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It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the boundary of a 4-dimensional closed, orientable manifold $M^3 = \partial W^4.$

Suppose that we are given an Heegaard diagram for $M$ or equivalently, a 3D Kirby diagram representing the handle decomposition of $M$, can we produce from this the Kirby diagram of $W^4$?

I tried to work out this for $M =\mathbb{T}^3$.

Can you explain to me how to get a Kirby diagram of $W$ in this case?

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    $\begingroup$ Have you read Lickorish's proof? The basic idea is you can exchange Dehn twists for little surguries along curves, until you've changed the attaching map to one that describes $S^3$, then you cap off with a $D^4$. $\endgroup$ – Ryan Budney Jun 15 '19 at 16:51
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(This answer is very similar to Ryan Budney's comment.)

Lickorish (https://www.jstor.org/stable/1970373?seq=1#page_scan_tab_contents) gives an algorithm for converting a Heegaard diagram into a Dehn surgery presentation of a 3-manifold (with integer surgery coefficients). This Dehn surgery diagram can then be reinterpreted as describing attaching maps for gluing 4-dimensional 2-handles to the 4-ball. The boundary of this 4-manifold is the closed 3-manifold you started with.

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