(This question is common to Steven Karp and Thomas Lam) We must use the following fact in our document:

Theorem 1.Let $ M ^ n $ to be a compact contractable topological variety with a limit, such as the limit $ partial M $ is homeomorphic to a sphere $ S ^ {n-1} $. then $ M $ is homeomorphic to a closed balloon.

**Q1.** Is there a reference where this is indicated with proof? A reference we found is the theorem (slightly stronger) 10.3.3 (ii) in the Davis book. He says this for all $ n $but only gives a sketch of proof for $ n geq $ 6.

**Q2.** What is the easiest way to prove Theorem 1? Here is an argument that we collected from various MO messages: The limit of $ M $ is collared by Brown's theorem. So, we can stick a $ n $-ball to $ M $and by van Kampen and Mayer – Vietoris, it follows that the resulting space is a merely connected sphere of homology. So it's a sphere of PoincarĂ©'s conjecture, so $ M $ is a ball closed by Brown's Schoenflies theorem.

Note that we do not need the interior of $ M $ to be an open ball (and we rather demand that the limit be a sphere), which is why this question does not duplicate this question.