Which topological counts constrain Hamiltonian fixed points?

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Family 347 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

A Hamiltonian system evolves positions and momenta according to an energy function. For a sufficiently small time-independent Hamiltonian on a compact symplectic manifold, points returning after one unit of time are exactly its critical points. Topology forces a minimum number of critical points, so it is natural to ask whether arbitrary Hamiltonian motions must retain the same minimum number of fixed points.

This idea leads to several versions of the Arnold fixed-point conjecture, with different hypotheses and different topological counts. They are not interchangeable. Counting homology generators is weaker than minimizing the critical points of actual smooth functions, and allowing degenerate fixed points changes the problem again.

Family 347 contains five manuscripts claiming counterexamples to several stronger formulations. Three concern the stable Morse number with all fixed points nondegenerate. One concerns the ordinary Morse number in dimension twelve. The last constructs a smooth map with only three fixed points on a six-dimensional quadric, where the unrestricted critical-number benchmark is four.

The family’s message is a separation of precise bounds, not a claim that all established Arnold-type lower bounds fail.

2. Background and essential definitions

Hamiltonian fixed points

A symplectic manifold $(M,\omega)$ has a closed nondegenerate two-form. A smooth time-dependent Hamiltonian $H_t$ determines a flow $\varphi_H^t$. A point $x$ is fixed by the time-one map if $\varphi_H^1(x)=x$; its trajectory over that time is a periodic loop.

The point is nondegenerate when the linear map $D\varphi_H^1(x)$ has no eigenvalue one. This is a condition on the return map, not a statement that its trajectory is constant. Some formulations count only fixed points whose trajectory loops can be contracted to a point. Choices of contracting disks do not create additional fixed points.

Three minima for smooth functions

A Morse function has only nondegenerate critical points: its Hessian is invertible at each critical point. Write $\operatorname{Morse}(M)$ for the least number of critical points of such a function on $M$.

The stable Morse number $\operatorname{SM}(M)$ permits auxiliary variables. One minimizes over Morse functions

$$F:M\times\mathbb R^k\longrightarrow\mathbb R$$

that agree with a nondegenerate quadratic form in the auxiliary variables outside a sufficiently large ball. The integer $k\geq0$ and the quadratic signature may vary. Thus

$$\operatorname{SM}(M)\leq\operatorname{Morse}(M).$$

This is a minimum over actual smooth functions, not merely algebraic chain complexes.

The critical number $\operatorname{Crit}(M)$ instead minimizes critical points of arbitrary smooth functions on $M$, allowing degeneracy. It is the benchmark relevant to one unrestricted formulation without a nondegeneracy assumption.

Homology counts can be smaller

Homology groups record topological cycles. Their free ranks are Betti numbers, while finite-order elements form the torsion subgroup. Let $b(M)$ be the total free rank, and let $d(T)$ be the least number of generators of a finite abelian group $T$. The degreewise quantity

$$\lambda(M)=b(M)+2\sum_i d(\operatorname{Tor}H_i(M;\mathbb Z))$$

is a lower bound for the geometric stable Morse number used in the manuscripts. In their simply connected high-dimensional examples, they invoke Smale’s handle theorem to attain it with an ordinary Morse function, so both geometric Morse minima equal $\lambda(M)$.

A cyclically graded Floer bound may combine different integer degrees. For minimal Chern number one, it retains only even and odd degrees. The minimal Chern number is the positive generator of the first Chern class evaluated on spherical homotopy classes. Grouping degrees can reduce the number of generators required by mixed-prime torsion.

3. Development history and previous partial results

Conley–Zehnder established fixed-point bounds for standard symplectic tori through variational methods. Floer then built homology from Hamiltonian trajectories and pseudoholomorphic curves, creating a systematic bridge between dynamics and topology. Later work of Hofer–Salamon, Ono, Fukaya–Ono, Liu–Tian, and Ruan extended the theory and the rational homological lower bound. Conley–Zehnder, Floer.

These results do not automatically identify fixed-point counts with the minimum for a Morse function. Damian showed that ordinary and stable Morse numbers can differ. Positive stable-Morse theorems hold under extra conditions: Dimitroglou Rizell–Golovko proved the bound for generic Hamiltonians when both symplectic area and the first Chern class vanish on spherical homotopy classes, with an independent approach due to Pöder Balkeståhl. The sphere and blowup factors in the new examples lie outside those assumptions. Dimitroglou Rizell–Golovko.

Recent refinements retain more homological information. Abouzaid–Blumberg obtained bounds over fields of positive characteristic. Bai–Xu developed an integral bound incorporating torsion at every prime, with degrees reduced modulo twice the minimal Chern number. The family explicitly compares its examples with this cyclic bound rather than treating integral torsion as an automatic stable-Morse obstruction. Bai–Xu.

