Notice a principal $S^{1}$ bundle is topologically determined by its first Chern class. It suffices to compare the first Chern classes of ${\mathcal{M}^{*}}$ and $\mathbb{L}$ over the sphere bundle $\mathcal{S}_{\epsilon}$ for a small $\epsilon$. As in the proof of Lemma 4.2 th Gysin sequence gives

(4.45)

$$0\rightarrow H^{2}(P;\mathbb{Z})\xrightarrow{p^{*}}H^{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\xrightarrow{\int}H^{0}(P;\mathbb{Z})\simeq\mathbb{Z}\rightarrow 0.$$

From the proof of Lemma 4.2 we know

(4.46)

$$\int c_{1}({\mathcal{M}^{*}})=\int_{\mathcal{S}_{\epsilon}(p)}\frac{1}{2\pi}\Upsilon=-1.$$

Also by (2.49) we have

(4.47)

$$\int c_{1}(\mathbb{L})=\int_{S^{2}\subset\mathbb{R}^{3}}\frac{1}{2\pi}\Upsilon_{0}=-1.$$

So

(4.48)

$${\mathcal{M}^{*}}=\mathbb{L}\otimes p^{*}L^{\prime}$$

for some $U(1)$ bundle $L^{\prime}$ over $P$. Now we restrict both ${\mathcal{M}^{*}}$ and $\mathbb{L}$ to the subset $H_{0}\subset\mathcal{U}$ where $y=0$ and $z=z_{0}$ for a fixed $z_{0}<0$. We can identify $H_{0}$ with $H$ by the projection map. Now we claim both restrictions have first Chern class equal to $k_{-}c_{1}(\mathbb{L}_{2})$. For ${\mathcal{M}^{*}}$ this follows from construction and for $\mathbb{L}$ we notice that $z=z_{0}<0$ implies that $s_{2}\neq 0$ and $s_{1}=0$, so the projection map $(s_{1},s_{2})\mapsto|2z_{0}|^{1/2}\cdot s_{2}$ gives an isomorphism between the restriction of $\mathbb{L}$ and the unit circle bundle in $\mathbb{L}_{2}$. This also explains the choice of the weight of the $S^{1}$ action in (4.42).

Now it follows from the claim that $L^{\prime}$ is indeed a trivial principal $S^{1}$ bundle, and this finishes the proof. ∎