Examples of Groups: Coxeter Groups

Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

Mike Davis

OSU

May 31, 2008 http://www.math.ohio-state.edu/ mdavis/

Mike Davis Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

1 Cell complexes Simplicial complexes Cubical cell complexes The complex PL The universal cover of PL and the gp WL

2 When Is PeL contractible?

3 A digression: moment angle complexes

4 Construction of examples The basic construction Cohomological dimension Aspherical mﬂds not covered by Rn The reﬂection group trick

Mike Davis Examples of Groups: Coxeter Groups Cell complexes Simplicial complexes When Is PeL contractible? Cubical cell complexes A digression: moment angle complexes The complex PL Construction of examples The universal cover of PL and the gp WL

Deﬁnition A cell complex is a union of convex polytopes (= “cells”) in some Euclidean space so that the intersection of any two is either empty or a common face of both.

Definition A simplex is the convex hull of a finite set T of affinely independent points in some Euclidean space. Its dimension is Card(T ) − 1. For S a finite set, ∆S, the simplex on S, is the convex hull of the standard basis of RS (where RS := {x : S → R}).

Example A 1-simplex is an interval; a 2-simplex is a triangle; a 3-simplex is a tetrahedron.

Deﬁnition An abstract simplicial complex consists of a set S (of vertices) and a poset S of ﬁnite subsets of S s.t. {s} ∈ S, ∀s ∈ S. If T ∈ S and U ⊂ T , then U ∈ S.

Deﬁnition A (geometric) simplicial complex is a cell complex in which all cells are geometric simplices.

Suppose L is a geometric simplicial cx. Put

S := Vert(L), and S(L) := {T ⊂ S | T is the vertex set of a simplex in L}.

Deﬁnition A geometric realization of an abstract simplicial cx S is a geometric simplicial cx L s.t. S = S(L).

Theorem Every abstract simplicial cx S has a geometric realization.

Given a set S and a function x : S → R, S Supp(x) := {s ∈ S | xs 6= 0}. R denotes the Euclidean space of ﬁnitely supported functions x : S → R and ∆S is the simplex on S. We want to prove:

Theorem Every abstract simplicial cx S has a geometric realization.

Proof. Given S, deﬁne a subcx L ⊂ ∆S by [ L := {x ∈ ∆S | Supp(x) ∈ S} = ∆T T ∈S

Clearly, S(L) = S.

Deﬁnition S S S The standard cube on a set S is: := [−1, 1] ⊂ R . Its dimension is Card(S).

For each T ⊂ S, put

T T S−T S := [−1, 1] × {0} ⊂ R .

The link of a vertex v in a cell is the intersection of a small sphere about the vertex with the cell. For example, the link of a vertex in a cube is a (spherical) simplex.

Lk(v) Lk(v)

Similarly, the link, Lk(v), of a vertex in a cubical cell complex is a simpicial cx.

Suppose v is a vertex in a cubical cell complex P. As an abstract simplicial complex Lk(v; P) is isomorphic to the poset of cells of P which properly contain v. A neighborhood of v in P is homeomorphic to the cone on Lk(v).

T T S−T S Recall := [−1, 1] × {0} ⊂ R . S T T A face of is parallel to if it has the form [−1, 1] × {ε} for some ε ∈ {±1}S−T .

The cubical complex PL Given a simplicial complex L with vertex set S, deﬁne a S subcomplex PL of by

[ T PL := all faces parallel to T ∈S(L)

Main Property

For each vertex v, Lk(v, PL) = L

Example Suppose S consists of 3 points and L is the union of an interval and a point. Then S is a 3-cube and P is indicated subcx. L L PL

More examples n−1 n If L = ∆ , then PL = n−1 n n−1 If L = ∂∆ , then PL = ∂ = S . n If L is a set of n points, then PL is the 1-skeleton of , 0 2 1 eg, if L = S , then PL = ∂ = S .

