Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic.
For domains with boundary, it's easy to construct examples where this statement is no longer true. For instance, consider a topological disk constructed as a CW complex with one cell in each dimension, i.e., take a 0-cell $x$, glue both endpoints of a 1-cell $\gamma$ to $x$, and finally glue in a 2-cell $A$ homeomorphically. From here it's easy to check that duality fails:
$H_2 = \mathrm{ker}\ \partial_2/\mathrm{im}\ \partial_3 = \mathrm{ker}\ \partial_2 = \emptyset,$
since every 2-cell (namely: $A$) has boundary; but we also have
$ H^0 = \mathrm{ker}\ \delta_0/\mathrm{im}\ \delta_{-1} = \mathrm{ker}\ \delta_0 = nx,\ n \in \mathbb{Z}$
since the 0-cell $x$ has empty coboundary, i.e., there is no 1-cell with boundary $x$ (consider that $\partial_1 \gamma = x - x = 0$). Hence, $H_2 \ne H^0$.
However, this example does not provide much insight -- we see that duality fails, but why? In other words, what is the essential difference between what is captured by homology and cohomology when the domain has boundary? Do homology and cohomology together capture everything about the topology of a manifold with boundary? What information can we read off from the homology versus the cohomology independently? (Maybe there's a nice categorical way of thinking about these questions?)
Thanks!