A $C^0$ Arnold-type criterion via spectral invariants (Question 3 of Buhovsky–Humilière–Seyfaddini)

The Arnold conjecture predicts that a Hamiltonian diffeomorphism of a closed symplectic manifold has at least $cl(M)$ fixed points, where $cl(M)$ is the cup length. Buhovsky–Humilière–Seyfaddini showed that no direct $C^0$ version survives: on every closed symplectic manifold of dimension at least four there is a Hamiltonian homeomorphism with a single fixed point. They asked (Question 3 of that paper) whether the Floer–Hofer criterion does survive: if a continuous Hamiltonian generating a Hamiltonian homeomorphism has fewer distinct spectral invariants than $cl(M)$, must its time-one map have infinitely many fixed points — a candidate $C^0$ Arnold conjecture. In the subsequent literature, the question is answered positively, in the stronger form that the fixed-point set is homologically non-trivial, on symplectically aspherical manifolds by the same authors, and in a corresponding quantum-product form on negative monotone manifolds and $\mathbb{CP}^n$ by Kawamoto. The problem remains open on general closed symplectic manifolds, where a positive answer would imply the still-open smooth cup-length Arnold conjecture.

Background

A closed symplectic manifold $(M,\omega)$ carries the group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms: time-one maps $\varphi_H^1$ of the flows generated by smooth time-dependent functions (Hamiltonians) $H:M\times[0,1]\to\mathbb{R}$. The Arnold conjecture predicts that fixed points of $\varphi_H^1$ abound; in its homological version it states $\#\mathrm{Fix}(\varphi_H^1)\geq cl(M)$, where the cup length $cl(M)$ is the largest $k+1$ for which $k$ cohomology classes of positive degree on $M$ have a nonzero cup product. This bound was proved for symplectically aspherical manifolds — manifolds on which both $\omega$ and the first Chern class $c_1$ vanish on $\pi_2(M)$ — by Floer and Hofer, and for $\mathbb{CP}^n$ by Fortune, but it remains open for general closed symplectic manifolds.

Passage to $C^0$ limits changes the picture drastically. A Hamiltonian homeomorphism is the time-one map naturally generated by a continuous Hamiltonian, in the sense of the group $\mathrm{Hameo}(M,\omega)$ (the more restrictive Müller–Oh definition adopted in the source paper; the related class of uniform limits of Hamiltonian diffeomorphisms is in general slightly larger); on closed surfaces such maps still satisfy the Arnold conjecture (Le Calvez), but Buhovsky–Humilière–Seyfaddini constructed, on every closed connected symplectic manifold of dimension at least four, a Hamiltonian homeomorphism with a single fixed point. Hence the Arnold conjecture admits no direct $C^0$ extension, and the question becomes which Floer-theoretic rigidity survives under $C^0$ limits.

The main rigid invariant is the system of spectral invariants introduced by Viterbo and developed by Oh and Schwarz: to each smooth Hamiltonian $H$ and each nonzero homology class $a\in H_*(M)$, filtered Hamiltonian Floer homology (the analogue of Morse homology for the action functional on the loop space of $M$) associates a number $c(a,H)\in\mathbb{R}$, which is always the action value of some fixed point of $\varphi_H^1$ and depends Lipschitz-continuously on $H$ in the supremum norm; this Lipschitz property extends $c(a,H)$ to continuous Hamiltonians. Floer and Hofer proved that if a Hamiltonian diffeomorphism of a symplectically aspherical manifold has fewer distinct spectral invariants than $cl(M)$, then it has infinitely many fixed points. In the continuous setting the correspondence between spectral invariants and fixed points breaks down completely: the single-fixed-point example of Buhovsky–Humilière–Seyfaddini is generated by a continuous Hamiltonian with at least $cl(M)$ distinct spectral invariants — indeed with the same Floer-theoretic barcode (the persistence-module encoding of the filtered Floer homology of $H$) as a $C^2$-small Morse function. Their Question 3 therefore asks whether the Floer–Hofer criterion survives $C^0$ limits exactly when its hypothesis is not excluded by the counterexample; a positive answer would be a $C^0$ version of the Arnold conjecture.

Problem Statement

Let $(M,\omega)$ be a closed connected symplectic manifold and let $H:M\times[0,1]\to\mathbb{R}$ be a continuous Hamiltonian generating a Hamiltonian homeomorphism $\varphi_H^1$ (the uniform limit of time-one maps of smooth Hamiltonians converging to $H$). Let $c(a,H)\in\mathbb{R}$, $a\in H_*(M)\setminus\{0\}$, denote the spectral invariants of $H$, extended to continuous Hamiltonians by Lipschitz continuation from the smooth theory, and let $cl(M)$ be the cup length of $M$ (one plus the maximal number of positive-degree cohomology classes on $M$ whose cup product is nonzero); as is standard in the theory, all homology and cohomology groups are taken with coefficients in a fixed field, and the question is posed for that choice. Question (Buhovsky–Humilière–Seyfaddini, Question 3): if the number of distinct values among the spectral invariants $c(a,H)$, as $a$ ranges over the nonzero homology classes, is strictly smaller than $cl(M)$, must $\varphi_H^1$ have infinitely many fixed points?

