# Obtaining the bounding 4-manifold from the Heegaard diagram

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?

• 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 $D^4$. Jun 15, 2019 at 16:51