I am trying to understand (and prove) why the following should be true: let $X$ be an $n$-dimensional closed manifold homotopy equivalent to $S^n$. If $D^n \subseteq X$ is an embedded disc, I would like to see why $X- \mathring{D^n}$ is homotopy equivalent to $S^n- \mathring{D^n}$.
Context. This step is needed in the proof of the generalised Poincaré conjecture using Smale's h-cobordism theorem. Removing two open discs yields an homotopy equivalence with $S^{n-1} \times I$ and one can apply the h-cobordism theorem. In the question I only removed one disc for simplicity.