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?