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The right-angled cube Davis complex and its boundary 2-sphere

Example

Let $S = {a, b, c}$ with $m (s, t) = 2$ for all distinct $s, t$ , let $W$ be the presented group with length $ℓ$ , let $V = R^{S}$ carry the Coxeter form $B$ , and let $S$ , $W S$ , $Σ = ∣ W S ∣$ , $K = ∣ S ∣$ be as in Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization. The diagram of $(S, m)$ then has no edges (Coxeter diagrams: edges, labels, components and finite type (1)); by the disconnected-diagram product theorem and the one-generator presentation, $W ≅ (Z / 2)^{3}$ (Disconnected diagrams, direct products, and comparison of invariant forms (1), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so every subset of $S$ generates a finite subgroup:

(i) For every $T \subseteq S$ one has $W_{T} ≅ (Z / 2)^{T}$ and $∣ W_{T} ∣ = 2^{∣ T ∣}$ , so every subset of $S$ is spherical; and in the $B$ -orthogonal decomposition $V_{T} = ⨁_{s \in T} R e_{s}$ (Disconnected diagrams, direct products, and comparison of invariant forms (1),(2)) the Coxeter cell of Finite Coxeter orbit polytopes, face isometries and their cocycle (1) is the rectangular box $C_{T} = \prod_{s \in T} [- d_{s} e_{s}, d_{s} e_{s}],$ a compact convex polyhedral cell of dimension $∣ T ∣$ whose nonempty face poset is the coset poset of $W_{T}$ : each nonempty face is indexed uniquely by a coset $u W_{U}$ , and face containment matches coset containment; it is a Euclidean cube when the numbers $d_{s}$ , $s \in T$ , are equal. For $T = S$ the cell is a (possibly rectangular) $3$ -cube.

(ii) The cellulation of $Σ$ has $8$ vertices, $12$ edges, $6$ rectangular $2$ -cells (combinatorial squares) and one $3$ -cell, hence $27$ cells in all (Equality, inclusion and intersection of spherical cosets, and the quotient poset (1),(3)); $Σ$ is homeomorphic to the barycentrically subdivided box $C_{S}$ (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (1)), which is convex and hence contractible, and the alternating cell count gives Euler characteristic $8 - 12 + 6 - 1 = 1$ .

(iii) The proper cells of the cellulation form the boundary $\partial Σ ≅ S^{2}$ , the cubical $2$ -sphere with $8$ vertices, $12$ edges and $6$ rectangular $2$ -faces (combinatorial squares). The Coxeter complex of The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere (4) is the dual cellulation of the same sphere: it has $6$ vertices, the codimension-one cosets $w W_{S ∖ {s}}$ , and $8$ triangles, one for each vertex of the box; it is an octahedron.

(iv) The chamber $K = ∣ S ∣$ is the Boolean-lattice order complex; under $T \mapsto 1_{T}$ it is the six-tetrahedron staircase triangulation of $[0, 1]^{S}$ , since its maximal chains are the $3! = 6$ chains $\emptyset \subset {s_{1}} \subset {s_{1}, s_{2}} \subset S$ . The quotient $W \ Σ ≅ K$ is compact (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (4)); the barycentric subdivision of $C_{S}$ has $8 \cdot 3 \cdot 2 = 48$ tetrahedra, matching the $∣ W ∣ = 8$ chambers, each with six tetrahedra.

(v) Thus the right-angled cells are boxes, cubes when the $d_{s}$ agree, in contrast to the dihedral hexagon and octagon of the companion examples.

Facts & Assumptions

Given: $S = {a, b, c}$ with $m (s, t) = 2$ for all distinct $s, t$ ; the presented group $W$ with length $ℓ$ ; the space $V = R^{S}$ with the Coxeter form $B$ , the canonical representation $ρ$ and the reflections $r_{a}$ ; the diagram $Γ$ ; positive numbers $(d_{s})_{s \in S}$ ; and the objects $S$ , $W S$ , $Σ$ , $K$ of Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization.

