(this question is joint with Steven Karp and Thomas Lam) We need to use the following fact in our paper:

Theorem 1.Let $M^n$ be a compact contractible topological manifold with boundary, such that the boundary $\partial M$ is homeomorphic to a sphere $S^{n-1}$. Then $M$ is homeomorphic to a closed ball.

**Q1.** Is there a reference where this is stated with a proof? One reference that we found is the (slightly stronger) Theorem 10.3.3(ii) in Davis's book. He states this for all $n$, but only gives a proof sketch for $n\geq 6$.

**Q2.** What is the simplest way to prove Theorem 1? Here is an argument that we gathered from various MO posts: The boundary of $M$ is collared by Brown's theorem. Thus we can glue an $n$-ball to $M$, and by van Kampen and Mayer–Vietoris, it follows that the resulting space is a simply connected homology sphere. Thus it is a sphere by the Poincaré conjecture, therefore $M$ is a closed ball by Brown's Schoenflies theorem.

Note that we do not require the interior of $M$ to be an open ball (and instead we require that the boundary is a sphere), which is why this question is not a duplicate of this question.