In Section 2.2 we introduced an invariant $[\rho]\in H^{1}(B^{sm},T^{{\bf Z}}\otimes{{\bf R}})$ of a ${\bf Z}$-affine structure. Here we will give an interpretation of $[\rho]$ for integrable systems.

Let us consider $X^{\prime}=\pi^{-1}(B^{sm})$ which is a Lagrangian torus fibration over $B^{sm}$ (i.e. fibers are Lagrangian tori such that the fiber over $x\in B^{sm}$ is isomorphic up to a shift to the torus $T_{x}^{*}B^{sm}/(T_{x}^{*}B^{sm})^{{\bf Z}}\,$).

Any singular closed $1$-chain $c$ on $B^{sm}$ with values in the local system

$$(T^{\ast}_{x}B^{sm})^{{\bf Z}}\simeq H_{1}(T_{x}^{*}B^{sm}/(T_{x}^{*}B^{sm})^{{\bf Z}},{{\bf Z}})$$

gives a $2$-chain $\overline{c}$ on $X^{\prime}$ with the boundary belonging to a finite collection of fibers $\pi^{-1}(x^{(i)}),1\leq i\leq N$ of the fibration $\pi:X^{\prime}\to B^{sm}$. Moreover, for every point $x^{(i)}$ the part of $\partial\overline{c}$ over $x^{(i)}$ is homologous to zero in $\pi^{-1}(x^{(i)})$. Therefore, there exists a collection of $2$-chains $\overline{c}_{i},1\leq i\leq N$ supportred on $\pi^{-1}(x^{(i)}))$ such that the $2$-chain $\overline{c}+\sum_{1\leq i\leq N}\overline{c}_{i}$ is closed. In this way we obtain a group homomorphism $J_{s}:H_{1}(B^{sm},(T^{\ast})^{{\bf Z}})\to H_{2}(X^{\prime},{{\bf Z}})/H_{2}^{0}(X^{\prime},{{\bf Z}})$, where $H_{2}^{0}(X^{\prime},{{\bf Z}})\subset H_{2}(X^{\prime},{{\bf Z}})$ denotes the sum of images of $H_{2}(\pi^{-1}(y),{{\bf Z}})$ where $y\in B^{sm}$ (it is enough to pick one base point $y$ for any connected component of $B^{sm}$). It is easy to see that $\langle[\rho],[c]\rangle=\langle[\omega],J_{s}([c])\rangle$, where $[\omega]$ is the class of the symplectic form $\omega$.