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?

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.

Let me make sure that I understand this right: consider the smooth algebraic curve $C: y^{3}+z^{6m+1}=\epsilon$. After taking double cover of the unit ball $B$ (inside $C^{2}$) branched over $B\cap C$, we get a Stein domain $W$ bounded by $\Sigma(2,3,6m+1)$. The homology of $W$ can be computed (for example, as in L. Kauffman, Open books, branched covers and knot periodicity) and one sees that intersection form of $W$ coincides with the Seifert form of $B\cap C$ (this surface can actually be pushed to a Seifert surface in $S^{3}$), which is not definite. Is this argument right?

When we consider the cobordism induced map $F$, don't we need to restrict to the canonical spin-c structure for the symplectic manifold $W$?