# Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$.

Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ copies of the 2-handle $B^2\times B^2$ to $B^4$ along the identifications of components of regular neighborhood and $\partial B^2 \times B^2$, we get a compact connected 4-manifold denoted by$W_L$.

On the other hand, we can glue $m$ copies of $B^2\times \partial B^2$ to $S^3 \setminus Int(U)$ along the boundary. Let us call the resulting 3-manifold $M$.

Question;

I think $M$ is equal (or homeomorphic) to $\partial W_L$. How can I prove it? Could you show me a proof or give me some references.

Also is there any way to visualize this?

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I think the main reason why it's not immediately clear to you is you're not being very explicit about what it is you're doing. Instead of saying "glue" and such, use actual operations on spaces. Things like disjoint unions, and quotient spaces. If you write out what you're doing you can analytically (rather quickly) determine the answer to your question. – Ryan Budney Feb 28 '12 at 12:15
@Bakhoda I see that you have created surgery-theory tag. It would be nice if you could also write a short tag-excerpt and tag-wiki with an explanation what type of questions is the tag intended for. You can have a look ta other tag wikis or on the guidelines here. – Martin Sleziak Jul 16 '13 at 11:56
@Martin Sleziak Thanks for the reminder. – Sepideh Bakhoda Jul 16 '13 at 19:46