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Math-2456 · mathematics · Topology

Arnold Fixed Point Bound by Cup Length

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does every Hamiltonian diffeomorphism of a closed connected symplectic manifold $M$ have at least $\operatorname{cl}(M)$ fixed points, where $\operatorname{cl}(M)$ denotes its cup length?

Formulation references: [S1], [S2].

Short English statement

Does every Hamiltonian diffeomorphism of a closed connected symplectic manifold $M$ have at least $\operatorname{cl}(M)$ fixed points, where $\operatorname{cl}(M)$ denotes its cup length?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2456-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2456

Degenerate Arnold fixed-point obstruction: The positive-degree-only convention would give three instead; this ambiguity prevents a full disproof label.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Arnold Fixed Point Bound by Cup Length

OpenAI's claim

Degenerate Arnold fixed-point obstruction: A Hamiltonian map of the complex quadric threefold is claimed to have exactly three fixed points although its critical number and rational cup-length lower bound are four. At least one fixed point is degenerate.

Mathematical statement of the claimed result

Mathematical claim 1.
For the complex quadric $Q^3=\{[z]\in\mathbb{CP}^4:\sum_{i=0}^4z_i^2=0\}$ with restricted Fubini--Study symplectic form, the claim constructs a smooth Hamiltonian diffeomorphism $\phi$ with \[\#\operatorname{Fix}(\phi)=3<4=\operatorname{Crit}(Q^3)=\operatorname{cuplength}(Q^3;\mathbb Q).\] At least one fixed point is degenerate. Here $\operatorname{Crit}$ minimizes over all smooth functions, including degenerate ones; the cup-length convention is the longest nonvanishing product of positive-degree classes, with one added to that count.

Outline of the argument

Degenerate Arnold fixed-point obstruction: A finite-order Hamiltonian map is perturbed by the flow of an invariant extension of a function on its fixed locus. A power identity excludes new fixed points elsewhere, and symmetric criticality identifies the remaining fixed points with critical points on that locus. A suitably chosen function on the quadric surface then supplies the three-point count.

Formal verification

Family 347 describes Lean coverage for the degenerate quadric critical-number obstruction; the nondegenerate ordinary and stable-Morse constructions are separate.

Degenerate Arnold fixed-point obstruction: The manuscript links ArnoldCounterexample.lean (declaration OAI.ArnoldCounterexample.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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