SOME REMARKS ON CENTRAL ELEMENTARY ABELIAN P-SUBGROUPS AND COHOMOLOGY OF CLASSIFYING SPACES

Carlos Broto and Hans-Werner Henn

1. Introduction

This note is concerned with the question how a central elementary abelian p-subgroup C of a compact Lie group G influences the mod p cohomology of its classifying space BG and, more generally, of the Borel construction EG ×_{G} X for a G space X on which C acts trivially. Here p is a prime number and an elementary abelian p-group is a group isomorphic to (Z/p)^{n} for some natural number n which is called its rank. Mod p cohomology of a space Y will always be denoted by H^{∗}Y while for a G-space X we write H_{G}^{∗}X for the mod p cohomology of the Borel construction EG ×_{G} X.

Our starting point is the simple observation that for a central elementary abelian psubgroup C of G and a G-space X with trivial C-action the multiplication C × G ⟶ G induces a map BC × (EG ×_{G} X) ⟶ EG ×_{G} X which in turn induces an H^{∗}BC comodule structure on H_{G}^{∗}X. This observation leads to a short proof of the following result on the depth of H_{G}^{∗}X (see section 2.)

1.1 Theorem. Let G be a compact Lie group, X a G-space and C a central elementary abelian p-subgroup of G acting trivially on X. If H^{∗}X is finite dimensional over Z/p then the depth of H_{G}^{∗}X is at least as big as the rank of C.

This theorem was first proved by Duflot [D] in case G is finite. Stong has shown how to generalize Duflot’s argument if G is a compact Lie group (unpublished.) The proof we give works also for discrete groups of finite virtual cohomological dimension or profinite groups (cf. Remark 2.3.)

Another simple application of the existence of this comodule structure is the following result (cf. [DW].)

1.2 Proposition. Suppose that C is a central elementary abelian p-subgroup of a topological group G and that X is a G-space on which C acts trivially. Then for any point in X the image of the resulting restriction map H_{G}^{∗}X ⟶ H^{∗}BC is a sub Hopf algebra of H^{∗}BC.

1.3 Remark. This result and standard techniques on Hopf algebras (see [AS] for example) give a very precise hold on the structure of the image I of the restriction map from H_{G}^{∗}X to H^{∗}BC: There are linearly independent elements u_{1}, …, u_{s} in H^{1}BC for a certain s ≥ 0 and integers 0 ≤ k_{1} ≤ k_{2} ≤ … ≤ k_{s} such that I is the polynomial sub Hopf algebra F_{2}[u_{1}^{2^{k_{1}}} , …, u_{s}^{2^{ks}}] if p = 2. For p odd I is as well a tensor product of sub Hopf algebras of the form either F_{p}[βu_{i}] ⊗ E[u_{i}] or F_{p}[(βu_{i})^{p^{k_{i}}}], where β denotes the Bockstein homomorphism and E[u_{i}] denotes the exterior algebra on u_{i}.

If G is a compact Lie group then s is equal to the rank of C because of 2.1 below. In the following it might be helpful to keep this in mind but in fact our proofs would not become shorter if we used this.

Proposition 1.2 will be used in the proof of the next theorem. In it we determine the structure of compact Lie groups with an elementary abelian p-subgroup C which is both central and detecting, the latter meaning that the restriction homomorphism H^{∗}BG ⟶ H^{∗}BC is a monomorphism.

More generally the theorem yields necessary group theoretic conditions on the structure of compact Lie groups G for which the restriction homomorphism from H^{∗}BG to the product ∏ H^{∗}BE of all elementary abelian p-subgroups E is a monomorphism. Results of Lannes [L] imply that this property is inherited by the centralizers C_{G}(E′) of all elementary abelian p-subgroups E′. If E′ is maximal in such a group G then E′ is central and detecting in C_{G}(E′) and hence C_{G}(E′) has the very special form described in the theorem.

1.4 Theorem. Let G be a compact Lie group and let C be a detecting and central elementary abelian p-group. Then G is isomorphic to a product E × N × S where E is an elementary abelian p-group and N is a central extension of a torus T by a finite group π of order prime to p, i.e. there is a central exact sequence 1 ⟶ T ⟶ N ⟶ π ⟶ 1. S is trivial for p > 2 and a product of three dimensional spheres S^{3} otherwise.

Clearly, if G is isomorphic to E × N × S as in the Theorem, then G has a detecting and central elementary abelian p-subgroup.

1.5 Corollary. If G is a finite group and C is a detecting central elementary abelian p-subgroup, then G is isomorphic to C × N where N is a group of order prime to p.

