Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 4 / c / Solution

Order the chambers $wK$ by nondecreasing Coxeter length, beginning with $K$. When $wK$ is attached, letbe its right descent set. Claim C2 applied to the coset $w{W}_{T(w)}$, followed by C1, shows that $T(w)$ is spherical. The part of $wK$ already present is exactlyIt is nonempty for $w\ne e$ and is contractible by C3. The chamber $wK$ is contractible as well, so C4 shows inductively that every finite length-ordered union of chambers is contractible.

The Davis complex is a CW complex and is the increasing union of these chamber unions. Every map from a sphere has compact image and therefore lies in a finite union; the next finite contractible union null-homotopes it. Thus every homotopy group of the Davis complex vanishes. Since it is connected, the Whitehead theorem implies