1 · The essentials
These manuscripts examine how many points must stay in place after a particular kind of motion on a geometric space. The proposed rules compare that count with the number of places where a smooth function on the same space must stop rising or falling. The main manuscript shows examples in 22 dimensions that fall below a proposed benchmark called the stable Morse number. The gap can grow without limit, even while the dimension stays fixed. In those examples, every fixed point satisfies a regularity condition that rules out degeneracy. A separate example on the complex quadric threefold has exactly three fixed points, although every smooth function on that space must have at least four critical points. That separate example has a Lean formalization, but it includes at least one degenerate fixed point.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine comparing the places that remain unmoved during a controlled rearrangement of a landscape with the places where its slope vanishes. These manuscripts show that proposed rules forcing the first count to reach the second count—or a related benchmark—can fail. The image is only a guide: the actual motions and spaces satisfy specific geometric conditions.
Why it matters
The results distinguish several proposed fixed-point bounds rather than treating them as interchangeable. In particular, the manuscripts show that excluding degenerate fixed points does not rescue the stable-Morse bound. A companion example also attains the cyclic integral Floer bound while falling below the stable Morse number, separating those two benchmarks.
2 · Going further
The central problem is whether the number of fixed points of a Hamiltonian diffeomorphism must be at least the stable Morse number of its manifold. Manuscript 1 shows that, for every integer , there is a simply connected closed Kähler manifold of real dimension with stable Morse number and a Hamiltonian diffeomorphism with exactly fixed points. All fixed points are nondegenerate, and their orbit loops are contractible. The deficit is , hence unbounded in this fixed dimension; the abstract also states that the fixed-point count is at most of the stable Morse number.
The supplied abstracts give construction details chiefly for Manuscript 3, which shows a deficit of sixteen. It combines a Hamiltonian involution with integral homology torsion at two different primes, using finite-dimensional Morse theory and complex blowups. The abstract of Manuscript 1 states the dimension-22 family but does not describe its argument further. Manuscript 2 supplies a different comparison: on a simply connected closed Kähler manifold of real dimension and minimal Chern number one, the fixed-point count is , while the stable Morse number is . Its nondegenerate fixed points have contractible orbit loops, and its count attains the cyclic integral Floer bound despite the deficit of .
The remaining manuscripts address other versions of the proposed bounds. Manuscript 4 shows that a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold can have a time-one map with fewer fixed points than the ordinary Morse number, with arbitrarily large deficits among twelve-dimensional examples; every fixed point is nondegenerate and has a contractible trajectory. Manuscript 5 gives a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one degenerate, while both the critical number and the unit-inclusive rational cup length are four. This is the Lean-formalized result identified in the source, and it addresses the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.
Useful notions
- Fixed points of a map
- Critical points of smooth functions
- Hamiltonian motions and their time-one maps
- Nondegenerate versus degenerate fixed points
- Contractible loops
What was known before / what is new
Before
The source presents stable-Morse, ordinary Morse-number, critical-number, and rational cup-length lower bounds as proposed constraints on Hamiltonian fixed-point counts. It also identifies a cyclic integral Floer bound for comparison. It supplies no further historical account of these proposals.
New
The main manuscript shows an unbounded deficit below the stable Morse number within fixed real dimension 22, with all fixed points nondegenerate and all orbit loops contractible. Companion manuscripts show a sharp cyclic integral Floer count below the stable Morse number and arbitrarily large ordinary Morse-number deficits in dimension twelve. The Lean-formalized quadric-threefold example gives three fixed points against critical-number and unit-inclusive rational cup-length benchmarks of four.
Limits and assumptions
The dimension-22 result concerns a constructed family of simply connected closed Kähler manifolds, not every manifold of that dimension. The three-fixed-point quadric example has at least one degenerate fixed point, so it does not by itself address bounds restricted to nondegenerate fixed points. The supplied verification label attaches to Manuscript 5; it does not establish Lean coverage of the dimension-22 family or the other companion results, which are reported here as shown in their manuscripts. The abstracts give only limited information about the arguments and identify no specific remaining open problem.
Mini glossary
- Fixed point —
- A point that a map sends back to itself.
- Critical point —
- A point where the derivative of a smooth function vanishes.
- Contractible orbit loop —
- A closed trajectory that can be continuously shrunk to a point within the manifold.
- Stable Morse number —
- The Morse-theoretic benchmark compared with Hamiltonian fixed-point counts in the first three manuscripts.
- Critical number —
- The minimum number of critical points required of a smooth function on the manifold.
Keywords : Hamiltonian fixed points · Stable Morse number · Arnold bounds
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two
05/10/2026We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number and a Hamiltonian diffeomorphism with exactly fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.
Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number
05/10/2026We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is . Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.
Hamiltonian Fixed Points Below the Stable Morse Number
05/10/2026We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
A nondegenerate counterexample to the Morse-number Arnold bound
23/09/2026We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.
Three fixed points on the symplectic quadric threefold
23/09/2026We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.