The answer is known to be yes — with the stronger conclusion that the fixed-point set is homologically non-trivial (for every open neighborhood $U$ of the fixed-point set, the map $H_j(U)\to H_j(M)$ induced by the inclusion is nontrivial for some $j>0$) — when $M$ is symplectically aspherical ($\omega$ and the first Chern class $c_1$ vanish on $\pi_2(M)$), and in a corresponding quantum-product form — if the spectral invariants $c(a,\cdot)$ and $c(a\ast b,\cdot)$, related through the quantum product $\ast$ on quantum homology, coincide for nonzero classes $a,b$ with $b\neq[M]$, then the fixed-point set is homologically non-trivial — when $M$ is negative monotone ($[\omega]$ a negative multiple of $c_1$ on $\pi_2(M)$) or $(\mathbb{CP}^n,\omega_{FS})$. The quantum-product criterion is not stated in the exact counting form of the hypothesis, so the counting-form question also remains open on negative monotone manifolds. The open core is the counting-form question on general closed symplectic manifolds, including monotone symplectic manifolds other than projective spaces ($[\omega]$ a positive multiple of $c_1$ on $\pi_2(M)$): determine whether the implication holds there, or else construct a continuous Hamiltonian with fewer distinct spectral invariants than $cl(M)$ whose time-one map has only finitely many fixed points. Any positive answer on a class of manifolds implies the smooth homological (cup-length) Arnold conjecture on that class, since for smooth Hamiltonians each spectral invariant is the action of a fixed point and therefore a finite fixed-point set forces the spectral invariants to take no more distinct values than there are fixed points.

References

1. L. Buhovsky, V. Humilière, S. Seyfaddini, 'A C^0 counterexample to the Arnold conjecture', Inventiones mathematicae 213 (2018) 759–809. DOI: 10.1007/s00222-018-0797-x. arXiv:1609.09192. https://arxiv.org/abs/1609.09192
2. L. Buhovsky, V. Humilière, S. Seyfaddini, 'The action spectrum and C^0 symplectic topology', Mathematische Annalen 380 (2021) 293–316. DOI: 10.1007/s00208-021-02183-w. arXiv:1808.09790. https://arxiv.org/abs/1808.09790
3. L. Buhovsky, V. Humilière, S. Seyfaddini, 'An Arnold-type principle for non-smooth objects', Journal of Fixed Point Theory and Applications 24 (2022), article 24. DOI: 10.1007/s11784-022-00934-z. arXiv:1909.07081. https://arxiv.org/abs/1909.07081
4. Y. Kawamoto, 'On C^0-continuity of the spectral norm for symplectically non-aspherical manifolds', International Mathematics Research Notices 2022 (2022) 17187–17230. DOI: 10.1093/imrn/rnab206. arXiv:1905.07809. https://arxiv.org/abs/1905.07809
5. W. Gong, 'Degenerate symplectic fixed points and Gromov-Witten invariants', arXiv:2507.04191 (2025). https://arxiv.org/abs/2507.04191
6. A. Floer, 'Cuplength estimates on Lagrangian intersections', Communications on Pure and Applied Mathematics 42 (1989) 335–356. DOI: 10.1002/cpa.3160420402. https://doi.org/10.1002/cpa.3160420402
7. H. Hofer, 'Lusternik-Schnirelman-theory for Lagrangian intersections', Annales de l'Institut Henri Poincaré C, Analyse non linéaire 5 (1988) 465–499. DOI: 10.1016/s0294-1449(16)30339-0. https://doi.org/10.1016/s0294-1449(16)30339-0
8. W. Howard, 'Action Selectors and the Fixed Point Set of a Hamiltonian Diffeomorphism', arXiv:1211.0580 (2012). https://arxiv.org/abs/1211.0580
9. B. Fortune, 'A symplectic fixed point theorem for CP^n', Inventiones Mathematicae 81 (1985) 29–46. DOI: 10.1007/bf01388770. https://doi.org/10.1007/bf01388770
10. P. Le Calvez, 'Periodic orbits of Hamiltonian homeomorphisms of surfaces', Duke Mathematical Journal 133 (2006) 125–184. DOI: 10.1215/s0012-7094-06-13315-x. https://doi.org/10.1215/s0012-7094-06-13315-x
11. C. Viterbo, 'Symplectic topology as the geometry of generating functions', Mathematische Annalen 292 (1992) 685–710. DOI: 10.1007/bf01444643. https://doi.org/10.1007/bf01444643
12. Y.-G. Oh, 'Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds', in: The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics 232, Birkhäuser (2005) 525–570. DOI: 10.1007/0-8176-4419-9_18. https://doi.org/10.1007/0-8176-4419-9_18
13. M. Schwarz, 'On the action spectrum for closed symplectically aspherical manifolds', Pacific Journal of Mathematics 193 (2000) 419–461. DOI: 10.2140/pjm.2000.193.419. https://doi.org/10.2140/pjm.2000.193.419
14. A. Kislev, E. Shelukhin, 'Bounds on spectral norms and barcodes', Geometry & Topology 25 (2021) 3257–3350. DOI: 10.2140/gt.2021.25.3257. arXiv:1810.09865. https://arxiv.org/abs/1810.09865
15. E. Shelukhin, 'Viterbo conjecture for Zoll symmetric spaces', Inventiones Mathematicae 230 (2022) 321–373. DOI: 10.1007/s00222-022-01124-x. arXiv:1811.05552. https://arxiv.org/abs/1811.05552
16. N. Hingston, 'Subharmonic solutions of Hamiltonian equations on tori', Annals of Mathematics 170 (2009) 529–560. DOI: 10.4007/annals.2009.170.529. https://doi.org/10.4007/annals.2009.170.529
17. V. L. Ginzburg, 'The Conley conjecture', Annals of Mathematics 172 (2010) 1127–1180. DOI: 10.4007/annals.2010.172.1127. https://doi.org/10.4007/annals.2010.172.1127