The strong Arnold conjecture: Morse-number lower bound for fixed points of Hamiltonian diffeomorphisms

The strongest form of Arnold's conjecture on Hamiltonian fixed points predicts that a generic (nondegenerate) Hamiltonian diffeomorphism of an arbitrary closed symplectic manifold $M$ has at least $\mathrm{Morse}(M)$ fixed points, where $\mathrm{Morse}(M)$ is the minimal number of critical points of a Morse function on $M$. This "strong Arnold conjecture" remains open. The best general bounds are homological (sums of Betti numbers, recently strengthened to mod-$p$ and integral counts), and the best Morse-theoretic bound — the stable Morse number $\mathrm{stabMorse}(M)$ — is proven only for symplectically aspherical manifolds. Because Damian's and Gompf's examples give closed symplectic manifolds with $\mathrm{Morse}(M) > \mathrm{stabMorse}(M)$, the stable bound does not imply the Morse-number bound; even for symplectically aspherical manifolds the Morse-number bound is open whenever $\mathrm{Morse}(M) > \mathrm{stabMorse}(M)$, since $\mathrm{Morse}(M) = \mathrm{stabMorse}(M)$ is not established for every such manifold. The problem asks to prove or refute the conjecture in full generality.

Background

A symplectic manifold $(M,\omega)$ is a smooth even-dimensional manifold equipped with a closed 2-form $\omega$ that is nondegenerate ($\omega^{\wedge n}$ is a volume form). A Hamiltonian diffeomorphism is the time-1 map $\varphi$ of the flow of the time-dependent vector field $X_{H_t}$ defined by $\iota_{X_{H_t}}\omega = -dH_t$ for a smooth periodic Hamiltonian $H:\mathbb{R}/\mathbb{Z}\times M\to\mathbb{R}$. Its fixed points are the 1-periodic orbits of the flow. The fixed-point data are called nondegenerate (or the diffeomorphism generic) when the graph of $\varphi$ meets the diagonal transversally; then the fixed points form a finite set, each carrying a Conley–Zehnder index playing the role of a Morse index. Let $\mathrm{Morse}(M)$ denote the minimal number of critical points of a Morse function on $M$, and $\mathrm{stabMorse}(M)$ the stable Morse number: the minimum number of critical points of a Morse function on $M\times\mathbb{R}^k$ (over all $k\geq 0$) that is almost quadratic at infinity (its differential differs from that of a fixed nondegenerate quadratic form on the $\mathbb{R}^k$-factor by a uniformly bounded amount). One always has $\mathrm{Morse}(M)\geq\mathrm{stabMorse}(M)\geq\sum_i b_i(M)$, where $b_i(M)$ is the $i$-th Betti number, with equality throughout in favorable cases: for simply connected closed manifolds of dimension at least 6, a classical theorem of Smale identifies $\mathrm{Morse}(M)$ with the minimal number of generators of a $\mathbb{Z}$-complex computing $H_\bullet(M)$. By contrast, Damian proved that closed manifolds with a 'complicated' fundamental group satisfy $\mathrm{Morse}(M)>\mathrm{stabMorse}(M)$, and Gompf showed every finitely presentable group occurs as $\pi_1$ of a closed symplectic manifold of dimension $2n\geq 4$, so the two numbers genuinely differ within the symplectic category.

