The Davis complex as a CW complex: disk cells and the Cayley skeleta

Fix a spherical $T$. If $T=\mathrm{\varnothing }$, then $V_T=\{0\}$, $x_T=0$, and $C_T=\{0\}$, so the unique map from this point to the closed zero-ball is a homeomorphism and both boundaries are empty. Suppose $T\mathrm{\ne }\mathrm{\varnothing }$. Write $C_T=\{v:{\mathrm{\ell }}_i(v)\ge 0(i\in I)\}$ as in [F1], with $I$ finite and ${\mathrm{\ell }}_i(0)>0$. For a unit vector $u\in V_T$, at least one index satisfies ${\mathrm{\ell }}_i(u)<{\mathrm{\ell }}_i(0)$: otherwise ${\mathrm{\ell }}_i(su)={\mathrm{\ell }}_i(0)+s({\mathrm{\ell }}_i(u)-{\mathrm{\ell }}_i(0))\ge 0$ for every $s\ge 0$ and every $i$, so the whole ray would lie in the bounded set $C_T$. Along that ray, each index with ${\mathrm{\ell }}_i(u)<{\mathrm{\ell }}_i(0)$ imposes $s\le {\mathrm{\ell }}_i(0)\mathrm{/}({\mathrm{\ell }}_i(0)-{\mathrm{\ell }}_i(u))$, while the other indices impose no upper bound. Thus $C_T\cap \{su:s\ge 0\}=\{su:0\le s\le t_{\max }(u)\}$, with $t_{\max }(u)$ the finite positive minimum in clause (1).

By [F18], the closed cells $v{W}_V$ and $w{W}_T$ meet exactly when the spherical coset subsets $v{W}_V$ and $w{W}_T$ meet. By [F15], this is equivalent to $w^{-1}v\in W_T{W}_V$, hence to $v\in w{W}_T{W}_V$; conversely, $v=wab$ with $a\in W_T$, $b\in W_V$ gives the common element $wa=v{b}^{-1}$. There are finitely many spherical $V\subseteq S$, and each product $w{W}_T{W}_V$ is finite because spherical $W_T,W_V$ are finite. Thus only finitely many cells meet a fixed closed cell, proving closure finiteness (C). The gluing definition [F17] tests openness cellwise; by taking complements this is exactly condition (W) in [F6].

Assume $T\mathrm{\ne }\mathrm{\varnothing }$. For each unit $u_0$, let $I_0=\{i:{\mathrm{\ell }}_i(u_0)<{\mathrm{\ell }}_i(0)\}$, which is nonempty by [step 1.1]. Every $f_i(u)={\mathrm{\ell }}_i(0)\mathrm{/}({\mathrm{\ell }}_i(0)-{\mathrm{\ell }}_i(u))$ for $i\in I_0$ is continuous near $u_0$. If $i\mathrm{\notin }{I}_0$ and ${\mathrm{\ell }}_i(u_0)>{\mathrm{\ell }}_i(0)$, that index remains inactive near $u_0$; if ${\mathrm{\ell }}_i(u_0)={\mathrm{\ell }}_i(0)$, its value tends to $+\mathrm{\infty }$ whenever it becomes active as $u\to u_0$. Choose $j\in I_0$; $f_j$ stays bounded on a sufficiently small neighborhood, so after shrinking that neighborhood no newly active equality index can attain the minimum. There $t_{\max }={\min }_{i\in I_0}{f}_i$, proving continuity at $u_0$. By [F1], choose $ϵ>0$ with $\{v:\mathrm{\parallel }v{\mathrm{\parallel }}_B<ϵ\}\subset C_T$ and $R<\mathrm{\infty }$ with $C_T\subseteq \{v:\mathrm{\parallel }v{\mathrm{\parallel }}_B\le R\}$. The ray description gives $ϵ\le t_{\max }(u)\le R$ for every unit $u$.