For possibly degenerate maps, cup-length and critical-number bounds have important positive cases, including tori, projective spaces, and suitable symplectically aspherical manifolds. Cup length, in the convention used here, is one plus the largest number of positive-degree cohomology classes with nonzero product. The added one includes the unit. Golovko’s survey of the different formulations.

Regularity also separates problems. Earlier Hamiltonian homeomorphism counterexamples do not themselves give smooth diffeomorphisms. All five manuscripts here claim smooth constructions. M5 also cites Jiao’s September 27, 2026 preprint reporting a two-fixed-point smooth map on the equal-area product of two spheres; that distinct construction is not an input to this family. Jiao.

4. What the new manuscripts claim

Stable Morse counterexamples: M1–M3

All three examples are simply connected closed Kähler manifolds. Every fixed point is nondegenerate, every orbit loop is contractible, and the count includes all fixed points.

The source column gives the manuscript and the real dimension in parentheses.

Source	Fixed points	Stable Morse number
M1 (22)	F	S
M2 (3332)	1,872,232	1,872,264
M3 (1412)	318,952	318,968

In M1, $m$ is any positive integer, and the table’s two counts are

$$\begin{aligned} F&=80+1952m,\\ S&=80+1968m. \end{aligned}$$

The deficit is $16m$, so it becomes arbitrarily large while the dimension stays twenty-two. The fixed-point count is at most $127/128$ of the stable Morse number. The underlying manifold is allowed to vary with $m$.

M2 has minimal Chern number one and attains the cyclic integral Floer bound exactly. Its count is therefore simultaneously optimal for that cited bound and thirty-two below the stable Morse number. M3 supplies a sixteen-point deficit through the original mixed-prime construction; M1 and M2 develop that mechanism in different directions.

Ordinary Morse counterexamples: M4

M4 asserts closed connected symplectic twelve-manifolds with

$$\#\operatorname{Fix}(\varphi_H^1) =\chi(M)+128<\operatorname{Morse}(M),$$

where $\chi(M)$ is the Euler characteristic. All fixed points are nondegenerate and have contractible trajectories. The deficit can be arbitrarily large among such twelve-dimensional examples, again allowing the manifold to vary.

These examples use a nontrivial fundamental group. They separate ordinary Morse theory from a stabilized construction; they are not the simply connected stable-Morse examples in the first table.

A degenerate example: M5

The complex quadric threefold is the smooth hypersurface

$$Q^3=\left\{[z_0:\cdots:z_4]\in\mathbb{CP}^4: \sum_{j=0}^4z_j^2=0\right\}.$$

With its restricted standard symplectic form, it has real dimension six. M5 claims a smooth Hamiltonian diffeomorphism $\varphi$ satisfying

$$\#\operatorname{Fix}(\varphi)=3 <4=\operatorname{Crit}(Q^3) =\operatorname{cuplength}(Q^3;\mathbb Q).$$

At least one fixed point is degenerate. The manuscript says that three attains Gong’s existing lower bound for this quadric. It does not violate a theorem whose hypotheses require every fixed point to be nondegenerate. Gong’s version cited by M5.

5. Why the advance matters

Importance: 5/5. If correct, the family would sharply redraw the relationship between Hamiltonian fixed points and geometric critical-point minima. The stable-Morse examples are especially strong because the ambient manifolds are simply connected and Kähler, and every fixed point is nondegenerate.

M1 makes the separation robust in a numerical sense: the dimension is fixed and the deficit grows without bound, with a uniform proportional gap. M2 supplies a different kind of precision by attaining the cyclic integral homological bound exactly while missing the geometric one.

The six-dimensional quadric example addresses the degenerate problem with a very small count. Together, the papers illustrate why an “Arnold bound” must always be stated with its benchmark, coefficient convention, grading, and nondegeneracy hypotheses.

This rating measures the significance of the claims if correct. It does not independently validate any construction or compare priority beyond the sources’ descriptions.

6. The main idea

Convert a fixed submanifold into finitely many fixed points

A common device in M1–M3 and M5 starts with a Hamiltonian map $A$ of finite order, whose fixed set is a smooth submanifold $F$. Its fixed tangent vectors must be exactly the tangent vectors to $F$.

Choose a function $f$ on $F$, extend it to an $A$-invariant Hamiltonian $K$, and let $B_s$ be its flow. If $A^r$ is the identity, commutation gives

$$(A\circ B_\epsilon)^r=B_{r\epsilon}.$$

A smooth vector field on a compact manifold has no nonconstant periodic trajectories of arbitrarily short period. For sufficiently small $\epsilon$, this identity excludes extra fixed points away from the desired equilibria. Symmetric criticality identifies the remaining ones with the critical points of $f$ on $F$.