Arnold conjectured in the 1960s (see Appendix 9 of his Mathematical Methods of Classical Mechanics and p. 284 of Arnold's Problems) that the number of fixed points of an arbitrary (resp. generic) Hamiltonian diffeomorphism of a closed symplectic manifold is at least the number of critical points of a smooth (resp. Morse) function on it. The generic clause with the Morse number is called the strong Arnold conjecture. The problem drove the creation of Floer homology: an infinite-dimensional Morse theory whose chain complex is generated by the fixed points and whose differential counts solutions of a perturbed Cauchy–Riemann equation (pseudoholomorphic strips). The homological version — $|\mathrm{Fix}(\varphi)|\geq\sum_i\dim H_i(M;\mathbb{Q})$ for nondegenerate $\varphi$ — is a theorem for all closed symplectic manifolds (Fukaya–Ono, Liu–Tian, Ruan, via virtual perturbation techniques), and has been sharpened to mod-$p$ Betti sums by Abouzaid–Blumberg using Morava $K$-theory (a family of generalized cohomology theories with mod-$p$ coefficients) and to an integral Betti count (torsion included) by Bai–Xu. On the strong side: the conjecture holds for tori (Conley–Zehnder), for surfaces of genus at least 2 and certain Kähler manifolds (Floer), and Floer's original theory covers the monotone setting (where $[\omega]$ is a real multiple of the first Chern class $c_1$ on $\pi_2(M)$, making the area and index of pseudoholomorphic spheres — holomorphic maps from the Riemann sphere into $M$ — compatible). For symplectically aspherical manifolds (those for which both the symplectic area class $A\mapsto\int_A\omega$ and the first Chern class $c_1$ vanish on $\pi_2(M)$), Dimitroglou Rizell and Golovko proved — via Lagrangian Floer homology of the diagonal (a Lagrangian submanifold is a half-dimensional submanifold on which $\omega$ restricts to zero) with $\mathbb{Z}[\pi_1]$-twisted coefficients, Sullivan's bifurcation analysis showing the Floer complex is simple homotopy equivalent — equivalent through elementary cell attachments and collapses, i.e. with vanishing Whitehead torsion — to the Morse complex, and Damian's realization theorem for complexes of stable Morse type — that a generic Hamiltonian diffeomorphism has at least $\mathrm{stabMorse}(M)$ fixed points; Pöder later gave an independent proof by action–energy arguments, extended to certain convex-at-infinity manifolds with vanishing first Chern class. In the degenerate (non-generic) direction, Rudyak–Oprea proved the $\mathrm{Crit}(M)$ bound ($\mathrm{Crit}(M)$ denoting the minimal number of critical points of an arbitrary smooth function on $M$) for symplectically aspherical manifolds, and Asselle–Starostka recently established cup-length bounds (the cup-length being the maximal number of positive-degree cohomology classes with nonvanishing product) for $\mathbb{T}^{2m}\times\mathbb{CP}^n$ near the identity. All of these remain below the Morse number in general: on manifolds with complicated fundamental group the stable Morse bound is strictly weaker than what the strong conjecture predicts, and beyond the aspherical (or monotone) classes even the stable Morse bound is not known.

Problem Statement

Prove or refute the strong Arnold conjecture: for every closed symplectic manifold $(M,\omega)$ and every generic (nondegenerate) Hamiltonian diffeomorphism $\varphi$ of $M$, the number of fixed points satisfies

$$|\mathrm{Fix}(\varphi)| \geq \mathrm{Morse}(M),$$

where $\mathrm{Morse}(M)$ is the minimal number of critical points of a Morse function on the underlying smooth manifold $M$. Equivalently, in the periodic-orbit formulation: any nondegenerate time-periodic Hamiltonian on a closed symplectic manifold has at least $\mathrm{Morse}(M)$ 1-periodic orbits. The bound is required uniformly over all closed symplectic manifolds, with no asphericity (vanishing of the symplectic area class and of $c_1$ on $\pi_2(M)$), monotonicity ($[\omega]$ a real multiple of $c_1$ on $\pi_2(M)$), semipositivity (a mild positivity condition on $c_1$ on $\pi_2(M)$ under which bubbling of pseudoholomorphic spheres is controllable), or fundamental-group restriction; known results cover only tori, surfaces of genus at least two, certain Kähler manifolds, monotone/high-dimensional simply connected cases, and — at the weaker stable Morse level — symplectically aspherical manifolds. A companion clause of Arnold's strongest formulation, $|\mathrm{Fix}(\varphi)| \geq \mathrm{Crit}(M)$ for arbitrary (possibly degenerate) Hamiltonian diffeomorphisms, where $\mathrm{Crit}(M)$ is the minimal number of critical points of an arbitrary smooth function on $M$, is likewise open in general (proved only for symplectically aspherical manifolds) and constitutes a formally stronger degenerate variant; the nondegenerate Morse-number clause above is the target problem. A refutation consists of a closed symplectic manifold $M$ and a nondegenerate Hamiltonian diffeomorphism with strictly fewer than $\mathrm{Morse}(M)$ fixed points.

References

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