Maps of BZ/pZ to BG

William G. Dwyer Clarence W. Wilkerson^{∗}

The purpose of this note is to give an elementary proof of a special case of the result of [Adams, Lannes_{2}, Miller-Wilkerson] characterizing homotopy classes of maps from the classifying space of an elementary p-group into the classifying space of a connected Lie group. Our result states

Theorem I: If G is a connected compact Lie group, then the natural map

α : Hom_{grp}(Z/pZ, G)/(∼) ