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

Here $W=⟨s,t:s^2=t^2=(st{)}^2=1⟩\cong C_2\times C_2$. In the Davis complex described by the basic construction of a Coxeter group, the fundamental chamber $K$ is the order complex ofit consists of two triangles sharing the edge from $\varnothing$ to $\{s,t\}$. Four labelled copies $wK$ are glued along their $s$- and $t$-mirrors. The result is a square subdivided from its centre to the midpoints and vertices of its boundary, with eight triangular chambers.

For the poset description, the spherical cosets consist of four singleton cosets $w{W}_{\varnothing }$, four cosets of rank-one parabolics, and the single coset $W_{\{s,t\}}=W$. Their flag realization has one vertex at each original square vertex, one at each edge midpoint, and one at the square centre; its flags are precisely the same eight triangles. Thus the two requested drawings are the chamber-gluing picture and the barycentric subdivision of a square, respectively.

Write the basic construction aswhere $(w,x)\sim (w^{\prime },x)$ exactly when $w^{-1}{w}^{\prime }$ belongs to the subgroup generated by the mirrors containing $x$. Let $v_T$ be the vertex of $K$ corresponding to a spherical subset of a Coxeter system $T$. Its stabilizer is $W_T$. ThereforeEvery conjugate of every spherical standard parabolic subgroup consequently occurs as a vertex stabilizer.

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