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«Right angularity, flag complexes, asphericity Mike Davis OSU Dubrovnik July 1, 2011 Mike Davis Right angularity, flag complexes, asphericity ...»

Introduction

Polyhedral products

Corners

Right angularity, flag complexes, asphericity

Mike Davis

OSU

Dubrovnik

July 1, 2011

Mike Davis Right angularity, flag complexes, asphericity

Introduction

A cubical complex

Polyhedral products

Its universal cover

Corners

I want to discuss three related constructions of spaces and give

conditions for each of them to be aspherical:

polyhedral product construction,

reflection trick applied to a “corner” of spaces, the pulback of a “corner” via a coloring of a simplicial complex.

All three are related to the construction of the Davis complex for a RACG Mike Davis Right angularity, flag complexes, asphericity Introduction A cubical complex Polyhedral products Its universal cover Corners Introduction A cubical complex Its universal cover Polyhedral products Graph products of groups The fundamental group of a polyhedral product When is a polyhedral product aspherical?

Corners Mirrored spaces The reflection group trick Pullbacks Mike Davis Right angularity, flag complexes, asphericity Introduction A cubical complex Polyhedral products Its universal cover Corners Notation Given a simplicial complex L with vertex set I, put S(L) := {simplices in L} (including ∅) Given σ ∈ S(L), I(σ) is its vertex set.

[−1, 1]I is the |I|-dim’l cube.

C2 (= {±1}) is the cyclic group of order 2. It acts on [−1, 1]. Hence, (C2 )I [−1, 1]I.

For x := (xi )i∈I, a point in [−1, 1]I, put Supp(x) := {i ∈ I | xi ∈ (−1, 1)}.

Mike Davis Right angularity, flag complexes, asphericity Introduction A cubical complex Polyhedral products Its universal cover Corners Define a subcx ZL (or ZL ([−1, 1], {±1})) of [−1, 1]I by ZL := {x ∈ [−1, 1]I | Supp(x) ∈ S(L)} Alternatively, suppose Zσ is the union of all faces parallel to [−1, 1]

–  –  –

ZL is CAT(0) iff L is a flag cx. (Pf: the link of each vertex is L. Then use Gromov’s Lemma.) (Recall L is a flag cx if it is obtained from L1 by filling in every complete subgraph with simplex.) If this is the case, ZL is the Davis complex associated to WL.

Next, we want to extend this to arbitrary pairs of spaces rather than ([−1, 1], {±1}).

–  –  –

If all (A(i), B(i)) are the same, say (A, B), then we omit underlining and write ZL (A, B) instead of ZL (A, B).

Example If L is a flag cx, thenZL (S 1, ∗) is the standard K (π, 1) for the RAAG associated to L1.

The space ZL (D 2, S 1 ) has been called the moment angle complex. It is simply connected and admits an (S 1 )I -action.

There has been a great deal of work lately on computing cohomology and stable homotopy type of polyhedral products, by Bahri-Bendersky-Cohen-Gitler, Denham-Suciu, Buchstaber-Panov and others.

–  –  –

Data A simplicial graph L1 with vertex set I.

A family of discrete gps G = {Gi }i∈I Definition The graph product of the Gi is the group Γ formed quotienting the free product of the Gi by the normal subgroup generated by all commutators of the form [gi, gj ] where {i, j} ∈ Edge(L1 ), gi ∈ Gi and gj ∈ Gj.

Example If all Gi = C2, then Γ is the RACG determined by L1.

If all Gi = Z, then Γ is the RAAG determined by L1.

–  –  –

If each A(i) is path connected & B(i) is a basepoint, ∗i, then π1 (ZL (A, B)) is the graph product of the Gi := π1 (A(i), ∗i ).

(Pf: van Kampen’s Theorem).

–  –  –

Relative graph products More data For each i ∈ I, suppose given a gp Gi & a Gi -set E(i).

Put (Cone E, E) := {(Cone E(i), E(i))}i∈I.

Form the polyhedral product ZL (Cone E, E). It is not simply connected if at least 1 E(i) has more than 1 element. Let

–  –  –

Example If each Gi = C2, then ZL (Cone C2, C2 ) is the space ZL ([−1, 1], {±1}) considered previously.

Remarks If each E(i) = Gi, then the group of lifts, Γ, agrees with the first definition of graph product.

The inverse image of i∈I E(i) in ZL (Cone E, E) is the set of (centers of) chambers in a “right-angled building” (a RAB).

If L is a flag complex, then ZL ([−1, 1], {±1}) is the Davis complex for the RACG W and ZL (Cone E, E) is the standard realization of the RAB.

–  –  –

Let (A, B) = (A(i), B(i))i∈I. Suppose each A(i) is path connected. Let pi : A(i) → A(i) be the univ cover.

Put Gi = π1 (A(i)) and let E(i) be the set of path components of pi−1 (B(i)) in A(i). So, E(i) is a Gi -set.

Proposition π1 (ZL (A, B)) = Γ, where Γ is the relative graph product of the (Gi, E(i)).

Remember: G = Gi acts on ZL (Cone E, E) and Γ is gp of lifts of G-action to ZL (Cone E, E).

–  –  –

Proof of Proposition.

(A, B) := {(A(i), pi−1 (B(i))}i∈I. ZL (A, B) → ZL (A, B) is an intermediate covering space and G is the gp of deck transformations. There is a G-equivariant map ZL (A, B) → ZL (Cone E, E) inducing an iso on π1. The univ cover ZL (A, B) → ZL (A, B) is induced from ZL (Cone E, E) → ZL (Cone E, E).





