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«Criteria for asphericity of polyhedral products: corrigenda to “Right-angularity, flag complexes, asphericity” Michael W. Davis Peter H. ...»

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Criteria for asphericity of polyhedral products:

corrigenda to “Right-angularity, flag

complexes, asphericity”

Michael W. Davis Peter H. Kropholler

January 12, 2015


Given a simplicial complex L with vertex set I and a family A =

{(A(i), B(i))}i∈I of pairs of spaces with base points ∗i ∈ B(i), there

is a definition of the “polyhedral product” AL of A with respect to

L. Sometimes this is called a “generalized moment angle complex”.

This note concerns two refinements to earlier work of the first author.

First, when L is infinite, the definition of polyhedral product needs clarification. Second, the earlier paper omitted some subtle parts of the necessary and sufficient conditions for polyhedral products to be aspherical. Correct versions of these necessary and sufficient conditions are given in the present paper.

AMS classification numbers. Primary: 20F65, 57M07, 57M10.

Secondary: 20F36, 20F55, 20J05.

Keywords: aspherical space, graph product, polyhedral product, right-angled Coxeter group, right-angled building.

The definition of a polyhedral product Let L be a simplicial complex.

Suppose that A = {(A(i), B(i))}i∈I is a collection of pairs of spaces, with base points ∗i ∈ B(i), indexed by the vertex set I of L. As a set, the polyhedral product of A with respect to L is the subset AL of the product i∈I A(i)

consisting of the points (xi )i∈I satisfing the following two conditions:

(I) xi = ∗i for all but finitely many i, (II) {i ∈ I | xi ∈ B(i)} is a simplex of L.

/ For pairs M M of subcomplexes of L, the base points determine natural inclusions AM → AM. When L is finite, the polyhedral product inherits its topology from the product i∈I A(i). In general AL acquires the topology of the colimit lim AM as M runs through the finite subcomplexes.

−→ The above definition is given in [5, §4.1]. When L is finite, condition (I) is vacuous and the definition reduces to the one in [4, §2.1]. The restriction to finite L in [4] is implicit but inessential.

In [1] and [4], the notation ZL (A, B) was used for the polyhedral product, here denoted simply by AL.

It is assumed that all the spaces A(i) and B(i) are CW-complexes. In particular, simply connected aspherical spaces are contractible.

Asphericity of polyhedral products We assume each A(i) is path connected. Let Gi denote the fundamental group π1 (A(i), ∗i ) and let Ei = π1 (A(i), B(i), ∗i ), so that Ei is a Gi -set. Define a subset I of I, by I = {i ∈ I | Card(Ei ) = 1} (1) and let L be the full subcomplex of L with vertex set I.

In the statement and proof of [4, Theorem 2.22]two implicit assumptions

are made:

(a) for each path component of B(i), the induced map of fundamental groups π1 (B(i)) → π1 (A(i)) is injective, and (b) when B(i) is path connected, the image of π1 (B(i)) is a proper subgroup of π1 (A(i)).

Under assumptions (a) and (b), it is shown in [4] that AL is aspherical if

and only if the following three conditions hold:

• Each A(i) is aspherical.

• If a vertex i is not joined to every other vertex by an edge, then (A(i), B(i)) is an aspherical pair. (See Definition 2, below, for the meaning of “aspherical pair.”)

• L is a flag complex.

In proving the necessity of these conditions, assumption (a) leads to a conclusion that certain pairs (A(i), B(i)) must be aspherical; without (a), the (A(i), B(i)) are only “nearly aspherical” (cf. Definition 2, below). Assumption (b) implies that I = I and hence L = L. When (b) does not hold further modifications are needed in [4, Theorem 2.22] (cf. Example 1 below).

Example 1. (cf.

[4, Example 2.1]). If for each i ∈ I, (A(i), B(i)) = ([0, 1], {1}), then the polyhedral product ([0, 1], {1})L is elsewhere called a chamber and denoted K(L). Since K(L) is isomorphic to the cone on the barycentric subdivision of L, the space ([0, 1], {1})L is contractible for any L.

Example 2. More generally, if, for each i, A(i) is contractible and B(i) is acyclic, then AL is simply connected and the the natural map AL → ([0, 1], {1})L induces an isomorphism on homology; hence, AL is contractible for any L.

Example 3. If L is a simplex, then AL = i∈I A(i).

Hence, AL is aspherical if and only if each A(i) is aspherical – no condition is needed on the B(i).

Definition 1. (cf. [4, p. 249]). A vertex of a simplicial complex L is conelike if it is joined by an edge to every other vertex of L Definition 2. (cf. [4, Definition 2.21]). Suppose (A, B) is a pair of spaces with A path connected. Let A denote the universal cover of A and let B be the inverse image of B in A. Then (A, B) is an aspherical pair if A is contractible and each component of B is contractible (in other words, A is aspherical, each path component of B is aspherical and the fundamental group of each such component maps injectively into π1 (A)). The pair (A, B) is nearly aspherical if A is contractible and if each path component of B is acyclic.