Thus the fixed-point count is controlled by a function on the fixed submanifold, which can require fewer critical points than a function on the entire ambient manifold.

Mixed primes distinguish integer degrees from parity

For finite abelian groups,

$$d\bigl((\mathbb Z/2)^a\oplus(\mathbb Z/3)^b\bigr)=\max\{a,b\}.$$

For example, $\mathbb Z/2\oplus\mathbb Z/3$ is cyclic of order six and needs only one generator. If the two summands occur in separate homological degrees, however, degreewise Morse theory must account for both separately.

M1–M3 build a simply connected Kähler manifold $X$ by blowing up projective space along carefully chosen centers carrying two-primary and three-primary torsion. A blowup replaces a submanifold by its projective normal directions and adds predictable shifted homology groups.

They multiply $X$ by a rational surface $Y$ whose Hamiltonian involution fixes four projective lines. The resulting manifold $M=X\times Y$ has four fixed components, each $X\times\mathbb{CP}^1$. The perturbation device realizes four times the Morse count of that component.

The free homology ranks of $\mathbb{CP}^1$ and $Y$, in even degrees, are respectively $(1,1)$ and $(1,6,1)$. Products combine the torsion sequences with these different patterns. Choosing where each prime dominates makes the component count smaller than the ambient degreewise count.

M1 embeds arbitrarily many disjoint centers in one fixed projective dimension, giving the dimension-twenty-two family. M2 chooses seven centers with a more elaborate degree pattern so that the component count equals the parity-grouped bound exactly. These refine M3’s mechanism; their constructions are not five copies of one identical example.

Group generators drive the twelve-dimensional construction

M4 uses the groups $G=A_5^k$, direct products of the alternating group on five letters. Their minimum number of group generators grows with $k$, while the augmentation ideal in the integral group ring has a generating list of length two. The augmentation ideal consists of formal integer combinations of group elements whose coefficients sum to zero; generating it as a module is different from generating $G$ as a group.

Gompf’s construction supplies a symplectic four-manifold $X$ with this fundamental group. Relative handle operations produce a Morse model on $X\times\mathbb R^8$ with a controlled count, exactly quadratic outside a compact region. The paper inserts its small-time dynamics into sixteen product-cap regions of $X\times(S^2)^4$, matching rotations outside the caps and excluding additional fixed points.

The resulting count is $\chi(M)+128$, while group generators force a larger ordinary Morse count. The distinction is geometric: a short algebraic complex alone would not produce the required closed Hamiltonian map.

Three points arise from a lower-dimensional fixed surface

For M5, a Hamiltonian involution of $Q^3$ has fixed set the quadric surface $Q^2$, diffeomorphic to $S^2\times S^2$. The paper supplies a function on that surface with exactly three critical points, one degenerate, and transfers them to fixed points of the ambient map.

Meanwhile, the hyperplane class $h$ of $Q^3$ satisfies $h^3\ne0$, forcing unit-inclusive rational cup length at least four. A rotation realizes four critical points on $Q^3$, giving the matching critical-number upper bound. The smaller fixed-set function is therefore the source of the discrepancy.

7. Scope, limitations, and open questions

The examples concern unrestricted strong formulations. They do not contradict stable-Morse theorems with symplectic-asphericity and Chern-class hypotheses, nor the cited nondegenerate homological inequalities. M2 is explicitly designed to attain one such integral inequality.

The degeneracy in M5 is essential to its comparison with nondegenerate bounds. Conversely, degeneracy cannot explain M1–M4 because all their fixed points are stated to be nondegenerate.

The unbounded-deficit families vary the manifold. The manuscripts do not claim an unbounded deficit on a single fixed compact manifold, a lowest possible counterexample dimension, or a complete replacement for every strong Arnold formulation.

All five constructions are reported as stated. The digest does not verify their handle realizations, invariant perturbations, torsion calculations, or exclusion of additional fixed points.

8. Sources and further reading

Golovko’s survey is useful for keeping the formulations separate. The primary sources linked above explain the homological lower bounds and special positive cases. Optional background appears under Hamiltonian vector field, Morse theory, and Floer homology.

Source manuscripts

• M1. OpenAI, Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two (October 5, 2026).
• M2. OpenAI, Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number (October 5, 2026).
• M3. OpenAI, Hamiltonian Fixed Points Below the Stable Morse Number (October 5, 2026).
• M4. OpenAI, A nondegenerate counterexample to the Morse-number Arnold bound (September 23, 2026).
• M5. OpenAI, Three fixed points on the symplectic quadric threefold (September 23, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.