Given a manifold $M$ with boundary $W = \partial M$, I know that having a handle decomposition of $M$ allows one to compute its homology, at least in nice cases, by - for example - using the Morse Homology of its critical points. Is it similarly easy to compute the homology of $W$, since the handle decomposition of $M$ provides a surgery description of $W$?

If it helps, I'm interested in a particularly simple case: $M$ is a smooth $2n$-manifold which is described by simultaneously adding some number of $n$-handles to $D^{2n}$. However, I am primarily concerned with the *integer* homology, rather than over $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Q}$.