Let $Y$ be a manifold of dimension $2n-1$. Let $\Sigma\subseteq Y$ be an oriented submanifold of codimension three and let $Y^{\prime}=Y-\Sigma$. Let $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}$ be a principal $S^{1}$-bundle over $Y^{\prime}$ with Chern class $c_{1}=\pm 1$. For each triple $(Y,\Sigma,\pi^{\prime})$ there is a unique compactification $X=X^{\prime}\cup\Sigma$ extending the topology of $X^{\prime}$, making $X$ into a manifold and such that

$$\begin{array}[]{ccc}X^{\prime}&\hookrightarrow&X\\ \downarrow&&\downarrow\\ Y^{\prime}&\hookrightarrow&Y\end{array}$$

commutes, with $\pi:X\rightarrow Y$ proper and $\pi|_{\Sigma}:\Sigma\rightarrow\Sigma$ the identity.