1.6 Remark. It is well known (and follows easily from Proposition 2.1 below) that a detecting elementary abelian p-subgroup is maximal. On the other hand a maximal and central elementary abelian p-subgroup of a compact Lie group G is not necessarily detecting although this is true in the connected case (see Proposition 3.3.)

The remainder of the paper is organized as follows. In section 2 we prove Theorem 1.1 and Proposition 1.2. Section 3 deals with the proof of Theorem 1.4 and Corollary 1.5.

The first author is grateful to D. Puppe and H.-W. Henn for their hospitality and to the DFG for financial support during the winter semester 1990–91.

2. Cohomological consequences of comodule structures

We start with having a closer look at the comodule structure of H_{G}^{∗}X mentioned in the introduction. H^{∗}BC is a Hopf algebra with commutative diagonal induced by multiplication in C. The coaction Δ_{X}: H_{G}^{∗}X ⟶ H^{∗}BC ⊗ H_{G}^{∗}X being induced by a continuous map is actually a map of algebras, so H_{G}^{∗}X is a comodule algebra in the sense of Milnor and Moore [MM]. If X is a point we obtain a coaction H^{∗}BG → H^{∗}BC ⊗ H^{∗}BG which we will denote by Δ for simplicity.

The map from X to a point induces an H^{∗}BC comodule algebra map H^{∗}BG ⟶ H_{G}^{∗}X. Also for any point in X the map BC ⟶ EG ×_{G} X given by the inclusion of this point into X and of C into G induces a comodule algebra map H_{G}^{∗}X ⟶ H^{∗}BC. Note that if X is a point this is the usual restriction map from H^{∗}BG to H^{∗}BC.

The following well known result will be used repeatedly.

2.1 Proposition (cf. [Q1]). If H is a closed subgroup of the compact Lie group G then the restriction map H^{∗}BG ⟶ H^{∗}BH makes H^{∗}BH into a finitely generated H^{∗}BG module. □

Before we start with the proof of Theorem 1.1 we recall the definition of the depth of H_{G}^{∗}X. We consider H_{G}^{∗}X as H^{∗}BG module via the algebra map H^{∗}BG ⟶ H_{G}^{∗}X induced by the projection from X to a point. A sequence of elements x_{1}, …, x_{n} of H^{∗}BG of positive degree is called regular on H_{G}^{∗}X if x_{1} is not a zero divisor on H_{G}^{∗}X and x_{i+1} is not a zero divisor on the quotient H_{G}^{∗}X/(x_{1}, …, x_{i})H_{G}^{∗}X. H^{∗}BG is well known to be noetherian (this follows for example from 2.1) and if H^{∗}X is finite dimensional over Z/p, then H_{G}^{∗}X is finitely generated as H^{∗}BG module. Under these conditions any two maximal regular sequences have the same finite length, called the depth of H_{G}^{∗}X(cf. [Mat]).

In the situation of Theorem 1.1 we will construct a regular sequence x_{1}, …, x_{n} if n = rank C, even if H^{∗}X is not finite dimensional over Z/p. For this we will need the following.

2.2 Lemma. Suppose C is a central elementary abelian p-subgroup of the compact Lie group G. Then for each element u ∈ H^{1}BC there is a natural number l and an element x ∈ H^{∗}BG such that the coaction Δ : H^{∗}BG → H^{∗}BC ⊗ H^{∗}BG satisfies

Δ(x) = u^{l} ⊗ 1 + 1 ⊗ x + ∑_{j=1}^{l−1} u^{j} ⊗ x_{j} for suitable elements x_{j} ∈ H^{∗}BG if p = 2,
Δ(x) = (βu)^{l} ⊗ 1 + 1 ⊗ x + ∑_{j=1}^{l−1}(βu)^{j} ⊗ x_{j} for odd primes.

Proof: If the rank of C is one this is an immediate consequence of Proposition 2.1. In the general case consider u as a homomorphism C ⟶ Z/p. Let C_{1} be its kernel. C_{1} is central in G, in particular normal. Hence G/C_{1} is a compact Lie group and C/C_{1} is central in G/C_{1}. We get a commutative diagram in which the vertical arrows are canonical projections and the horizontal arrows are given by multiplication:

C × G  ⟶ G
↓  ↓
C/C_{1} × G/C_{1} ⟶ G/C_{1}.

