Strong Weinstein conjecture in dimension three: homologically trivial weighted cycles of closed Reeb orbits (Abbas–Cieliebak–Hofer conjecture)

Every Reeb vector field on a closed 3-manifold has a closed orbit (the Weinstein conjecture, proved by Taubes via Seiberg–Witten Floer theory), but it is unknown whether the conclusion can always be strengthened homologically: must every contact form admit a finite non-empty set of closed Reeb orbits with positive integer weights whose weighted sum vanishes in first homology? This 'strong Weinstein conjecture', posed by Abbas, Cieliebak and Hofer, is known for overtwisted forms (Hofer; Taubes), for planar contact structures (Abbas–Cieliebak–Hofer), and for contact structures with torsion Euler class (a direct consequence of Taubes's Seiberg–Witten theorem), but remains open for tight structures with non-torsion Euler class, such as the canonical contact structure on the unit cotangent bundle of a surface of genus at least two. The general statement remains open.

Background

A contact form on a closed oriented 3-manifold $M$ is a 1-form $\lambda$ with $\lambda\wedge d\lambda$ nowhere zero; its kernel $\xi$ is a contact structure and there is a unique vector field $R$, the Reeb vector field, characterized by $d\lambda(R,\cdot)=0$ and $\lambda(R)=1$. Its flow lines are the closed objects of study: a closed Reeb orbit is a periodic trajectory of $R$. The Weinstein conjecture (Weinstein, 1979) predicted that every Reeb field on a closed odd-dimensional manifold has at least one closed orbit; in dimension three this was proved by Taubes using Seiberg–Witten Floer homology (also called monopole Floer homology), a gauge-theoretic invariant built from solutions of the Seiberg–Witten equations associated with each $Spin^c$ structure on $M$ (a lift of the structure group of the tangent frame bundle from $SO(3)$ to $Spin^c(3)\cong U(2)/\{\pm 1\}$, the datum needed to write the equations for spinors). Abbas, Cieliebak and Hofer strengthened the conjecture in dimension three: not only must closed Reeb orbits exist, but finitely many of them, counted with positive integer multiplicities, should represent the zero class in $H_1(M;\mathbb{Z})$ — equivalently, the orbits form a homologically trivial 1-cycle. The question is non-vacuous exactly when $b_1(M)\ge 1$: if $H_1(M;\mathbb{Z})$ is torsion, the torsion-order multiple of any single closed orbit is already null-homologous, so Taubes's existence theorem settles that case. Two dichotomies organize what is known. First, a contact structure is overtwisted if it contains an overtwisted disk (an embedded disk whose boundary is tangent to $\xi$) and tight otherwise; Hofer proved in 1993 that overtwisted Reeb fields always have a contractible closed orbit, which by itself yields the desired null-homologous weighted cycle. Second, the Euler class $c_1(\xi)\in H^2(M;\mathbb{Z})$ (the first Chern class of $\xi$ viewed as a complex line bundle) may be torsion or non-torsion; Taubes's companion theorem (via Seiberg–Witten Floer theory in the canonical $Spin^c$ structure) realizes every class Poincaré dual to $e$ with $e-\tfrac{1}{2}c_1$ torsion by a positive-integer-weighted orbit set, and taking $e=0$ settles all torsion-Euler-class structures. Related technology includes embedded contact homology (ECH), whose chain complexes are generated by orbit sets (finite sets of closed Reeb orbits with positive integer multiplicity) in a fixed total homology class $\Gamma$, and which Taubes proved isomorphic to monopole Floer homology; this yields orbit-count refinements (at least two embedded Reeb orbits on any closed 3-manifold, by Cristofaro-Gardiner and Hutchings) but, as explained below, not yet the strong Weinstein conclusion in general. What survives is the tight, non-torsion Euler class case with $b_1\ge 1$.

Problem Statement

Let $M$ be a closed oriented 3-manifold and $\lambda$ a contact form with Reeb vector field $R$. Must there exist a finite non-empty set of closed $R$-orbits $\gamma_1,\dots,\gamma_n$ and positive integers $m_1,\dots,m_n$ such that $\sum_i m_i[\gamma_i]=0\in H_1(M;\mathbb{Z})$ (the strong Weinstein conjecture of Abbas–Cieliebak–Hofer)? The conjecture is posed for every contact form, regardless of tightness or Euler class; by the results above its remaining open cases lie among tight contact structures with non-torsion first Chern class on manifolds with $b_1(M)\ge 1$. Equivalently, in embedded-contact-homology terms: for every contact form, must the chain group of total homology class $\Gamma=0$ contain a non-empty generator, given that the empty orbit set always generates trivially?

References

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