[F1]

The diagram has vertex set $S$ and an edge between distinct $s, t$ exactly when $m (s, t) \geq 3$ , so here $Γ$ has no edges and its components are the singletons ${a}, {b}, {c}$ ; moreover $W_{T} = ⟨ s : s \in T ⟩$ is a Coxeter system for the restricted matrix. (Coxeter diagrams: edges, labels, components and finite type (1),(2), Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1),(2), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

[F2]

For every $T \subseteq S$ , $W_{T} = ⟨ T ⟩$ has the Coxeter presentation restricted to $T$ ; in particular $W_{{s}}$ has the single generator $s$ and the single relator $s^{2} = 1$ . (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[F3]

If the diagram is disconnected with components $S_{1}, \dots, S_{k}$ , the multiplication map $W_{S_{1}} \times \dots \times W_{S_{k}} \to W$ is a group isomorphism. (Disconnected diagrams, direct products, and comparison of invariant forms (1)).

[F4]

The Coxeter cell: for spherical $T$ the point $x_{T} = \sum_{s \in T} d_{s} v_{s}^{(T)} \in V_{T}$ lies at distance $d_{s}$ from the simple mirror of $s$ , and $C_{T} = conv (W_{T} x_{T})$ is a compact convex polyhedral cell of dimension $∣ T ∣$ whose nonempty faces are exactly the sets $conv (u W_{U} x_{T})$ , $u \in W_{T}$ , $U \subseteq T$ , each occurring for exactly one coset $u W_{U}$ , with face inclusion agreeing with coset inclusion. (Finite Coxeter orbit polytopes, face isometries and their cocycle (1)).

[F5]

Reflection formula and invariance: for $a \in V$ with $B (a, a) \neq 0$ the map $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ is linear and involutive, preserves $B$ , fixes ${v : B (v, a) = 0}$ pointwise and satisfies $r_{a} (a) = - a$ ; moreover $ρ (s) = r_{e_{s}}$ . (The real Coxeter form, its radical, reflections, and form-preserving maps (2),(3), Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2), The canonical reflection homomorphism, roots, reflections, and the positive cone (1)).

[F6]

The projection $π ⁣ : W S \to S$ , $w W_{T} \mapsto T$ , is well-defined; the members of $W S$ are the left cosets of the subgroups $W_{T}$ , and each fixed-type coset family partitions $W$ . (Equality, inclusion and intersection of spherical cosets, and the quotient poset (1), Left and right cosets $g H$ and $H g$ of a subgroup).

[F7]

The canonical barycentric-subdivision map $∣ W S ∣ \to X$ is a homeomorphism $Σ ≅ X$ carrying the subposet below each cell address $q$ onto the barycentric subdivision of $C_{q}$ . (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (1)).

[F8]

Under this identification, cells indexed by $w W_{T}$ have dimension $∣ T ∣$ , and there is one $W$ -orbit of cells for each spherical type. (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (2)).

[F9]

The quotient $W \ Σ$ is compact and $K$ is a strict fundamental domain. (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (4)).

[F10]

The chamber is $K = ∣ S ∣$ , the order complex of the poset of spherical subsets; since $S$ is finite, $K$ is finite and compact. (Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization (3)).

[F11]

The chambers of $Σ$ are the images $w K$ and the map $w \mapsto w K$ is injective. (Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization (4)).

[F12]

Coxeter complex: in finite type the assignment $w W_{I} \mapsto w \overline{C_{I}}$ is a bijection from the proper spherical cosets onto the proper faces of the chamber decomposition with $w W_{I} \subseteq v W_{J} ⟺ w \overline{C_{J}} \subseteq v \overline{C_{I}}$ , and the abstract simplicial complex with vertices the cosets $w W_{S ∖ {s}}$ and simplices the sets ${w W_{S ∖ {s}} : s \notin I}$ , $I ⊊ S$ , is a triangulation of $S^{∣ S ∣ - 1}$ . (The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere (3),(4)).

[F13]

The Euler characteristic of a finite CW complex is the alternating sum of its numbers of cells by dimension. (Euler characteristic of a finite CW complex).

[F14]

When $W_{T} ≅ (Z / 2)^{n}$ , the Coxeter polytope is a product of intervals and is a regular $n$ -cube when the generating point is equidistant from the bounding hyperplanes (Davis, The Geometry and Topology of Coxeter Groups, Example 7.3.2(iii), p. 129).

[F15]

For disconnected diagram components, the spaces $V_{i}$ form a $B$ -orthogonal direct sum, and each factor action preserves its own component space and fixes the other component spaces pointwise. (Disconnected diagrams, direct products, and comparison of invariant forms (2)).

[F16]

For a finite group $G$ and a subgroup $H$ , $∣ G ∣ = [G : H] ∣ H ∣$ . (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[F17]

Coset inclusion satisfies $w W_{U} \subseteq w^{'} W_{T}$ if and only if $U \subseteq T$ and $w^{- 1} w^{'} \in W_{T}$ . (Equality, inclusion and intersection of spherical cosets, and the quotient poset (2)).

[F18]

The Coxeter form has $B (e_{s}, e_{s}) = 1$ for every $s \in S$ . (The real Coxeter form, its radical, reflections, and form-preserving maps (2)).

Verification

technique · direct construction in the orthogonal decomposition

1.1F1F2F3algebra

The diagram $Γ$ has no edges by [F1], so its components are the singletons ${a}, {b}, {c}$ . For each singleton $s$ , [F2] gives the restricted presentation with only the relator $s^{2} = 1$ : every word reduces to $1$ or $s$ , while the map sending $s$ to the nonidentity element of $Z / 2$ respects the relator and separates them. Thus $W_{{s}} = {1, s}$ has order $2$ . By [F3], multiplication is an isomorphism $W_{{a}} \times W_{{b}} \times W_{{c}} \to W$ , so $W ≅ (Z / 2)^{3}$ and $∣ W ∣ = 8$ . For $T = \emptyset$ , $W_{T} = {1}$ ; for $∣ T ∣ = 1$ the order-two calculation applies; and for $∣ T ∣ \geq 2$ the restricted diagram has singleton components, so [F3] gives $W_{T} ≅ (Z / 2)^{T}$ and $∣ W_{T} ∣ = 2^{∣ T ∣}$ . Hence every subset of $S$ is spherical and $S$ is the full power set.

2.1step 1.1F4F5F15F18algebra

For $T = \emptyset$ , step 1.1 and the convention $⟨ \emptyset ⟩ = {1}$ give $W_{T} = {1}$ , while $V_{T} = {0}$ , $x_{T} = 0$ , and $C_{T} = {0}$ , the product over the empty set. For every nonempty $T$ , step 1.1 gives sphericality; fix such a $T$ and let $x_{T} = \sum_{s \in T} d_{s} v_{s}^{(T)}$ be as in [F4]. By [F15] applied to $T$ , the sum $V_{T} = ⨁_{s \in T} R e_{s}$ is $B$ -orthogonal, and by [F18] $B (e_{s}, e_{s}) = 1$ , so $B (v_{s}^{(T)}, e_{t}) = δ_{s t}$ forces $v_{s}^{(T)} = e_{s}$ and $x_{T} = \sum_{s \in T} d_{s} e_{s}$ . For $s \in T$ the reflection $ρ (s) = r_{e_{s}}$ fixes $e_{t}$ pointwise for $t \in T ∖ {s}$ , because $B (e_{t}, e_{s}) = 0$ and the reflection formula of [F5] reduces to $r_{e_{s}} (v) = v$ when $B (v, e_{s}) = 0$ , while $r_{e_{s}} (e_{s}) = - e_{s}$ since $B (e_{s}, e_{s}) = 1$ ; hence $ρ (W_{T}) x_{T}$ is exactly the set of points $\sum_{s \in T} ε_{s} d_{s} e_{s}$ with $ε_{s} \in {+ 1, - 1}$ , the vertex set of the box of (i).

2.2step 1.1F6F7F8F13F16algebra

By step 1.1 every $T \subseteq S$ is spherical. The cells of the cellulation are the cosets $w W_{T}$ , one for each element of $W S$ , and have dimension $∣ T ∣$ by [F7,F8]. The cosets of each $W_{T}$ partition $W$ by [F6], and Lagrange's formula [F16] gives their number as $∣ W ∣ / ∣ W_{T} ∣ = 8 / 2^{∣ T ∣}$ : there are $8$ cosets of $W_{\emptyset}$ (the vertices), $3 \cdot 4 = 12$ of the rank-one parabolics (the edges), $3 \cdot 2 = 6$ of the rank-two parabolics (the rectangular $2$ -cells, combinatorial squares) and one coset $W_{S}$ (the top $3$ -cell), so there are $27$ cells in all. Since $W_{S} = W$ is the maximum coset, every cell is a face of the top cell, and $Σ = ∣ W S ∣ = ∣ W S_{\leq W_{S}} ∣$ is the barycentric subdivision of $C_{S}$ by [F7]; the contraction $H (t, x) = (1 - t) x$ of the convex box $C_{S}$ to $0$ transfers through this homeomorphism to a contraction of $Σ$ , while the alternating count of [F13] gives $χ = 8 - 12 + 6 - 1 = 1$ .

