A pair-topology Arnold conjecture for $b^m$-symplectic manifolds in dimensions at least four

On a compact $b^m$-symplectic (log-symplectic) manifold the symplectic form degenerates along a critical hypersurface $Z$, and only a restricted class of admissible $b^m$-Hamiltonians admits a meaningful count of 1-periodic orbits. Brugués–Miranda–Oms proved that for $b^{2m}$-symplectic manifolds of any dimension — under the hypotheses of their Corollary 5.1, namely a critical set given by a global defining function together with either an acyclic associated graph or a unimodular Hamiltonian — the number of nondegenerate 1-periodic orbits of an admissible Hamiltonian is at least the sum of the mod-2 Betti numbers of the ambient manifold, and for orientable $b^m$-symplectic surfaces they obtained the sharper, optimal bound $\sum_v(2g_v+|\deg(v)-2|)$ depending on the topology of the pair $(M,Z)$; they also constructed a Floer homology $^{b^m}HF_*(M,\omega,H)$ for admissible Hamiltonians. The open problem, stated as Open Questions 7.2 and 7.3 of that work, is whether these surface results extend beyond dimension 2: does a compact $b^m$-symplectic manifold of dimension at least 4 admit a lower bound on the number of nondegenerate 1-periodic orbits of admissible Hamiltonians that is computable purely from the topology of the pair $(M,Z)$, in the spirit of the surface formula — and can the Floer homology $^{b^m}HF_*(M,\omega,H)$ be computed in terms of the topology of $M$ and $Z$? No progress on either formulation has been found in the subsequent literature; the problem is open.

Background

A symplectic manifold is a smooth even-dimensional manifold $(M,\omega)$ carrying a closed 2-form $\omega$ that is nondegenerate; the Hamiltonian vector field of a function $H$ is defined by $\iota_{X_H}\omega=dH$, and its flow is the basic object of Hamiltonian dynamics. Arnold's conjecture predicts that for a time-dependent Hamiltonian on a closed symplectic manifold whose 1-periodic orbits are all nondegenerate (the linearized time-one return map at each orbit fails to have 1 as an eigenvalue), the number of such orbits is at least the sum of the mod-2 Betti numbers $\sum_i\dim H_i(M;\mathbb{Z}_2)$. Floer proved the conjecture in foundational cases by constructing Hamiltonian Floer homology, an infinite-dimensional Morse theory on the loop space generated by the 1-periodic orbits, whose homology is that of $M$ and thus forces the bound (Floer 1986; the general case was settled only decades later). A $b^m$-symplectic manifold $(M,Z,\omega)$ (log-symplectic when $m=1$) is the even-dimensional singular analogue introduced by Guillemin–Miranda–Pires: $\omega$ is symplectic on the complement of a fixed hypersurface $Z$, the critical set, and is allowed a pole of order $m$ transverse to $Z$, i.e. it is a section of $\Lambda^2({}^{b^m}T^*M)$, the second exterior power of the bundle dual to vector fields tangent to $Z$ up to order $m$. Such structures arise as regularized phase spaces of the restricted three-body problem (a $b^3$-structure), in the space of geodesics of the Lorentz plane, and in the study of Painlevé transcendents. Along $Z$ the structure induces a cosymplectic structure — a pair consisting of a closed 1-form $\alpha$ and a closed 2-form $\beta$ with $\alpha\wedge\beta^{n-1}$ a volume form — which foliates $Z$ by symplectic leaves of codimension one. For smooth Hamiltonians the Hamiltonian vector field of a $b^m$-symplectic form is tangent to these leaves, producing typically infinitely many periodic orbits accumulating near $Z$, so the naive Arnold count is meaningless. Brugués–Miranda–Oms therefore restrict to admissible Hamiltonians: $b^m$-functions, i.e. functions with a $\log|z|$-type singularity along $Z$ (for $m=1$) or a $z^{1-m}$-type singularity (for $m>1$), that are linear along a transverse symplectic vector field and invariant under the Reeb flow of the cosymplectic structure on $Z$ (the Reeb vector field is the unique vector field $R$ on $Z$ with $\iota_R\beta=0$ and $\iota_R\alpha=1$, and its flow preserves the symplectic leaves); for these only finitely many periodic orbits sit near the critical set. Their key tool is the desingularization of Guillemin–Miranda–Weitsman, which for $b^{2m}$-structures produces a smooth symplectic form on the same manifold, but for $b^{2m+1}$-structures produces only a folded symplectic form (nondegenerate up to a sign flip across $Z$), blocking direct transport of Floer theory in odd order. Within this framework the source paper proves: (i) Theorem A (Corollary 5.1): for any compact $b^{2m}$-symplectic manifold whose critical set is given by a global defining function and whose associated graph is acyclic or whose Hamiltonian is unimodular (a global condition on the Hamiltonian's transverse part), $\#P(H)\ge\sum_i\dim H_i(M;\mathbb{Z}_2)$; (ii) Theorem B (Theorem 5.4): for a compact orientable $b^m$-symplectic surface, $\#P(H)\ge\sum_{v\in V}(2g_v+|\deg(v)-2|)$, where $V$ indexes the connected components of $\Sigma\setminus Z$ as vertices of the graph of the $b$-manifold (critical curves are edges), $g_v$ is the genus and $\deg(v)$ the valence — a bound depending on the topology of the pair $(\Sigma,Z)$, shown optimal in the paper's Appendix A; (iii) Theorem 5.6: for higher-dimensional $b^m$-symplectic manifolds whose critical set is a union of trivial mapping tori (products $S^1\times L$ with $L$ a symplectic leaf), a bound of the form $\sum_{v\in V}\max(\sum_{i=0}^{2n}\dim H_i(M^v;\mathbb{Z}_2)-\sum_{e_j}\#P(X_h|_{Z_j}),0)$, in which the $\max(\cdot,0)$ truncation is applied vertex by vertex, $M^v$ is the compactification of the component of $M\setminus Z$ corresponding to the vertex $v$, and the second sum runs over the edges $e_j$ incident to $v$; this is not purely topological because it subtracts a count of periodic orbits of an induced Hamiltonian $h$ on the components of $Z$; and (iv) Theorem C plus Section 6: a Floer complex for admissible Hamiltonians on $b^m$-symplectic manifolds whose solutions obey a minimum principle keeping them away from $Z$ (constructed in Section 6.2 under the assumption that $(M,\omega)$ is aspherical, i.e. that $[\omega]$ vanishes on $\pi_2(M)$, with the grading additionally requiring $c_1(\omega)$ to vanish on $\pi_2(M)$), yielding a Floer homology $^{b^m}HF_*(M,\omega,H)$ which the authors could neither prove independent of the auxiliary choices (Hamiltonian, compatible almost complex structure) nor compute. The unresolved target is the extension of the surface picture, in which the pair $(M,Z)$ — not the ambient manifold alone — controls the orbit count, to dimension at least 4.