–  –  –

When is a polyhedral product aspherical?

Suppose L is a flag cx. ZL (A, B) is aspherical in the following

cases:

ZL (S 1, ∗) = BAL, where AL is the associated RAAG.

ZL ([−1, 1], {±1}) = Bπ, where π = Ker(WL → (C2 )m ).

–  –  –

Definition A pair of CW complexes (A, B) is aspherical, if A is aspherical, each path component of B is aspherical and the fundamental gp of any such component injects into π1 (A).

Definition A vertex i of a simplicial cx L is conelike if it is connected by an edge to every other vertex.

–  –  –

Corollary If (A(i), B(i)) = (BGi, ∗) and L is a flag cx, then ZL (A, B) = BΓ, the classifying space for the graph product Γ.

Corollary Suppose each (A(i), B(i)) = (Mi, ∂Mi ) is a mfld with bdry and an aspherical pair. Also suppose L is a flag triangulation of a sphere. Then ZL (A, B) ⊂ Mi is a closed aspherical mfld.

–  –  –

Ingredients for the proof Retraction Lemma Suppose L ⊂ L is a full subcx on vertex set I. Then the map r : ZL (A, B) → ZL (A, B) induced by i∈I A(i) → i∈I A(i) is a retraction.

RAB Lemma Suppose E = (E(i))i∈I is a collection of sets (each with the discrete topology). Then ZL (Cone E, E) is contractible ⇐⇒ L is a flag complex. Moreover, if this is the case, then ZL (Cone E, E) is the “standard realization” of a RAB of type WL.

–  –  –

Restatement of Theorem ZL (A, B) is aspherical ⇐⇒ Each A(i) is aspherical.

For each non-conelike vertex i ∈ I, (A(i), B(i)) is

–  –  –

Comment What is the point of Condition (ii)? If L is flag, then the set of conelike vertices spans a simplex ∆ and L decomposes as a

join, L = L ∗ ∆, and ZL as a product:

–  –  –

Restatement of Theorem ZL (A, B) is aspherical ⇐⇒ Each A(i) is aspherical.

For each non-conelike vertex i ∈ I, (A(i), B(i)) is

–  –  –

Definition A mirror structure M on a space X is a collection{Xi }i∈I of closed subspaces. Its nerve is the simplicial cx, N(M), with vertex set I and σ ≤ I is a simplex iff Xσ = ∅, where Xσ := i∈σ Xi.

–  –  –

Let W be a finite Coxeter group of rank n with fund set of generators indexed by [n], eg, W = (C2 )n.

Define equiv. relation ∼ on W × X by (w, x) ∼ (w, x ) iff x = x and w WI(x) = w WI(x), where I(x) := {i ∈ [n] | x ∈ Xi }. The basic construction is the W -space, U(W, X ) := (W × X )/ ∼.

Question: When is U(W, X ) aspherical?

–  –  –

Let p : X → X be the univ cover. For i ∈ [n], let Ei be the set of path components of p−1 (Xi ) and let E = i∈[n] Ei.

There is an induced mirror structure, M = {Xe }e∈E, where Xe := e. Let N (= N(M)) denote its nerve.

There is a“coloring” f : N → ∆n−1 defined by Ei → i and an induced Coxeter group WN with set of fund generators indexed by E.

–  –  –

When do these conditions hold?

Example (Products) Suppose (A, B) = (A(i), B(i))i∈[n], where A(i) is aspherical and each component of B(i) is aspherical and π1 -injective. Put X = i∈[n] A(i). Define a corner structure on X by Xi = {x ∈ A(i) | xi ∈ B(i)}. Let Ei be the set of path components of p−1 (Xi ) in X. Since N(M) is the n-fold join, ∗ Ei, it is a flag cx.

–  –  –

Example (Borel-Serre compactifications) Suppose Γ is a torsion-free arithmetic lattice in the real points of an algebraic Lie group G of Q-rank n 0. The quotient of the symmetric space by Γ can be compactified to a manifold with corners X (which is a corner) so that each stratum is aspherical, π1 -injective. The nerve N(M) is the spherical bldg associated to G(Q); hence, a flag cx.

The idea of applying the reflection gp trick to these examples was explained to me by Tam Phan, who has recently written a preprint on the subject.

–  –  –

f : L → ∆n−1 a nondegenerate simplicial surjection (a “coloring). It induces f : S(L) → S(∆n−1 ) = P([n]). For τ ≤ [n], let τˇ:= [n] − τ M = {Xi }i∈[n], a corner structure on X.

Want to define a new space f ∗ (X ). For each σ ∈ S(L), define Q(σ) ≤ S(L) × X, by Q(σ) := (σ, Xf (σ)ˇ).

There is an obvious equiv relation on Q := σ∈S(L) Q(σ) which identifies (σ, Xf (σ )ˇ) with the corresponding face of (σ, Xf (σ)ˇ), whenever σ ≤ σ. Put

–  –  –

Sketch.

Let Y be cubical cx associated to N(M). L a flag cx =⇒ f ∗ (Y ) is locally CAT(0). Moreover, f ∗ (X ) is homotopy equivalent to f ∗ (Y ); hence, univ cover of f ∗ (X ) is contractible.

Remark If X is a mfld with corners and L is a triangulation of sphere (or generalized homology sphere) then f ∗ (X ) is a mfld. So, these methods can be used to construct aspherical mflds.

–  –  –





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