Let $f:X\rightarrow B$ be a stitched fibration and let $\gamma^{\pm}$ be bases of $H_{1}(X,\mathbb{Z})$ satisfying the above conditions. Then the restrictions of $\gamma^{\pm}$ to $H_{1}(X^{\pm},\mathbb{Z})$ induce embeddings,

$$\Lambda^{\pm}\hookrightarrow T^{\ast}_{B^{\pm}}.$$

Let $\alpha^{\pm}:B^{\pm}\rightarrow\mathbb{R}^{n}$ be the corresponding action coordinates satisfying $\alpha^{\pm}(b)=0$ for some $b\in\Gamma$. Then the map

$$\alpha=\begin{cases}\alpha^{+}&\textrm{on}\ B^{+}\\ \alpha^{-}&\textrm{on}\ B^{-}\end{cases}$$

is an admissible change of coordinates. If $b_{1},\ldots b_{n}$ denote the action coordinates on $B$ given by $\alpha$, then $\{db_{1},\ldots db_{n}\}$ is a basis of $\Lambda^{+}$ and $\Lambda^{-}$. Furthermore, the reduced space $\bar{Z}$ can be identified with $T^{\ast}\Gamma/\penalty\langle db_{2},\ldots,db_{n}\rangle_{\mathbb{Z}}$ and the reduced fibration $\bar{f}$ can be identified with the standard projection $\bar{\pi}$. Moreover $\ell_{1}$ satisfies

$$\int_{[db_{j}]}\ell_{1}=m_{j},\quad j=2,\ldots,n$$

(49)

where $[db_{j}]\in H_{1}(\bar{Z},\mathbb{Z})$ is the class represented by $db_{j}$.