Problem Statement

This is Open Questions 7.2 and 7.3 of Brugués–Miranda–Oms, which ask whether the surface results extend beyond dimension 2. Let $(M,Z,\omega)$ be a compact $b^m$-symplectic manifold of dimension $2n\ge 4$ (no parity restriction on $m$), and let $H$ be a time-dependent admissible $b^m$-Hamiltonian whose 1-periodic orbits are all nondegenerate. (1) Dynamical form: is the number $\#P(H)$ of 1-periodic orbits of $X_H$ bounded below by an explicit quantity depending only on the topology of the pair $(M,Z)$ — the higher-dimensional analogue of the surface formula $\sum_v(2g_v+|\deg(v)-2|)$, accounting for the topology of $Z$ and its relative position in $M$, and reducing the dynamical correction term of Theorem 5.6 to topology — valid in dimension at least 4, including odd $m$ where desingularization currently yields only a folded symplectic structure? The bound is required to be genuinely pair-sensitive: neither the constant $0$ nor a restatement of the ambient mod-2 Betti bound of Theorem A answers the question, the source itself regarding that bound as unsatisfactory here and expecting an optimal bound to take into account not only the topology of $M$ but also that of $Z$ and its relative position in $M$. (2) Floer-theoretic form: can the Floer homology $^{b^m}HF_*(M,\omega,H)$ defined by the source paper for admissible Hamiltonians (in the source, under the asphericity hypothesis recalled in the Background) be computed in terms of the topology of $M$ and $Z$, after establishing whatever invariance under the choices of Hamiltonian and compatible almost complex structure is needed? A complete solution is either a theorem supplying such a bound (respectively such a computation, with the bound as its dimensional shadow), ideally with sharpness examples as in the surface case, or a rigorous negative answer — a class of compact $b^m$-symplectic manifolds of dimension at least 4 and admissible Hamiltonians for which no bound of this character exists (for instance, where the infimum of $\#P(H)$ over admissible Hamiltonians with nondegenerate orbits equals the ambient Arnold bound of Theorem A despite nontrivial topology of $Z$, or, in classes not covered by Theorem A, where no positive topological lower bound exists at all). The sibling questions from the same list (celestial-mechanics applications, enlarging the admissible class, and leaf-wise or foliated Arnold bounds for smooth Hamiltonians) are separate open problems and are not part of this problem's completion boundary.

