Along an edge $\mathfrak{\Delta}\subset\mathfrak{D}$, the singular fibres have the topology of $T^{3}$ with $T^{2}$ collapsed to $S^{1}$, alternatively written as $I_{1}\times S^{1}$, where $I_{1}$ refers to the nodal elliptic curve or equivalently $S^{2}$ with two points identified. Notice the singularity on the fibre is not isolated. These singular fibres have Betti numbers $(b_{1},b_{2})=(2,2)$ and Euler characteristic 0. The $T^{3}$-fibration is locally described as the Kodaira type $I_{1}$ degenerating family of elliptic curves over a disc $D^{2}\subset\mathbb{C}$, Cartesian product with the trivial $S^{1}$-bundle $S^{1}\times\mathbb{R}\to\mathbb{R}$. The monodromy around the edge $\mathfrak{\Delta}\subset\mathfrak{D}$ acting on $H_{1}(T^{3})\simeq\mathbb{Z}^{3}$ can be written in a suitable basis as

$$\begin{bmatrix}1&1&0\\ 0&1&0\\ 0&0&1\end{bmatrix}.$$

For an alternative viewpoint which ties in better with the Gibbons-Hawking construction (see later Sections 1.2), the base is $B\subset\mathbb{R}^{3}$, and the total space $M$ is a singular $S^{1}$-bundle over $S^{1}\times B\times S^{1}$, where the $S^{1}$-fibres collapse to points along the codimension 3 locus $\{0\}\times\mathfrak{\Delta}\times S^{1}\subset S^{1}\times B\times S^{1}$. In the 3 transverse directions, the singular $S^{1}$-bundle structure is topologically modelled on the Hopf map

(1.1)

$$\pi_{\mathbb{C}^{2}}:\mathbb{C}^{2}\to\mathbb{R}\times\mathbb{C},\quad(z_{0},z_{1})\mapsto\left(\frac{1}{2}(|z_{1}|^{2}-|z_{0}|^{2}),z_{0}z_{1}\right).$$

The Chern class $c_{1}\in H^{2}(S^{1}\times B\times S^{1}\setminus(\{0\}\times\mathfrak{\Delta}\times S^{1}),\mathbb{Z})$ evaluates to 1 on a suitably oriented $S^{2}$-cycle linking $\{0\}\times\mathfrak{\Delta}\times S^{1}$ inside $S^{1}\times B\times S^{1}$.