For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such that $X$ is transverse to $\partial M$, outward pointing along $V_+$ and inward pointing along $V_-$. In particular this means that there cannot be critical points of $f$ in a small neighborhood of the boundary. The resulting homology is usually noted $H(M,V_+)$, I assume this is to suggest an isomorphism with relative singular homology but I have never seen a proof of this fact, how does one prove this statement?
Also, how would one define the "absolute" homology of $M$, meaning not relative to a subset of the components of the boundary?