References

1. Joaquim Brugués, Eva Miranda, Cédric Oms, 'The Arnold conjecture for singular symplectic manifolds', Journal of Fixed Point Theory and Applications 26 (2024), no. 2, Article 16. DOI: 10.1007/s11784-024-01105-y, https://doi.org/10.1007/s11784-024-01105-y. Preprint: arXiv:2212.01344, https://arxiv.org/abs/2212.01344 (source of the open questions).
2. Andreas Floer, 'Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds', Duke Mathematical Journal 53 (1986), no. 1, 1–32. DOI: 10.1215/s0012-7094-86-05301-9, https://doi.org/10.1215/s0012-7094-86-05301-9.
3. Victor Guillemin, Eva Miranda, Ana Rita Pires, 'Symplectic and Poisson geometry on b-manifolds', Advances in Mathematics 264 (2014), 864–896. DOI: 10.1016/j.aim.2014.07.032, https://doi.org/10.1016/j.aim.2014.07.032.
4. Victor Guillemin, Eva Miranda, Jonathan Weitsman, 'Desingularizing $b^m$-symplectic structures', International Mathematics Research Notices 2019 (2019), no. 10, 2981–2998. DOI: 10.1093/imrn/rnx126, https://doi.org/10.1093/imrn/rnx126. Preprint: arXiv:1512.05303, https://arxiv.org/abs/1512.05303.
5. Charlotte Kirchhoff-Lukat, 'Log Floer cohomology for oriented log symplectic surfaces', Journal of Geometry and Physics 209 (2025), Article 105412. DOI: 10.1016/j.geomphys.2024.105412, https://doi.org/10.1016/j.geomphys.2024.105412. Preprint: arXiv:2207.06894, https://arxiv.org/abs/2207.06894.
6. Josep Fontana-McNally, Eva Miranda, Cédric Oms, Daniel Peralta-Salas, 'A counterexample to the singular Weinstein conjecture', Advances in Mathematics 458 (2024), Article 109998. DOI: 10.1016/j.aim.2024.109998, https://doi.org/10.1016/j.aim.2024.109998. Preprint: arXiv:2310.19918, https://arxiv.org/abs/2310.19918.
7. Kai Cieliebak, Urs Frauenfelder, Eva Miranda, Jagna Wiśniewska, 'The two-boost problem and Lagrangian Rabinowitz Floer homology', preprint (2024). arXiv:2412.08415, https://arxiv.org/abs/2412.08415.
8. Eva Miranda, Pablo Nicolás, 'Which singular tangent bundles are isomorphic?', Journal of the London Mathematical Society 113 (2026), no. 5, Article e70550. DOI: 10.1112/jlms.70550, https://doi.org/10.1112/jlms.70550. Preprint: arXiv:2502.18602, https://arxiv.org/abs/2502.18602.
9. Michael Vogel, Jagna Wiśniewska, 'b-Contact structures on tentacular hyperboloids', Journal of Geometry and Physics 191 (2023), Article 104867. DOI: 10.1016/j.geomphys.2023.104867, https://doi.org/10.1016/j.geomphys.2023.104867. Preprint: arXiv:2303.01164, https://arxiv.org/abs/2303.01164.
10. Jagna Wiśniewska, 'Two-boost problem for the Newtonian potential at the infinity', preprint (2025). arXiv:2512.13373, https://arxiv.org/abs/2512.13373.
11. Kerr Maxwell, 'Non-Symplectic Deformations of Geometric Quantisation', preprint (2025). arXiv:2511.18549, https://arxiv.org/abs/2511.18549.