For any subset J ≤ I, write J for the full subcomplex of L spanned by

J. Given a field F define disjoint subsets of I − I :

I1 := {i ∈ I − I | B(i) is acyclic with coefficients in Z} I2 := {i ∈ I − I | B(i) is not acyclic with coefficients in F}, Put I = I2 ∪ I and L = I. (2)

From now on we assume the following:

(∗) for some field F, I1 ∪ I2 is a partition of I − I.

Although this assumption seems fairly mild, without it there is a problem finding a clean statement of our theorem (cf. Example 4 below).

The corrected statement of Theorem 2.22 in [4] is the following.

Theorem 1. Suppose that I is the vertex set of a simpliciial complex L, that A = {(A(i), B(i))}i∈I is a family of pairs and that condition (∗) holds.

The following four conditions are necessary and sufficient for AL to be aspherical.

(i) Each A(i) is aspherical.

(ii) If i ∈ I is not conelike in L, then (A(i), B(i)) is nearly aspherical.

(iii) The vertices of I2 are conelike in L. Moreover, I2 is a simplex and L is the join I2 ∗ L (iv) L is a flag complex.

Remark. When I = I, Theorem 2.22 of [4] is almost correct as stated, except that in condition (ii) the phrase “aspherical pair” should be replaced by “nearly aspherical pair.” Condition (iii) of Theorem 2.22 reads L instead of L (as in condition (iv) above). The point is that for AL to be aspherical it is not necessary that L be a flag complex but only that L be a flag complex.

This should have been obvious from Example 2.1 in [4] (which is essentially Example 1, above).

An important observation in the proof of Theorem 2.22 of [4] is that if J I, then the natural map r : AL → A J induced by the projection, i∈I A(i) → i∈J A(i) is a retraction (cf. [4, Lemma 2.5]). Applying this to the case where J is a singleton, we see that if AL is aspherical, then so is each A(i).

As in [4], Theorem 1 is proved by reducing to the special case where each pair (A(i), B(i)) has the form (Cone(Ei ), Ei ), for some discrete set Ei, where Cone(Ei ) means the cone on Ei. Put C := {(Cone(Ei ), Ei )}i∈I. Consider the polyhedral product CL and its universal cover CL. A key point is that CL is the standard realization of a right-angled building (abbreviated as RAB)

if and only if the following two conditions hold:

• for each i ∈ I, Card(Ei ) ≥ 2, and

• L is a flag complex.

This is stated in [4, Example 2.8] and a proof is given in [3]. Let W denote the right-angled Coxeter group associated to the 1-skeleton of L and let C denote the underlying chamber system of the RAB.

One flaw in [4] traces back to the assertion in the proof of Lemma 2.11, when L is a flag complex, the chamber system C (obtained, as in [3], via a covering of a nonstandard realization of i∈I Ei ) is a RAB. However, this is only true when each Ei has more than one element, i.e., when the statement after the first bullet point holds.

Let I I be the subset defined by (1). As before, put L = I. The correct version of Lemma 2.11 in [4] is the following.

Lemma 1. Let C := {(Cone(Ei ), Ei )}i∈I.

The polyhedral product CL is aspherical if and only if L is a flag complex.

Proof. First observe that the inclusion CL → CL is a homotopy equivalence. The reason is that there is a retraction r : CL → CL induced by the projection i∈I Cone(Ei ) → i∈I Cone(Ei ); moreover, r is a deformation retraction. (Since, for each i ∈ I −I, Ei is a singleton, Cone(Ei ) deformation retracts onto Ei.) The family {Ei }i∈I and the right-angled Coxeter group W associated to the 1-skeleton of L define a right-angled RAB with underlying chamber system C (defined as in [3] by taking the universal cover of a nonstandard realization of the product building, i∈I Ei ). Suppose L is a flag complex.

As was pointed out above, CL is the standard realization of C. Therefore, it is CAT(0) and hence contractible (cf. [2, Thm.18.3.1, p.338]). So, CL is aspherical and by the previous paragraph, CL also is aspherical.

Conversely, suppose L is not a flag complex. Then CL is a nonstandard realization of C, i.e., CL = U(C, K(L )), where K(L ) is the chamber dual to L and where U(C, K(L )) means the “basic construction” defined as in [2]. An apartment in this nonstandard realization is U(W, K(L )). Since L is not a flag complex, the realization of the apartment, U(W, K(L )), is not contractible (see [2, Thm.9.1.4, p.167]). Since U(C, K(L )) retracts onto the (nonstandard) realization of any apartment, we see that CL cannot be contractible and hence, CL also cannot be contractible.

