modules

The theory is built on contact spectral invariants derived from contact Hamiltonian Floer groups. For each admissible contact Hamiltonian function $h$ (typically Reeb-invariant), the associated spectral invariant $c_\alpha(h)$ is defined relative to the contact boundary $M = \partial W$. For $\theta \in SH_*(W)$,

$$c(h, \theta) = -\inf \{ \eta \in \mathbb{R} \mid \theta \text{ lies in the image of } HF_*(\eta \# h) \to SH_*(W) \}$$

where $\eta \# h$ is the shifted Hamiltonian (Djordjević et al., 17 Jul 2025).

A key algebraic property is duality: passing from $h$ to its time-reversed dual $\bar{h}(t,x) = -h(-t,x)$ causes the spectral invariant to flip sign,

$c(h, \theta) = -c(\bar{h}, PD(\theta))$

with $PD$ the Poincaré duality on symplectic homology (Djordjević et al., 17 Jul 2025). The theory formalizes the Floer groups as gapped persistence modules over partially ordered sets, encoding the filtration structure (non-traditional in contact geometry).

Spectral selectors (e.g., $C_\pm^\alpha$ on strongly orderable manifolds (Arlove, 16 Sep 2025)) and their analogues for lens spaces extract “spectral” data from contact isotopies, satisfying normalization, homogeneity, monotonicity, and a triangle inequality derived via pair-of-pants constructions.

3. Partial Contact Quasi-States and Quasi-Measures

A partial contact quasi-state $\zeta_\alpha : C^\infty(M) \to \mathbb{R}$ is constructed by asymptotically rescaling spectral invariants,

$$\zeta_\alpha(h) = \lim_{k \to \infty} \frac{c_\alpha(kh)}{k}$$

with properties analogous to Entov–Polterovich quasi-states in symplectic topology: normalization ($\zeta_\alpha(1) = 1$), stability for strict (Reeb-invariant) functions, vanishing on contact displaceable sets, and triangle inequality (Uljarević et al., 24 Mar 2025).

The induced contact quasi-measure $\tau_\alpha$ on closed subsets $A \subset M$ is defined

$$\tau_\alpha(A) = \inf \{ \zeta_\alpha(h) \mid h \in C^\infty(M),\ h|_A = 1,\ h \text{ strict} \}$$

It is monotone, normalized ($\tau_\alpha(M) = 1$), and, crucially, vanishes for Reeb-invariant displaceable sets.

4. Rigidity Phenomena and Proof Outline

The alternative proof of the theorem (Uljarević et al., 24 Mar 2025), in the spirit of Entov–Polterovich, proceeds by contradiction:

1. Assume every fiber of a contact involutive map $F : M \to \mathbb{R}^N$ is displaceable.
2. Since fibers are Reeb-invariant, the quasi-measure $\tau_\alpha$ vanishes on each fiber.
3. Cover $M$ by a finite union of fibers; subadditivity yields $\tau_\alpha(M) \le 0$.
4. Normalization mandates $\tau_\alpha(M) = 1$, contradiction.

Thus, at least one fiber must be contact non-displaceable. This rigidity extends to various subclasses of contact manifolds: strongly orderable manifolds (with spectral selectors) (Arlove, 16 Sep 2025), lens spaces, and prequantization bundles under quantum cohomology restrictions.

5. Applications: Non-Squeezing, Orderability, and Dynamics

Consequences of the Contact Big Fiber Theorem span several domains:

• Contact non-squeezing: If a Reeb-invariant Hamiltonian partitions $M$ into $\{h > 0\}$ and $\{h \le 0\}$, at least one set contains a non-displaceable fiber (Sun et al., 6 Mar 2025).
• Orderability: Algebraic properties of spectral invariants and selectors yield sufficient conditions for orderability of the contactomorphism group (Djordjević et al., 17 Jul 2025).
• Conjugation invariant norms: In strongly orderable cases and with periodic Reeb flow, there exist stably unbounded conjugation-invariant norms on the universal cover of the contactomorphism group (Arlove, 16 Sep 2025).
• Dynamics: The Reeb flow is shown to be geodesic for the discriminant and oscillation norms when all its orbits have the same period (in lens spaces and certain other cases) (Arlove, 16 Sep 2025).
• Legendrian intersection rigidity: For prequantization bundles, Legendrian lifts of Lagrangians cannot be displaced from their Reeb closures when quantum multiplication by the Euler class fails to be invertible (Sun et al., 6 Mar 2025).

6. Relations to Symplectic Big Fiber Theorems and Broader Significance

The Contact Big Fiber Theorem is directly analogous to the symplectic big fiber theorem of Entov–Polterovich, but with vital differences due to the contact setting: the need for a Liouville filling with nonzero symplectic homology, dependence on the topology (e.g., the quantum cohomology condition for Boothby–Wang bundles), and the subtle behavior of Reeb-invariant objects. The theory expands the toolkit of contact topology, introducing new invariants and algebraic structures (quasi-states, quasi-measures, spectral selectors, gapped persistence modules), and informs rigidity results, non-squeezing phenomena, and open problems in the quantitative geometry of contactomorphism groups.

7. Mathematical Formulas and Constructs

Object	Definition	Reference
Spectral invariant	$$c_\alpha(h) = -\inf\{\eta | e \in \mathrm{im}\ HF_*(\eta \# h)\to SH_*(W)\}$$	(Djordjević et al., 17 Jul 2025, Uljarević et al., 24 Mar 2025)
Quasi-state	$$\zeta_\alpha(h) = \lim_{k\to\infty} \frac{c_\alpha(kh)}{k}$$	(Uljarević et al., 24 Mar 2025)
Quasi-measure	$$\tau_\alpha(A) = \inf \{ \zeta_\alpha(h) \mid h|_A = 1 \}$$	(Uljarević et al., 24 Mar 2025)
Spectral selector $C_\pm$	$$C_{±}^\alpha([φ_t]) = ℓ_{±}^{β_\alpha}(G̃_\alpha([φ_t]) \cdot \tildeΔ)$$	(Arlove, 16 Sep 2025)

This structural approach, foundational in recent contact geometry, enables precise analysis of rigidity, uniqueness, and topological constraints for fibers under contact involutive maps.