For $T\mathrm{\ne }\mathrm{\varnothing }$, define $\theta :\bar{B}(0,1)\to C_T$ by $\theta (0)=0$ and, for $y\mathrm{\ne }0$, $t=\mathrm{\parallel }y{\mathrm{\parallel }}_B$, $u=y\mathrm{/}t$, and $\theta (y)=t{t}_{\max }(u)u$. The ray description shows $\theta (y)\in C_T$. If $x=su\in C_T$ with $u$ unit, then $\psi (x)=(s\mathrm{/}{t}_{\max }(u))u$ and $\theta (\psi (x))=x$; conversely, for $y=tu\mathrm{\ne }0$, $\psi (\theta (y))=tu=y$, and both composites fix $0$. Away from $0$ both maps are continuous by continuity of $t_{\max }$; at $0$, $\mathrm{\parallel }\theta (y){\mathrm{\parallel }}_B\le R\mathrm{\parallel }y{\mathrm{\parallel }}_B$ and $\mathrm{\parallel }\psi (x){\mathrm{\parallel }}_B\le \mathrm{\parallel }x{\mathrm{\parallel }}_B\mathrm{/}ϵ$, so both are continuous there. Thus $\theta$ and the stated $\psi$ are mutually inverse homeomorphisms by [F5]. For $s<t_{\max }(u)$ all inequalities defining $C_T$ are strict at $su$, so continuity of the finite affine list makes $su$ an interior point; at $s=t_{\max }(u)$ at least one inequality is equality, and for every larger $s$ that inequality fails. Thus the boundary consists exactly of $t_{\max }(u)u$ for unit $u$, and $\psi$ maps it onto the unit sphere.

For nonempty $T$, the map $\theta$ of [step 3.1], followed by the cell chart $C_T\to C_{w{W}_T}\subseteq \mathrm{\Sigma }$ of [F3], is a characteristic map for the cell indexed by $w{W}_T$. Its interior maps homeomorphically onto the open cell by [F16]. If $x$ is a boundary point, some defining inequality ${\mathrm{\ell }}_i(x)$ is $0$, since otherwise the finite affine list stays positive in a neighborhood of $x$; then $C_T\cap \{{\mathrm{\ell }}_i=0\}$ is a face: if a strict convex combination has ${\mathrm{\ell }}_i$-value $0$, both endpoint values are $0$. It is proper because ${\mathrm{\ell }}_i(0)>0$; by [F2] it is a lower-dimensional cell. Thus the boundary maps into the preceding skeleton. For $T=\mathrm{\varnothing }$, the one-point chart is the characteristic map of a zero-cell, with empty boundary.

Each cell boundary is a union of finitely many proper nonempty faces by [F2], and each such face is indexed by $w^{\mathrm{\prime }}{W}_U$ with $U⊊T$, so it lies in the preceding skeleton since its dimension is $\mathrm{\mid }U\mathrm{\mid }<\mathrm{\mid }T\mathrm{\mid }$. For a zero-cell this is the empty union. The zero-skeleton is the discrete set $W$. Attach the characteristic disks of [step 4.1] in increasing dimension; every attaching map has finite boundary support, and the weak attachment topology agrees with the gluing topology from [F17] because it tests openness on closed cell images, and each characteristic map is a homeomorphism onto its closed cell by [F3],[F5]. Since $S$ is finite, there are finitely many dimensions. The choice-free attachment lemma [F8] therefore gives the asserted CW structure with the given cells and topology, including its Hausdorff condition.