Joins If T and U are disjoint sets, then ∆T ∗ ∆U = ∆T ∪U . (Note: dim(∆T ∗ ∆U ) = Card(T ∪ U) − 1 = dim(∆T ) + dim(∆U ) + 1.) Similarly, if L1 and L2 are simplicial complexes, then L1 ∗ L2 is deﬁned by taking the joins of simplices in L1 with those in L2 (including the two empty simplices). We have: S(L1 ∗ L2) = S(L1) × S(L2)

More 0 0 P(L1∗L2) = PL1 × PL2 , eg, if L = S ∗ S , then 1 1 2 0 0 PL = S × S = T , or if L is the n-fold join S ∗ · · · ∗ S n (the bdry of an n-dim octahedron), then PL = T .

If L is a k-gon, then PL is the orientable surface of Euler characteristic 2k−2(4 − k)

n−1 If L is a triangulation of S , then PL is an n-mﬂd.

The (Z/2)S-action S Let {rs}s∈S be standard basis for (Z/2) . Represent rs as S reﬂection on across the hyperplane xs = 0, ie, rs changes the sign of the sth-coordinate. This deﬁnes a S S (Z/2) -action on . S The subcomplex PL is (Z/2) -stable. [0, 1]S is a (strict) fundamental domain for (Z/2)S-action on S S (= [−1, 1] ). S K := PL ∩ [0, 1] is a (strict) fundamental domain for S (Z/2) -action on PL.

The universal cover of PL is denoted PeL. The cubical structure on PL lifts to one on PeL.

L PL

˜ PL

S Let WL be the gp of all lifts of elements of (Z/2) to PeL. S Let ϕ : WL → (Z/2) be the projection. We have the short exact sequence S 1 → π1(PL) → WL → (Z/2) → 1. S (Z/2) acts simply transitively on Vert(PL), so, WL acts simply transitively on Vert(PeL). Let v ∈ K be the vertex (1,..., 1). Choose a lift Ke of K in PeL (N.B. K is a cone) and let v˜ be the lift of v in Ke. The 1-cells at v or v˜ correspond to elements of S. The reﬂection rs ﬂips the 1-cell at v labeled by s. Let s˜ be the unique lift of rs which stabilizes the corresponding 1-cell at v˜. (Eventually, I will drop the ˜ from s˜.)

A presentation for WL 2 Since s˜ ﬁxes v˜ and covers the identity on PL, it follows that s˜2 = 1.

Since WL is simply transitive on Vert(PeL), the 1-skeleton of PeL is the Cayley graph of (WL, Se). Suppose {s, t} is an edge of L. The corresponding 2-cell at v˜ has edges labeled successively by s˜,˜t, s˜,˜t. It follows that (s˜˜t)2 = 1.

Since the 2-skeleton of PeL is simply connected, it is the Cayley 2-complex of a presentation. Therefore, WL has a presentation with generating set Se = {s˜}s∈S and relations: s˜2 = 1 and (s˜˜t)2 = 1, ∀{s, t} ∈ Edge(L).

WL is a right-angled Coxeter group.

Mike Davis Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

Definition A simplicial cx L is a flag complex iff any finite set of vertices which are pairwise connected by edges spans a simplex of L.

Examples ∂∆n is not a flag cx for n ≥ 2 A k-gon (i.e. a triangulation of S1) is a flag cx iff k ≥ 4 The barycentric subdivision of any cell complex is a flag cx. (This shows that the condition of being a flag cx does not restrict the topological type of L: it can be any polyhedron.)

Mike Davis Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

Theorem

PeL is contractible iff L is a ﬂag cx.

First Proof.

One shows that He∗(PeL) = 0 (ie, PeL is acyclic). PeL is tessellated by translates of K (each of which is a “chamber”). Order these by word length and glue on one at a time. Mayer-Vietoris sequence shows that at each stage result is acyclic.

Second Proof (Gromov).

As a cubical cx, PeL has a piecewise Euclidean metric. Gromov: this is CAT(0) ⇐⇒ it is simply connected and the link of each vertex is a ﬂag cx.

Mike Davis Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

Here is a generalization of the construction of PL which has received a good deal of recent interest. Let (X, A) be a pair of spaces and L a simplicial cx with Vert(L) = S. We deﬁne certain subspaces of the product Q s∈S X. T For each T ∈ S(L)>∅, let X be the set of (xs)s∈S in the product deﬁned by ( xs ∈ X if s ∈ T ,

xs ∈ A if s ∈/ T .

