Top homology of an oriented, compact, connected smooth manifold with boundary

Just to complete the details of Dan's hint. The top homology of a non-compact manifold vanishes, e.g. by Poincare duality $H_n(M) \cong H^0_c(M)$ where $H^*_c(M)$ is cohomology with compact support. But $H^0_c(M)$ is just the kernel of $d : C^0_c(M) \to C^1_c(M)$. Therefore $C^0_c(M)$ consists of constant functions and because $M$ is non-compact the only such function is the zero function.

Alternatively, one can see the result directly from Lefschetz duality, a generalization of Poincare duality. This gives $H_n(M) \cong H^0(M,N)$ where $N$ is the boundary of $M$. But $H^0(M,N) \cong \tilde H^0(M/N) = 0$.

I'm sure there are many other and probably simpler ways to see this. But the upshot is that non-compact manifolds and manifolds with boundary are simpler from the point of view of homotopy theory than their closed cousins (just thinking of ${\bf R}^n$ which has homototy type of a point).

The top dimensional cycle of an orientable triangulated manifold without boundary is an oriented sum of its n-simplices. Each n-simplex shares each of its n-1-faces with exactly one other n-simplex and these two n-1 faces cancel each other out under the boundary operator. One can show that this cycle is unique up to orientation.

With a manifold with boundary the n-1 simplices on the boundary are not paired off and can not cancel out. Thus any cycle must exclude them. But if one removes the n-simplices that contain them then none of their interior n-1 faces can be cancelled. Continue removing n-simplices until there is nothing left.

This shows that the top homology is zero.

A good reference is Spanier's Algebraic Topology

You can prove that $H_n(M, \mathfrak d M)\cong \mathbb Z$ in essentially the same way as the special case $\mathfrak d M=\emptyset$, and also that the connecting homomorphism $\mathfrak d\colon H_n(M, \mathfrak d M)\to H_{n-1}(\mathfrak d M)$ takes a relative orientation cycle to an orientation cycle of the boundary. In particular, it is injective if $\mathfrak dM\ne\emptyset$ (and an isomorphism if the boundary is connected). The result now follows from looking at the long exact sequence $$0=H_n(\mathfrak d M)\to H_n(M)\to H_n(M, \mathfrak d M)\xrightarrow{\text{inj.}} H_{n-1}(\mathfrak d M).$$