3.1F4F17F14step 2.1algebra

Write $I_{s} : = [- d_{s} e_{s}, d_{s} e_{s}]$ and $A_{s} : = {- d_{s} e_{s}, d_{s} e_{s}}$ . The convex hull of a product of finite sets is the product of the convex hulls: a convex combination $\sum_{i} λ_{i} (a_{i}^{1}, \dots, a_{i}^{k})$ of product points has coordinates $\sum_{i} λ_{i} a_{i}^{s} \in conv (A_{s})$ , and conversely a tuple of convex combinations with coefficient vectors $λ^{1}, \dots, λ^{k}$ is the convex combination of product points with weights $λ_{i_{1}}^{1} \dots λ_{i_{k}}^{k}$ ; therefore $C_{T} = conv (ρ (W_{T}) x_{T}) = \prod_{s \in T} I_{s}$ by [step 2.1], a box of dimension $∣ T ∣$ , agreeing with the product-of-intervals description [F14]. Its nonempty faces are products with a set $U \subseteq T$ of free coordinates and signs fixed on $T ∖ U$ ; the face indexed by $u W_{U}$ has the signs of $u$ on $T ∖ U$ . For $u, v \in W_{T}$ these faces satisfy $F (u, U) \subseteq F (v, V)$ exactly when $U \subseteq V$ and $u, v$ have the same signs outside $V$ , which, by [F17], is equivalent to $u W_{U} \subseteq v W_{V}$ . Thus every nonempty box face is indexed by exactly one parabolic coset, and face-containment and coset-containment agree (so the reverse-inclusion nonempty face and coset posets agree as well); the additional empty face has no coset index.

4.1step 1.1step 2.2step 3.1F4F7F13algebra

By step 1.1 all proper subsets are spherical, and they index exactly the proper nonempty faces of the top cell $C_{S}$ by [F4]; by [F7] those faces are the boundary cells of $Σ$ , so they are the $8$ vertices, $12$ edges and $6$ rectangular $2$ -faces of [step 2.2]. By step 3.1, $C_{S} = \prod_{s \in S} [- d_{s} e_{s}, d_{s} e_{s}]$ in the orthonormal coordinates $e_{a}, e_{b}, e_{c}$ ; radial projection $x \mapsto x / ∥ x ∥$ sends $\partial C_{S}$ to $S^{2}$ and has continuous inverse $u \mapsto u / \max_{s \in S} (∣ u_{s} ∣ / d_{s})$ , so this boundary is a $2$ -sphere; the alternating count gives $8 - 12 + 6 = 2$ by [F13].

4.2F9F10F11step 1.1step 2.2step 3.1algebra

The chamber $K = ∣ S ∣$ is the order complex of the full power set of $S$ by [F10] and [step 1.1]. Every chain extends to a maximal chain. Map each vertex $T$ to its characteristic vector $1_{T} \in [0, 1]^{S}$ . For a permutation $(s_{1}, s_{2}, s_{3})$ , the chain $\emptyset \subset {s_{1}} \subset {s_{1}, s_{2}} \subset S$ gives a tetrahedron with vertices $0, e_{s_{1}}, e_{s_{1}} + e_{s_{2}}, 1_{S}$ . Every point $y \in [0, 1]^{S}$ lies in one of these tetrahedra: choose an ordering $y_{s_{1}} \geq y_{s_{2}} \geq y_{s_{3}}$ and write it as the convex combination with coefficients $1 - y_{s_{1}}$ , $y_{s_{1}} - y_{s_{2}}$ , $y_{s_{2}} - y_{s_{3}}$ , and $y_{s_{3}}$ on those four vertices. The coefficients are nonnegative and sum to one; strict coordinate orders give disjoint tetrahedron interiors, while ties make the corresponding coefficient differences zero and put the point in a shared face. Hence these six characteristic-vector tetrahedra triangulate the cube and give a homeomorphism $K ≅ [0, 1]^{S}$ ; $K$ has six top simplices. By [F9] the quotient $W \ Σ$ is homeomorphic to $K$ , and by [F11] the chambers are the eight distinct translates $w K$ [step 1.1]. By step 3.1, $C_{S}$ is a box; its barycentric subdivision has $8 \cdot 3 \cdot 2 = 48$ top simplices, by choosing a box vertex, an incident edge and an incident $2$ -face; this agrees with the $8 \cdot 6 = 48$ tetrahedra in the chamber translates.

