I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 and page 116)
Now, we consider compact 4-manifolds, and they have handle decomposition. 0-handle and $m$ 1-handles form $X_{1}$, which is diffeomorphic to $\natural m S^1 \times D^3$. Let $X_{2}$ be $X_{1}\cup $ 2-handles. I understand those facts, but I can't understand $\partial X_{2}=\partial(\natural m S^1 \times D^3).$ If you can help me, then please teach me the reason for this.