[ Z (L; X, A) := X T

T ∈S(L)>∅

Mike Davis Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples

Examples

(X, A) = ([−1, 1], {±1}). Then Z(L;[−1, 1], {±1}) = PL. (X, A) = (S1, {1}). Then the fundamental gp of Z(L; S1, {1}) is the right-angled Artin gp determined by the 1-skeleton of L. If L is a ﬂag cx, then Z(L; S1, {1}) is the standard K (π, 1) for the Artin gp. (X, A) = (D2, S1). Then Z (L; D2, S1) is the moment angle cx of L. The group (S1)S acts on (Z (L; D2, S1). The quotient space is the same space K (⊂ [0, 1]S) considered earlier. If K is a n-dim convex polytope and L is the bdry cx of its dual, then Z(L; D2, S1) is a smooth mﬂd, and if T is an appropriate subgp of codim n in (S1)S, then Z(L; D2, S1)/T is a “toric variety”.

Mike Davis Examples of Groups: Coxeter Groups Cell complexes The basic construction When Is PeL contractible? Cohomological dimension A digression: moment angle complexes Aspherical mﬂds not covered by Rn Construction of examples The reﬂection group trick

L is a ﬂag cx with vertex set S and WL is associated right-angled Coxeter gp. S is its fundamental set of generators..

The basic construction A mirror structure on a space X is a family of closed subspaces {Xs}s∈S. For x ∈ X, put S(x) = {s ∈ S | x ∈ Xs}. Deﬁne

U(W , X) := (W × X)/ ∼, where ∼ is the equivalence relation: (w, x) ∼ (w 0, x0) ⇐⇒ 0 −1 0 x = x and w w ∈ WS(x) (the subgp generated by S(x)). U(W , X) is formed by gluing together copies of X (the chambers). The gp WL (= W ) acts on it.

Another construction of PeL

S Recall K := PL ∩ [0, 1] . For each s ∈ S, Ks is the intersection of K with the hyperplane xs = 0. This is a mirror structure on K .

Theorem S The natural maps U((Z/2) , K ) → PL and U(WL, K ) → PeL are homeomorphisms.

The basic idea The topology of the simplicial cx L is reﬂected in properties of the gp WL.

Suppose π is a torsion-free gp. Its cohomological dimension, cd(π) is deﬁned to be the maximum integer k st Hk (π; M) 6= 0 for some π-module M. Its geometric dimension, gd(π) is the smallest dimension of a K (π, 1) complex. Obviously, cd(π) ≤ gd(π). Eilenberg-Ganea proved equality if cd(π) ≥ 3 and Stallings, Swan proved it for cd(π) = 1.

The Eilenberg-Ganea Problem Is there a gp π with cd(π) = 2 and gd(π) = 3?

Conjectured answer Yes.

Suppose L is a ﬂag triangulation of an acyclic 2-complex S with π1(L) 6= 1. Put πL := π1(PL) = Ker(ϕ : WL → (Z/2) ). πL is torsion-free. It is a conjectured Eilenberg-Ganea counterexample. ◦ Put ∂K := K − K . In our case it is acyclic. It follows that U(WL, ∂K ) is acyclic (but not simply connected). Hence, cd(πL) = 2.

dim PeL = dim L + 1 = 3 and the only contractible complex which πL seems to act on is PeL (= U(WL, K )).

Remark

Brady, Leary, Nucinkis proved these WL are counterexamples to the version of the Eilenberg-Ganea Problem for groups with torsion.

Example (Different cd over Z than Q) Suppose L is a ﬂag triangulation of RP2. Then 3 3 ∼ 2 2 H (πL; ZπL) = H (PeL; Z) = Hc (RP ) = Z/2. 3 2 H (PeL; Q) = 0 and H (PeL; Q) is a countably generated Q vector space. Hence,

cdZ(πL) = 3 and cdQ(πL) = 2

Facts A closed m-mfld Mm, m ≥ 3, with the same homology as Sm need not be homeomorphic to Sm, because it need not be simply connected. However, m m ∼ m If π1(M ) = 1, then M = S (Poincare´ Conjecture). Similarly, a contractible open mfld Y m, m ≥ 3, is homeomorphic to Rm iff it is simply connected at ∞. (Stallings, Freedman, Perelman). Every such homology m-sphere Mm (simply connected or not) bounds a contractible (m + 1)-mfld.

