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Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $0$. Consider the minimal resolution of $r:\widetilde{V}\rightarrow V$ of this singularity. Then $\Sigma:=r^{-1}(0)$ is a collection of (-2)-rational curves $\{S_{i}\}$ that intersect each other transversely in a pattern of the corresponding Dynkin diagram. Let $D_{i}$ be the disk bundle of the normal bundle of $S_{i}$ in $\widetilde{V}$. By plumbing $\{D_{i}\}$ together, one gets a smooth manifold $N$ (with corner). (Plumbing means that we glue $\{D_{i}\}$ together following the Dynkin diagram in a way that switches the fiber and base direction.)

The following results seem to be well-known:

(1) There exists a $\textbf{homeomorphism}$ $f:\widetilde{V}\rightarrow N$ that sends $\Sigma$ to the union of the 0-sections in $D_{i}$.

(2) Consider the circle action on $\mathbb{C}^{2}$ by complex scalar multiplications. It induces a circle action on $\widetilde{V}$ and hence a circle action on $\Sigma$. This further induces a circle action on $N$. Then $f$ can be chosen to respect this circle action.

$\textbf{Question}$: I can not find a proof of these facts. Any reference will be very helpful.

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    $\begingroup$ Did you look at Lamotke's book on regular solids and isolated singularities? He's discussing plumbing in some details, for example I think (1) follows from theorem IV.10.3. $\endgroup$
    – Nicolas Hemelsoet
    Commented Jan 20, 2022 at 7:41
  • $\begingroup$ Thanks! Theorem IV.10.3 in Lamotke's book states that a "suitable neighborhood" of the exceptional set can be obtained from plumbing. But what I need is a little stronger I want a specific neighborhood coming from the quotient of the unit ball. $\endgroup$
    – user44651
    Commented Jan 20, 2022 at 8:03
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    $\begingroup$ but the exceptional set is compact no? So by Lesbegue covering lemma or that kind of thing, you can shrink a neighborhood of the exceptional set to the preimage of a small neighborhood zero which is certainly contained in a neighborhood of the unit ball quotiented by $G$. Or do I misunderstand? $\endgroup$
    – Nicolas Hemelsoet
    Commented Jan 20, 2022 at 8:10
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    $\begingroup$ Sure. I can shrink a plumbing neighborhood so that it is contained in a quotient of a ball. I can also pick a ball with small radius so that the quotient is contained in a plumbing neighborhood. But how do I show that the quotient of the ball itself is homeomorphic to a plumbing neighborhood? $\endgroup$
    – user44651
    Commented Jan 20, 2022 at 8:18

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