Let $(M,\omega)$ be a closed smooth symplectic manifold, and let $\operatorname{Symp}_0(M,\omega)$ denote the identity component of the group of symplectomorphisms of $(M,\omega)$.

Let $\operatorname{Ham}(M,\omega)\subseteq\operatorname{Symp}_0(M,\omega)$ denote the subgroup of time-one maps of smooth Hamiltonian isotopies. Here a Hamiltonian isotopy is a smooth symplectic path generated by a time-dependent vector field $X_t\in\mathfrak X(M)$ for which there is a smooth function $H:[0,1]\times M\to\mathbb R$ such that, with $H_t(x)=H(t,x)$,

\begin{align*} \iota_{X_t}\omega=dH_t\quad\text{for every }t\in[0,1]. \end{align*}

Let $\widetilde{\operatorname{Symp}}_0(M,\omega)$ be the universal cover represented by fixed-endpoint homotopy classes of smooth symplectic paths $\varphi:[0,1]\to\operatorname{Symp}_0(M,\omega)$ with $\varphi_0=\operatorname{id}_M$, with group operation represented by pointwise composition of paths. Let $\widetilde{\operatorname{Ham}}(M,\omega)$ denote the universal cover of $\operatorname{Ham}(M,\omega)$, mapped into $\widetilde{\operatorname{Symp}}_0(M,\omega)$ by inclusion.

For such a symplectic path $\varphi$, define its generating vector field $X_t\in\mathfrak X(M)$ by

\begin{align*} \frac{d}{dt}\varphi_t=X_t\circ\varphi_t. \end{align*}

Define the flux class by

\begin{align*} \operatorname{Flux}(\varphi)=\left[\int_0^1\iota_{X_t}\omega\,d\mathcal L^1(t)\right]\in H^1(M;\mathbb R), \end{align*}

where the bracket denotes the de Rham cohomology class of the resulting closed $1$-form. Then this assignment descends to a well-defined [group homomorphism](/page/Group%20Homomorphism)

\begin{align*} \operatorname{Flux}:\widetilde{\operatorname{Symp}}_0(M,\omega)\to H^1(M;\mathbb R). \end{align*}

Assume the Banyaga flux exactness theorem for closed symplectic manifolds in the following form:

\begin{align*} \operatorname{Flux}([\varphi])=0\quad\Longleftrightarrow\quad[\varphi]\text{ is represented by a Hamiltonian isotopy}. \end{align*}

Then

\begin{align*} \ker(\operatorname{Flux})=\operatorname{im}\!\left(\widetilde{\operatorname{Ham}}(M,\omega)\to\widetilde{\operatorname{Symp}}_0(M,\omega)\right), \end{align*}

equivalently the kernel is the set of fixed-endpoint homotopy classes represented by Hamiltonian isotopies.