Suppose Ln−1 is a non simply connected homology (n − 1)-sphere triangulated as a ﬂag cx.

Then PeL is a contractible n-dim homology mﬂd (the non manifold points are the vertices) and PeL is not simply connected at ∞. (Its fundamental gp at ∞ is the inverse limit of free products of an increasing number of copies of π1(L).)

PeL can be modified to be a contractible n-mfld. Let C be a ◦ contractible n-mfld bounded by L (= ∂K ). Remove K and ◦ n replace it by C. Then Y := U(WL, C) is a contractible ∼ n n n n-mfld =6 R and M := Y /π (where π = π1(PL)) is a closed aspherical mfld with universal cover Y n.

Reflection Group Trick Given a group π which has a finite K (π, 1) complex, this is a technique for constructing an aspherical mfld M which retracts back onto K (π, 1).(Aspherical means its universal cover is contractible.) In a nutshell the trick goes as follows: Thicken K (π, 1) to X, a compact mfld with bdry. (X is homotopy equivalent to K (π, 1).)

Put L := ∂X. Triangulate L as a flag cx and let W (= WL) be the corresponding right-angled Coxeter gp. As before ◦ modify PL to a mfld by removing each copy of K and ◦ replacing it by X, ie, form M := U((Z/2)S, X), the desired aspherical mfld.

Sample applications Point: many interesting gps have ﬁnite K (π, 1)-complexes (even 2-dimensional ones).

By choosing π a Baumslag-Solitar gp, we can get π1(M) - to be non-residually ﬁnite, or - to have an inﬁnitely divisible subgp (=∼ Z[1/2]). By choosing π to have unsolvable word problem (can do this with a 2-dim K (π, 1)), we get π1(M) with unsolvable word problem.

M := U((Z/2)S, X). As before, we can construct U(WL, X). It is not contractible. But it is aspherical. (Pf: It is a union of copies of X glued together along contractible pieces.) Hence, M is aspherical (since it is covered by U(WL, X)). Me = (univ cover of M). We can explicitly describe Me as follows. Let Xe = (univ cover of X) and π = π1(X). Le is the induced triangulation of ∂Xe and Se = Vert(Le). Wf (= W ) the corresponding right-angled Coxeter gp. Give X eL e the induced mirror structure (indexed by Se). Then Me = U(Wf, Xe).

The gp Wf o π acts on U(Wf, Xe) with quotient space X and if Γ is the inverse image of the commutator subgp of WL in Wf, then Γ o π acts freely with quotient space M.

Exercise 1

Prove the following formula for the Euler characteristic of PL:

|T | X 1 χ(P ) = − . L 2 T ∈S(L)

The Hopf Conjecture If M2n is a closed, aspherical mﬂd, then (−1)nχ(M2n) ≥ 0.

The Charney-Davis Conjecture 2n−1 n If L is a ﬂag triangulation of S , then (−1) χ(PL) ≥ 0, ie, if κ(L) denotes the RHS of the formula in EXercise 1, then (−1)nκ(L) ≥ 0.

Exercise 2 Prove the Charney-Davis Conjecture for ﬂag triangulations of S1. Do any nontrivial example of the conjecture for S3. For example, calculate κ(L) for L the barycentric subdivision of ∂∆4.

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York and Berlin, 2002. R. Charney and M.W. Davis, The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold, Pac. J. Math. 171 (1995), 117-137. , Finite K (π, 1) s for Artin groups, Annals of Math Studies, vol 138, pp. 110–124, Princeton Univ Press, 1995. M.W. Davis, Exotic aspherical manifolds, School on High-Dimensional Topology, ICTP, Trieste, 2002. , The Geometry and Topology of Coxeter Groups, London Math. Soc. Monograph Series, vol. 32, Princeton Univ. Press, 2007.

Mike Davis Examples of Groups: Coxeter Groups