The diagram shows that it is enough to consider the rank 1 case C/C_{1} ↪ G/C_{1} and hence we are done. □ Proof of Theorem 1.1: We give the proof for p=2 and leave the easy modifications in the case of an odd prime to the reader. We choose a basis u_{1}, …, u_{n} of H^{1}BC. Lemma 2.2 provides us with elements x_{1}, …, x_{n} in H^{∗}BG such that:

Δ(x_{i}) = u_{i}^{l_{i}} ⊗ 1 + 1 ⊗ x_{i} + ∑_{j=1}^{l_{i}−1} u_{i}^{j} ⊗ x_{i,j}

for suitable natural numbers l_{i}, 1 ≤ i ≤ n and elements x_{i,j} ∈ H^{∗}BG, 1 ≤ i ≤ n, 1 ≤ j ≤ l_{i} − 1. We claim that the elements x_{1}, …, x_{n} form a regular sequence on H_{G}^{∗}X.

The special form of Δ(x_{i}) implies that the coaction Δ_{X}: H_{G}^{∗}X ⟶ H^{∗}BC ⊗ H_{G}^{∗}X induces for 1 ≤ i ≤ n a homomorphisms of algebras

Δ_{X,i}: \frac{H_{G}^{∗}X}{(x_{1}, …, x_{i−1})H_{G}^{∗}X} ⟶ \frac{H^{∗}BC}{(u_{1}, …, u_{i−1})H^{∗}BC} ⊗ \frac{H_{G}^{∗}X}{(x_{1}, …, x_{i−1})H_{G}^{∗}X}

Now consider a non-trivial element y ∈ H_{G}^{∗}X/(x_{1}, …, x_{i−1})H_{G}^{∗}X. We have to show that x_{i} · y is non-trivial and for this it suffices to show that Δ_{X,i}(x_{i} · y) is non-trivial.

For this we write Δ_{X,i}(y) = ∑_{r≥0} y_{r} and Δ_{X,i}(x_{i} · y) = ∑_{r≥0} z_{r} with y_{r} and z_{r} in (H^{∗}BC/(u_{1}, …, u_{i−1})H^{∗}BC)^{r} ⊗ H_{G}^{∗}X/(x_{1}, …, x_{i−1})H_{G}^{∗}X. Then y_{0} = 1 ⊗ y is nontrivial. Let r_{1} be maximal such that y_{r_{1}} is non-trivial. Since Δ_{X,i}(x_{i} · y) can be calculated from Δ(x_{i}) and Δ_{X,i}(y), the formula for Δ(x_{i}) implies that for r_{2} = r_{1} + l_{i} we get z_{r_{2}} = (u_{i}^{l_{i}} ⊗ 1) · y_{r_{1}}. This is clearly non-trivial and hence we are done. □

2.3 Remark. Theorem 1.1 remains true in the following cases (cf. [Q2]):

(a) G is a discrete group of finite virtual cohomological dimension.

(b) G is a profinite group, X is a point and H^{∗}BG is replaced by the continuous mod p cohomology of G.

The proof is the same as before. We only have to note that Proposition 2.1 remains true in these cases if H is isomorphic to Z/p ([Q2].)

Proof of Proposition 1.2: As we observed above the restriction map H_{G}^{∗}X ⟶ H^{∗}BC is a map of H^{∗}BC comodule algebras hence its image I is a sub comodule algebra of H^{∗}BC. Then the commutativity of the diagonal in H^{∗}BC implies that its restriction to I has image in I ⊗ I and therefore I is a sub Hopf algebra of H^{∗}BC. □

3. Proof of Theorem 1.4

We introduce the following notation. The dimension of a maximal torus T_{max} of a compact Lie group G will be called the Lie-rank of G and denoted by r_{L}(G), the maximal rank of the elementary abelian p-subgroups E of G will be called the p-rank of G and denoted by r_{p}(G).

The following lemma reduces the proof of Theorem 1.4 to a purely Lie group theoretical question.

3.1 Lemma. Let G and C be as in Theorem 1.4. Then G is isomorphic to a product E ×K where E is an elementary abelian p-subgroup and K is a compact Lie subgroup of G with the following properties:

(a) C ∩ K is central and detecting in K and contains all elements of K of order p.

(b) H^{∗}BK is concentrated in even degrees.

(c) The Lie-rank of K equals the p-rank of K.

Proof: (a) The following sequence of isomorphisms

(3.1.1)  H^{1}BG ≅ Hom(H_{1}BG, Z/p) ≅
≅ Hom(π_{1}BG, Z/p) ≅ Hom(π_{0}G, Z/p) ≅ Hom(G, Z/p)

shows that there is a homomorphism G ⟶^{φ} V for some elementary abelian p-group V such that φ induces an isomorphism H^{1}BV ⟶^{≅} H^{1}BG. Because C detects H^{∗}BG there is a subgroup E of C such that the composition E ↪ G ⟶^{φ} V is an isomorphism. Let K = ker φ. Then the centrality of E implies G ≅ E × K and C ∩ K is clearly central and detects H^{∗}BK. By Remark 1.6 it is also maximal and hence contains each element of K of order p.

