TITLE: Top homology of a manifold with boundary
QUESTION [6 upvotes]: As stated in the title, my question is the following:
Let $M$ be a compact orientable manifold with boundary $\partial M$. Is it true that $H_n(M;\mathbb{R})$ is always zero? 
In the trivial case for compact surfaces in $\mathbb{R}^3$, filling up the interior always gives something homotopically equivalent to an object of lesser dimension. However, I don't see a way of using this method for the general case. Manipulation with Stokes theorem doesn't seem to work either.

REPLY [5 votes]: I just found out this is a simple application of Lefschetz Duality.(See Hatcher, Chap. 3 Sec.2) By this duality,  $H_n(M)=H^0(M,\partial M)=0$.