Flux group - Knowlpedia

Definition

For a closed connected symplectic manifold $(M,\omega)$ , the flux group is

\Gamma_\omega =\operatorname{Flux}\!\left( \pi_1(\operatorname{Symp}_0(M,\omega),\operatorname{id}_M) \right) \subseteq H^1_{\mathrm{dR}}(M;\mathbb R).

In words, it is the subgroup of the first de Rham cohomology group consisting of the flux classes of loops of symplectomorphisms based at the identity. It is the ambiguity encountered when one tries to assign flux to the endpoint of an isotopy: two paths from the identity to the same symplectomorphism differ by a loop, so their fluxes differ by an element of $\Gamma_\omega$ .

Therefore flux descends to

\overline{\operatorname{Flux}}: \operatorname{Symp}_0(M,\omega) \longrightarrow H^1_{\mathrm{dR}}(M;\mathbb R)/\Gamma_\omega.

Its kernel is the group of Hamiltonian diffeomorphisms, giving the exact sequence

1\longrightarrow\operatorname{Ham}(M,\omega) \longrightarrow\operatorname{Symp}_0(M,\omega) \xrightarrow{\ \overline{\operatorname{Flux}}\ } H^1_{\mathrm{dR}}(M;\mathbb R)/\Gamma_\omega \longrightarrow 0.

Discreteness

For closed symplectic manifolds, $\Gamma_\omega$ is discrete. This is the $C^1$ -flux conjecture proved by Ono. As a consequence, $\operatorname{Ham}(M,\omega)$ is $C^1$ -closed in $\operatorname{Symp}_0(M,\omega)$ .

If $H^1_{\mathrm{dR}}(M)=0$ , then $\Gamma_\omega=0$ and every symplectic isotopy beginning at the identity is Hamiltonian up to its path class. On a torus, nontrivial translation loops and isotopies illustrate why the flux group and the quotient cannot generally be omitted.

Scope

The definition and exact sequence above use a closed manifold. Compactly supported versions for open manifolds replace the groups and cohomology by support-sensitive variants; their flux group is a different object and should be named with its support convention.

References

Kaoru Ono, “Floer–Novikov Cohomology and the Flux Conjecture,” Geometric and Functional Analysis 16 (2006), 981–1020. DOI record.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §10.2, the flux group and exact sequence.