# reference query – Controllable topological topological variety with boundary = a sphere is a ball

(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 Poincaré'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.