Recall that the fibration produced in Lemma 7.6 is a perturbation of the one in Example 5.8, whose total space is an open set of $\mathbb{C}^{3}$ with standard symplectic form, which is exact. One can see that the successive perturbations of this fibration have not modified the cohomology class of $\omega$. ∎

We apply a similar argument to the one used in the case of the positive fibre (see Proposition 4.11). Clearly, since $\gamma_{1}$ is represented by the orbits of the $S^{1}$ action

$$-\int_{\gamma_{1}(b)}\eta=b_{1},$$

which is continuous. We now prove that, for $j=2,3$

$$\alpha_{j}(b)=-\int_{\gamma_{j}(b)}\eta$$

(77)

extends continuously to points in $\Gamma_{d}$ or in $\Gamma_{e}$. As we did in Proposition 4.11, we can think of $\alpha_{j}(b)$ as

$$\alpha_{j}(b)=\int_{S}\omega,$$

where $S$ is a surface spanned by the cycles $\gamma_{j}(b^{\prime})$ as $b^{\prime}$ moves along a curve joining $\bar{b}$ and $b$. Suppose $b\in\Gamma_{e}$ (or $\Gamma_{d}$), then we need to show that $\alpha_{j}(b)$ is independent of the curve from $\bar{b}$ to $b$, or equivalently that

$$\int_{S_{1}-S_{2}}\omega=0,$$

where $S_{1}$ and $S_{2}$ are the surfaces corresponding to two different paths from $\bar{b}$ to $b$. The boundary $\partial(S_{1}-S_{2})$ is determined by monodromy. It is easy to see that $\partial(S_{1}-S_{2})$ is a multiple of $\gamma_{1}(b)$, therefore for some integer $k$ we have

$$\int_{S_{1}-S_{2}}\omega=-\int_{\partial(S_{1}-S_{2})}\eta=k\int_{\gamma_{1}(b)}\eta=0,$$

where the last equality follows from the fact that $b\in\Gamma_{d}$ or $\Gamma_{e}$. To show that $\alpha$ extends continuously also to points of $\Delta$ we can argue that (77) makes sense also over singular fibres, since both $\eta$ and $\gamma_{j}(b)$ are well defined when $b\in\Delta$. ∎

Since $\alpha_{1}(b)=b_{1}$, it is enough to show that, if for fixed $t\in\mathbb{R}$ we let $U_{t}=\{b_{1}=t\}$, then $\alpha_{t}=\alpha|_{U_{t}}$ is a bijection onto its image. If $\lambda_{2}$ and $\lambda_{3}$ are the periods of the fibration corresponding to $\gamma_{2}$ and $\gamma_{3}$, then $\alpha_{t}$ is computed by taking primitives of $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$. If we let $X_{t}$ denote the symplectic reduction of $X$ at $t$ and $G_{t}:X_{t}\rightarrow\mathbb{R}^{2}$ the reduced fibration, then it is not difficult to see that $\lambda_{2}|_{U_{t}}$ and $\lambda_{3}|_{U_{t}}$ are in fact periods of $G_{t}$ (cf. [3]Lemma 5.9). Now the conclusion follows by simply observing that $G_{t}$ is a proper Lagrangian submersion, i.e. an integrable system. The argument works also when $t=0$.

An explicit computation of the periods was done in [3]Proposition 5.10 for the fibration in Example 5.8. There we found that

	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle\beta_{1}\,db_{1}-e^{2b_{2}}db_{2},$
	$\displaystyle\lambda_{3}$	$\displaystyle=$	$\displaystyle\beta_{2}\,db_{1}-e^{2b_{3}}db_{3},$		(78)

where $\beta_{1}$ and $\beta_{2}$ are functions depending only on $b_{1}$. The periods of the perturbed fibration obtained in Lemma 7.6 will have this same expression away from where the perturbation took place (i.e. away from the white region in Figure 15), for example in a neighborhood of the codimension 1 part of $\Delta$. It is easy to see from this expression of the periods that $\alpha$ extends continuously to $\Delta$ and that it is a bijection. ∎

The fibrations constructed in Theorem 6.19 have smooth Lagrangian sections and the action coordinates extend continuously to the whole base. Since $\mathcal{F}|_{U}$ also has a Lagrangian section (cf. Remarks 7.7) and the action coordinates extend continuously to the whole base, the statement easily follows from the results on stitched fibrations such as the existence of a normal form. The latter is found extending the maps $f^{+}$ and $f^{-}$ beyond all connected components of the seam and then using the Lagrangian section to normalize with the period map.

∎