From a Heegaard diagram $(\Sigma, \alpha,\beta)$, the usual way to construct a 3-manifold is to look at $\Sigma \times [0,1]$ and attach 3D 2-handles to each of the beta curves in $\beta \times \{1\}$ and 3D 2-handles to the alpha curves in $\alpha \times \{0\}$. I don't understand what the thickening up of $\Sigma$ is doing? Why can't we just attach the handles to $\Sigma$?
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