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Examples of Groups: Coxeter Groups Cell complexes When Is PeL contractible? A digression: moment angle complexes Construction of examples Examples of Groups: Coxeter Groups 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. 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 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 := fx : S ! Rg). Example A 1-simplex is an interval; a 2-simplex is a triangle; a 3-simplex is a tetrahedron. 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 An abstract simplicial complex consists of a set S (of vertices) and a poset S of finite subsets of S s.t. fsg 2 S, 8s 2 S. If T 2 S and U ⊂ T , then U 2 S. Definition A (geometric) simplicial complex is a cell complex in which all cells are geometric simplices. 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 Suppose L is a geometric simplicial cx. Put S := Vert(L); and S(L) := fT ⊂ S j T is the vertex set of a simplex in Lg: 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. 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 Given a set S and a function x : S ! R, S Supp(x) := fs 2 S j xs 6= 0g. 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 := fx 2 ∆S j Supp(x) 2 Sg = ∆T T 2S Clearly, S(L) = S. 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 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] × f0g ⊂ R : 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 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. 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 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). 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 T T S−T S Recall := [−1; 1] × f0g ⊂ R : S T T A face of is parallel to if it has the form [−1; 1] × f"g for some " 2 {±1gS−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 2S(L) Main Property For each vertex v, Lk(v; PL) = L 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 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 . 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 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) 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 n−1 If L is a triangulation of S , then PL is an n-mfld. The (Z=2)S-action S Let frsgs2S 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. 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 The universal cover of PL is denoted PeL. The cubical structure on PL lifts to one on PeL. L PL ˜ PL 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 S Let WL be the gp of all lifts of elements of (Z=2) to PeL. S Let ' : 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 2 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~.) 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 A presentation for WL 2 Since s~ fixes v~ and covers the identity on PL, it follows that s~2 = 1.
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