Top homology of a manifold with boundary

Solution 1:

If your manifold is triangulated then if you want to find an $n$-cycle $c$, it is a linear combination of the $n$-simplices $\sigma_i$, $c=\sum a_i \sigma_i$. As in the case of a closed manifold, $\partial c=0$ implies that $a_i$'s are all equal. But it you have a boundary, you also get that $a_i=0$ for every $\sigma_i$ which is at the boundary. (I suppose that $M$ is connected, otherwise one needs to consider every component separately)

Solution 2:

I just found out this is a simple application of Lefschetz Duality.(See Hatcher, Chap. 3 Sec.2) By this duality, $H_n(M)=H^0(M,\partial M)=0$.