The zero-cells are $w{W}_{\mathrm{\varnothing }}=\{w\}$ by [F3] and [F9], so ${\mathrm{\Sigma }}^0=W$. Each one-cell is $w{W}_{\{s\}}=\{w,ws\}$ and its boundary vertices are $w$ and $ws$; its label is $s$, giving exactly the undirected Cayley graph by [F11]. Now fix distinct $s,t$ and put $m=m(s,t)$. By [F10], $W_{\{s,t\}}$ has presentation $⟨s,t\mid s^2=t^2=1,(st{)}^m=1⟩$ if $m<\mathrm{\infty }$, and omits the last relation if $m=\mathrm{\infty }$. When $m<\mathrm{\infty }$, writing $r=st$ and using $srs=r^{-1}$ reduces every word to $r^k$ or $s{r}^k$, $0\le k<m$, so the group has at most $2m$ elements. The map to the group of pairs $D_m=\mathbb{Z}\mathrm{/}m\times \{\pm 1\}$ with multiplication $(a,ϵ)(b,\delta )=(a+ϵb,ϵ\delta )$, $s\mapsto (0,-1)$ and $t\mapsto (1,-1)$, is onto: these images are involutions and their product $(-1,1)$ generates the rotation subgroup; hence $\mathrm{\mid }{D}_m\mathrm{\mid }=2m$ gives $\mathrm{\mid }{W}_{\{s,t\}}\mathrm{\mid }=2m$. When $m=\mathrm{\infty }$, the maps $s(x)=-x$, $t(x)=2-x$ on $\mathbb{R}$ satisfy the involution relations and make $st$ a nonzero translation, so $W_{\{s,t\}}$ is infinite. Hence $\{s,t\}$ is spherical exactly when $m<\mathrm{\infty }$. For finite $m$, the boundary walk of $w{W}_{\{s,t\}}$ alternates the $s$- and $t$-edges and has vertices $w(st{)}^k$ and $w(st{)}^{k}s$ ($0\le k<m$), all distinct by the dihedral normal forms; it closes at $w(st{)}^m=w$. By [F2] these alternating rank-one cosets are edges of the cell, so the closed walk through all $2m$ vertices is its polygon boundary. Translates of this circuit are indexed by the left cosets $w{W}_{\{s,t\}}$, since its vertices are exactly that coset and its cyclic order is the unique alternating circuit in the rank-two Cayley graph. By [F12], the 2-cells are precisely these circuits: the relators $(st{)}^m$ attach polygonal cells and the relators $s^2$ add none.

For $T=\{s\}$, $v_s^{(T)}=e_s$ because $B(e_s,e_s)=1$, so $x_T=d_s{e}_s$ by [F2]; since $r_s=r_{e_s}$ by [F20] and $\rho (s)=r_s$ by [F21], [F19] gives $s{x}_T=-d_s{e}_s$ and $C_T=[-d_s{e}_s,d_s{e}_s]$. For $T=\{s,t\}$ with finite $m$, [step 6.1] gives the $2m$-gon. By [F13], its generating mirrors bound a sector of angle $\pi \mathrm{/}m$; equality $d_s=d_t$ means $x_T$ is equidistant from those walls, hence lies on their angle bisector. The product of the two wall reflections rotates by $2\pi \mathrm{/}m$, so the dihedral orbit has arguments $\theta +2k\pi \mathrm{/}m$ and $-\theta +2k\pi \mathrm{/}m$ with $\theta =\pi \mathrm{/}(2m)$, which are the $2m$ equally spaced arguments $\theta +j\pi \mathrm{/}m$; their convex hull is regular. Finally, cells of dimension at least $3$ correspond exactly to spherical subsets of size at least $3$ by [F2], [F3], and [F16]; any such subset contains a spherical three-element subset because its parabolic subgroup is finite by [F9], and every spherical three-element subset gives a three-dimensional cell. Thus there are no cells of dimension $\ge 3$ exactly when no three-element spherical subset exists.

Clauses (1)–(4) follow from [step 3.1] with [step 4.1], [step 1.2] with [step 5.1], [step 6.1], and [step 7.1], respectively. No Choice is used: each exit parameter is a minimum over a specified nonempty finite set, and all cell maps and attachments are explicitly supplied; the finite-dimensional disk identifications require only finite-dimensional Euclidean bases.