Lemma 2. Suppose L is a simplicial complex consisting of two vertices.

Let I = {1, 2} be its vertex set and let A = {(A(i), B(i))}i∈I be two pairs of spaces such that A(i) path connected, neither B(1) nor B(2) is acyclic (i.e., I1 = ∅), and condition (*) holds. Then AL is aspherical if and only if, for i = 1, 2, (A(i), B(i)) is nearly aspherical.

Proof. For i = 1, 2, let Ei denote the set of path components B(i) and let C denote the family {(Cone(Ei ), Ei )}i∈I. Then CL is a complete bipartite graph Ω with vertex set E1 E2. Similarly, AL is a graph of spaces for the same graph Ω: the set of vertex spaces for one type is {B(1)e × A(2)}e∈E1 and for the other type it is {A(1) × B(2)f }f ∈E2. Here B(1)e (resp. B(2)f ) means the path component corresponding to e ∈ E1 (resp. f ∈ E2 ). Since A(i) is simply connected, π1 (AL ) ∼ π1 (Ω). (The reason is that the cover of = L A corresponding to the universal cover of Ω is a tree of spaces with simply connected vertex spaces.) Suppose AL is aspherical. Since A(i) is a retract of AL, it also is aspherical; so its universal cover A(i) is contractible. Since the universal cover of AL is contractible, the natural map AL → Ω is a homotopy equivalence. In particular, it induces an isomorphism on homology. On the other hand, when each A(i) is contractible, there is a formula for the homology of any polyhedral product (cf. [1, Thm.2.21]). In the case at hand, it gives that AL has the same homology as the suspension of a smash product, S 0 ∗ (B(1) ∧ B(2)).

So, B(1) ∧ B(2) must have the same homology as the discrete space E1 ∧ E2.

By condition (*) there is a field F so that I2 ∪ I is a partition of {1, 2}. With coefficients in F, the homology of a smash product is concentrated in degree 0 if and only if both factors are concentrated in degree 0 or if one of them is acyclic over F. If I2 = {1, 2}, then both B(1) and B(2) are connected and have nontrivial homology in positive degrees with F coefficients, which is impossible. If I2 is a singleton, then the other element of {1, 2} is in I (since I1 = ∅) and we again get a contradiction. So, I2 = ∅, i.e., neither B(1) nor B(2) is connected. In the case at hand, this means that, for i = 1, 2, each component of B(i) is acyclic (over Z). Hence (A(i), B(i)) is nearly aspherical.

Conversely, suppose each (A(i), B(i)) is nearly aspherical. There is a π1 (Ω)-equivariant map of trees of spaces, from the universal cover of AL to CL, which induces an isomorphism on homology. Hence the universal cover of AL is contractible.

Example 4. For i = 1, 2, suppose A(i) is contractible and that the reduced homology of B(i) is pi -torsion for distinct primes p1 and p2.

Then B(1)∧ B(2) is acyclic over Z. Hence, for L = {1, 2} as in Lemma 2, AL is contractible.

Proof of Theorem 1. First suppose AL is aspherical. As noted previously, this implies (i). Suppose i ∈ I is not conelike in L. Then there is another vertex j ∈ I not connected by an edge to i. Since A {i,j} is a retract of AL, it is also aspherical. Applying Lemma 2, we see that (ii) holds. For each i ∈ I2, B(i) is path connected and not acyclic, so, (A(i), B(i)) is not a nearly aspherical pair; hence, i must be conelike. By assumption, there is a field F so that for each i ∈ I2 ∪ I, B(i) is not acyclic with coefficients in F. (If i ∈ I, then B(i) is not connected.) Suppose there is an “empty simplex” in I2 ∪I, i.e., a subset J I so that J is isomorphic to the boundary of a ksimplex, k ≥ 2. Since the reduced homology (with coefficients in F) of smash product of {B(i)}i∈J is nonzero, it follows from the formula in [1, Thm. 2.21] that this homology, shifted up in degree by k, appears as a direct summand of H∗ (A I2 ∪I ; F). This shows, first of all, that J cannot be a subset of I2. (Since A I2 is aspherical, A I2 must be contractible.) Hence, I2 is a simplex. Furthermore, J cannot decompose as J2 ∪ J, where J2 = J ∩ I2 and J = J ∩ I are both nonempty. (This is because A J is aspherical and has the same fundamental group as C J ; hence, the homology of A J must vanish in degrees ≥ Card(J ).) Therefore, L is the join I2 ∗ L, i.e., (iii) holds. A similar argument shows that I cannot contain an empty k-simplex for k ≥ 2 with vertex set J, for then the universal cover of A J has a nonzero homology in degree k (cf. the second paragraph of the proof of Theorem 2.22 in [4, p. 249]). Therefore, L and L are both flag complexes, i.e., (iv) holds.

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