Let $X$ be a compact, oriented, connected $n$-manifold without boundary. Let $D$ be a $n$-disk in some chart domain of $X$. Then $X-D\simeq X-\{p\}$ is a non-compact, oriented, connected manifold without boundary, and $X-$int$(D)$ is a compact, oriented, connected manifold with boundary $\partial X\simeq S^{n-1}$.
I read in many places that $X-D\simeq X-\{p\}$ has top cohomology group zero (here). As for $X-$int$(D)$, Bredon's Topology and Geometry's page 356 has
where $M$ is a compact, connected, oriented manifold with boundary $\partial X$. This means $\mathrm{H}^n(X-\text{int}(D))\simeq \mathrm{H}_0(X-\text{int}(D),S^{n-1})$, which is nonzero.
But since int$(D)$ can retract to a smaller closed disk $D'$, we should have $X-D$ is homotopic to $X-$int$(D)$, and their cohomology groups isomorphic. Where did I go wrong?