5.1step 1.1step 3.1step 4.1F12algebra

Write an element of $W$ as its sign vector $(ε_{a}, ε_{b}, ε_{c}) \in {\pm 1}^{3}$ under the direct-product isomorphism of step 1.1. A Coxeter-complex vertex of type $S ∖ {s}$ is the coset $w W_{S ∖ {s}}$ ; it is determined exactly by the $s$ -coordinate $ε_{s}$ , since that parabolic changes the other two coordinates freely. Thus the six Coxeter-complex vertices are the three opposite pairs $(s, + 1), (s, - 1)$ for $s \in S$ . The chamber indexed by $w$ is the triangle with vertices $(a, ε_{a}), (b, ε_{b}), (c, ε_{c})$ , so the eight chambers are precisely all choices of one vertex from each opposite pair. This is the octahedral triangulation: its vertices are the six signed coordinate directions and its triangles choose one from each opposite pair. By step 3.1 these labels are the coordinate faces of the box; a rectangular $2$ -face with fixed $s$ -coordinate $ε_{s}$ corresponds to the Coxeter vertex $(s, ε_{s})$ ; a box vertex $(ε_{a}, ε_{b}, ε_{c})$ corresponds to the Coxeter triangle with those three vertices; a box edge with two fixed coordinates corresponds to the Coxeter edge joining the two matching signed vertices; and their incidences are reversed. This proves that the cubical boundary and the Coxeter complex are dual cellulations of the same $2$ -sphere, not the same cellulation.

6.1step 2.1step 2.2step 3.1step 4.1step 4.2step 5.1given∎

The clauses are proved: (i) is [step 2.1] with [step 3.1], (ii) is [step 2.2], (iii) is [step 4.1] with [step 5.1], (iv) is [step 4.2], and (v) restates (i). No Choice is used: all groups, hulls and cell families here are finite, and the only identifications are the explicit ones of the cited clauses.

Remarks

The item remains escalated while these in-run suppliers require current decisions or audits. Consumer ex-cg-right-angled-cube-davis-complex uses def-cg-spherical-nerve-coset-poset-and-davis-realization in step 4.2; lem-cg-spherical-coset-inclusion-and-intersection in steps 2.2 and 3.1; lem-cg-finite-coxeter-orbit-polytopes-and-face-metrics in steps 2.1, 3.1, and 4.1; and thm-cg-davis-complex-cell-incidence-and-stabilizers in steps 2.2, 4.1, and 4.2. Its cross-batch suppliers are def-cg-coxeter-diagram-components-and-finite-type (step 1.1); lem-cg-diagram-products-and-invariant-form-comparison (steps 1.1 and 2.1); def-cg-real-coxeter-form-and-reflection, lem-cg-reflection-form-invariance-and-rank-two-orders, and def-cg-canonical-reflection-homomorphism (step 2.1); thm-hh-parabolic-minimal-representatives-and-length-additivity and def-hh-coxeter-matrix-word-group-and-length (step 1.1); and thm-cg-finite-chamber-tiling-and-coset-face-identification (step 5.1). These supplier statements were inspected provisionally; keep each obligation open until its current Step-3 decision and this exact use are reconciled.

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Sources

M. W. Davis, The Geometry and Topology of Coxeter Groups (MSC lecture slides, Tsinghua, 2013) (standard reference, not scraped)
M. W. Davis, The Geometry and Topology of Coxeter Groups, author manuscript of the first edition (Princeton Univ. Press, 2008) (standard reference, not scraped)