Let $M$ be a connected topological $n$-manifold (not assumed to be compact or boundaryless) and let $D$ an embedded closed $n$-disc. In this situation, there is an inclusion map $S^{n-1} = \partial D \hookrightarrow M\setminus D^{\circ}$.

For which $M$ is the inclusion nullhomotopic?

One obvious example of such a manifold is $S^n$. Are there others?

If the inclusion is nullhomotopic, then $M$ is homotopy equivalent to $M\setminus D^{\circ}$ with an $n$-cell attached by a constant map, i.e. $M$ is homotopy equivalent to $(M\setminus D^{\circ})\vee S^n$. Therefore $H^n(M) \cong H^n((M\setminus D^{\circ})\vee S^n) = H^n(M\setminus D^{\circ})\oplus H^n(S^n) = \mathbb{Z}$ so $M$ is necessarily closed and orientable.

There is also an isomorphism of rings $H^*(M) \cong H^*(M\setminus D^{\circ})\oplus H^*(S^n)$ from which it follows that $H^i(M)$ has no free part for $0 < i < n$ (otherwise Poincaré duality would not hold). By using $\mathbb{Z}_m$ coefficients, the same argument shows that $H^i(M)$ also has no $\mathbb{Z}_m$ part for $0 < i < n$. Therefore $M$ is an integral homology sphere.

Are there any (non-trivial) integral homology spheres with the desired property?