(b) We denote C ∩ K by D. By Proposition 1.2 H^{∗}BK is a sub Hopf algebra of H^{∗}BD. If H^{∗}BK is not concentrated in even degrees then any non-trivial element x of smallest odd degree is primitive in H^{∗}BK, hence also in H^{∗}BD. Consequently x has degree one. However, H^{1}BK is trivial by construction and we arrive at a contradiction.

(c) Clearly r_{L}(K) ≤ r_{p}(K) and it suffices to show that the rank of D is less than or equal to r_{L}(K). From (b) and the universal coefficient theorems we conclude that the rational cohomology ring H^{∗}(BK; Q) and H^{∗}BK have the same Poincaré series, i.e. P_{H^{∗}(BK;Q)}(t) := ∑ dim_{Q} H^{m}(BK; Q) · t^{m} and P_{H^{∗}BK}(t) := ∑ dim_{F_{p}} H^{m}(BK; F_{p}) · t^{m} agree.

Furthermore H^{∗}(BK; Q) embedds into H^{∗}(BT_{max}; Q) (see [Bor] Prop. 20.4.) and H^{∗}BD is finitely generated over H^{∗}BK by Proposition 2.1. This allows us to compare the Poincaré series of H^{∗}BD and H^{∗}(BT_{max}; Q), namely P_{H^{∗}BD}(t) = p(t) · P_{H^{∗}BK}(t) ≤ p(t) · P_{H^{∗}(BT_{max};Q)}(t), where p(t) is a polynomial, and the inequality is understood in the sense that each coefficient in the left hand side series is smaller than or equal to the corresponding coefficient in the right hand side. It follows that the rank of D is less than or equal to the dimension of T_{max}. □

3.2 Remark. Corollary 1.5 is obviously implied by Lemma 3.1 (c).

Because of Lemma 3.1 and 3.1.1 Theorem 1.4 is implied by the following two results.

3.3 Proposition. Suppose G is a compact Lie group with the following property:

(a) Each element of order p is central.

Then the connected component G_{1} of the unit in G is isomorphic to a product T × S of a torus T and a product S of three dimensional spheres (which is trivial in case p is odd.)

3.4 Proposition. Suppose G is as in 3.3 and in addition satisfies:

(b) There are no non-trivial homomorphisms from G to Z/p.

(c) The Lie-rank of G equals the p-rank of G.

Then G is isomorphic to a product N × S where N and S are as in Theorem 1.4.

Proof of Proposition 3.3: Note that (a) is clearly inherited by G_{1}. Now consider the Weyl group W of G_{1}. This is isomorphic to a finite subgroup of GL_{n}(Z) if n is the dimension of T_{max}. Because W fixes the elements of order p in T_{max}, W is in the kernel of the reduction homomorphism GL_{n}(Z) ⟶ GL_{n}(Z/p). Hence W is trivial or an elementary abelian 2-group (if p=2) by the following result of Minkowski [Min1,2]. For the convenience of the reader we will provide a short proof of it.

3.5 Lemma. Suppose M is an element of finite order in GL_{n}(Z) whose reduction to GL_{n}(Z/p) is the identity. If M ≠ 1 then p = 2 and M has order two.

Proof: If M is non-trivial we write M = 1 + p^{r}N for some integer r ≥ 1 and some integral matrix N with N ≢ 0 mod p. If q is a prime then the binomial theorem shows that M^{q} = 1 is only possible if p^{r} = q = 2, in which case we have M^{2} = 1 + 4(N + N^{2}). In particular, M^{2} ≠ 1 implies M^{4} ≠ 1 and hence the order of M must be two. □

We return to the proof of 3.3 and use some structure theory for compact connected Lie groups (cf. [BtD] Chap. V and [Bou] Chap. VI.) If W is isomorphic to (Z/2)^{k} then G_{1} has a finite cover G̃_{1} which is isomorphic to a product T × S where T is an (n − k) dimensional torus and S is a k-fold product of three dimensional spheres. By the Lemma G̃_{1} ≅ T if p is odd.

We know at this point that G_{1} is isomorphic to the quotient of T ×S by a finite central subgroup K. We will show that K is contained in T and hence G_{1} ≅ T /K × S ≅ T × S as claimed.

