As mentioned in the beginning of this section, we identify a tubular neighborhood of $P$ in $Q$ with a neighborhood of the zero section in its normal bundle $N=N_{0}\oplus\mathbb{R}$. For simplicity we may assume this neighborhood is given by $\mathcal{B}_{\epsilon}$, the 2-ball bundle over $P$ consisting of the set of all elements in $N_{0}\oplus\mathbb{R}$ with norm smaller than or equal to $\epsilon$, and we denote by $\mathcal{S}_{\epsilon}$ the boundary of $\mathcal{B}_{\epsilon}$.

Fix $z_{0}>0$, then the composition of the natural maps

(4.22)

$$D\simeq D\times\{z_{0}\}\hookrightarrow Q\setminus P\hookrightarrow Q\rightarrow D$$

is the identity map, which implies that for all $k$, the map $H_{k}(Q\setminus P;\mathbb{Z})\rightarrow H_{k}(Q;\mathbb{Z})$ is surjective and we have a natural splitting

(4.23)

$$H_{2}(Q\setminus P;\mathbb{Z})=H_{2}(D;\mathbb{Z})\oplus K$$

for some $K$. By assumption for $z>0$,

(4.24)

$$[\partial_{z}\tilde{\omega}(z)]=[\partial_{z}\psi(z)]=k_{+}[\omega_{D}]=2\pi k_{+}c_{1}(L),$$

so $\frac{1}{2\pi}[\Upsilon]|_{D\times\{z_{0}\}}=k_{+}c_{1}(L)$ is integral. Hence it suffices to show the integral of $\frac{1}{2\pi}\Upsilon$ over any element in $K$ is also an integer.

By the Mayer-Vietoris sequence applied to $Q=(Q\setminus P)\cup\mathcal{B}_{\epsilon}$, we get

(4.25)

$$0\rightarrow H_{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(Q\setminus P;\mathbb{Z})\oplus H_{2}(\mathcal{B}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(Q;\mathbb{Z})\simeq H_{2}(D;\mathbb{Z})\rightarrow 0.$$

So we obtain the exact sequence

(4.26)

$$0\rightarrow K\rightarrow H_{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\rightarrow H_{2}(\mathcal{B}_{\epsilon};\mathbb{Z})\simeq H_{2}(P;\mathbb{Z}).$$

On the other hand, by the Gysin sequence applied to the 2-sphere bundle $p:\mathcal{S}_{\epsilon}\rightarrow P$ we get

(4.27)

$$0\rightarrow H^{2}(P;\mathbb{Z})\xrightarrow{p^{*}}H^{2}(\mathcal{S}_{\epsilon};\mathbb{Z})\xrightarrow{\int}H^{0}(P;\mathbb{Z})\xrightarrow{\wedge e}H^{3}(P;\mathbb{Z})\rightarrow\cdots$$

where $\int$ denotes integration over the 2-sphere fibers, and $\wedge e$ denotes the wedge product with Euler class of $\mathcal{S}_{\epsilon}$. Since the Euler class $e$ of $N_{0}\oplus\mathbb{R}$ vanishes, the above becomes

(4.28)

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

(4.26) and (4.28) together imply that modulo torsion, $K$ is generated by the homology class of a 2-sphere fiber of $p$. So we just need to show $\int\frac{1}{2\pi}[\Upsilon]|_{\mathcal{S}_{\epsilon}}$ is an integer.

By the expansion of $\psi$ and $h$ in Proposition 3.24 and Proposition 3.28, it is easy to check that by restricting to the fiber of $N$ over $p$, we have

(4.29)

$$\Upsilon|_{N(p)}=-\frac{\sqrt{-1}}{4r^{3}}(zdyd\bar{y}+(yd\bar{y}-\bar{y}dy)dz)+O(1).$$

Further restricting to the $2$-sphere with radius $\epsilon$, we get

(4.30)

$$\Upsilon|_{\mathcal{S}_{\epsilon}(p)}=-\frac{1}{2\epsilon^{2}}\dvol_{S^{2}_{\epsilon}}+O(1),$$

where $\dvol_{S^{2}_{\epsilon}}$ is the area form of the standard $\epsilon$-sphere in $\mathbb{R}^{3}$. Taking the integral and let $\epsilon\rightarrow 0$ gives that

(4.31)

$$\int_{\mathcal{S}_{\epsilon}(p)}\Upsilon=-2\pi.$$

∎