If p is odd there is nothing to be shown. So assume p = 2. The connectivity of G̃_{1} implies that the preimage of the center Z(G_{1}) under the covering projection is contained in the center Z(G̃_{1}) of G̃_{1}. Now the center of S^{3} is {±1} so Z(G̃_{1}) is isomorphic to T ×(Z/2)^{k}. If K is not contained in T then there is an element x ∈ K such that x has a non-trivial component in (Z/2)^{k}. Now T × S is a divisible group, hence there is an element y ∈ T × S such that y^{2} = x. Then the projection ȳ of y to G_{1} has order two, but ȳ is not central because y is not central. This contradicts (a) and finishes the proof of 3.3. □

Proof of Proposition 3.4: By 3.3 we know that G sits in an exact sequence 1 ⟶ T × S ⟶ G ⟶ π ⟶ 1 where π is a finite group which acts trivially on the elements of order p in T × S.

Claim 1. T is central in G and each factor S^{3} is normal in G. Proof: Certainly T is normal in G because it is the connected component of the center of T × S. The conjugation action of G on T defines a homomorphism π ⟶^{θ} Aut T ≅ GL_{n}(Z) and by Lemma 3.5 this can be non-trivial only if p = 2 and the image of θ is an elementary abelian 2-group. However, this contradicts the assumption that for p = 2 there are no non-trivial homomorphisms from G to Z/2.

Now we consider S. Because there are no non-trivial homomorphisms from S^{3} to S^{1} we see that S is normal. It is not hard to show that the group Aut(S) of automorphisms of S is canonically isomorphic to the wreath product Aut(S^{3}) ∫ Σ_{k}, Σ_{k} denoting the symmetric group on k letters and k being the number of factors S^{3} in S. By condition 3.3 (a) the conjugation action of G on S fixes the center of S and hence each factor S^{3} is normal. □

Claim 2. The order of π is prime to p. Proof: We discuss the simpler case of an odd prime first. Assume x̄ is an element of order p in π and let x be a preimage of x̄ in G_{1} ≅ T. Because x commutes with T, the subgroup H of G generated by T and x is abelian. Furthermore, T beeing divisible is an injective abelian group. Hence H is isomorphic to T × Z/p contradicting condition (c).

Now consider the case p = 2 and assume that x̄ is an element of π of order two. If x is any lift of x̄ to G then x need not leave a maximal torus T_{max} invariant. However, since all maximal tori are conjugate in G_{1} we can modify the lift so that T_{max} will be invariant under x. Next we take our maximal torus to be a product of T and maximal tori in each of the S^{3} factors. By claim 1 we know that x fixes T and normalizes the torus of each factor S^{3}. It follows that x acts like an element of the normalizer of T_{max} in G_{1} and hence we can find a lift x of x̄ which fixes T_{max}. Then x^{2} centralizes T_{max} and it is in G_{1}, hence x^{2} is in T_{max}. Now we continue as in the case of an odd prime and see that the subgroup H generated by T_{max} and x is isomorphic to T_{max} × Z/2 contradicting (c). □

Note that claim 1 and 2 settle the case of an odd prime. So from now on we can concentrate on the case p = 2. By claim 1 we know that T is a normal subgroup of G.

Claim 3. G/T is isomorphic to S × π.

Proof: Clearly G/T sits in an exact sequence 1 ⟶ S ⟶ G/T ⟶ π ⟶ 1. Furthermore we know that π has odd order and acts trivially on the center Z(S) of S which agrees with the elements of order two in S. In particular we find H^{2}(π; Z(S)) = 0. Furthermore the homomorphism from π to the group Out(S) of outer automorphims of S associated to the exact sequence is trivial because π fixes Z(S) and Out(S) = Σ_{k} (use Out(S^{3}) = 1 and Aut(S) ≅ Aut(S^{3}) ∫ Σ_{k}.) Now the proof of the classification theorem of extensions with non abelian (discrete) kernel ([Mac] Chap. IV, Thm. 8.8) works equally well in this situation and we conclude that the extension splits as claimed. □

Now we finish the proof of Proposition 3.4. By claim 1 we know that S is normal and by claim 3 we have a retraction ρ: G ⟶ S which shows that G is isomorphic to S × ker ρ. Finally ker ρ sits in an exact sequence 1 ⟶ T ⟶ ker ρ ⟶ π ⟶ 1. By claim 1 this is a central extension and by claim 2 the order of π is odd. □

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Carlos Broto  Hans-Werner Henn
Departament de Matemàtiques  Mathematisches Institut
Universitat Autònoma de Barcelona  Universität Heidelberg
08193 Bellaterra (Barcelona) Spain.  Im Neuenheimer Feld 288

6900 Heidelberg, Germany.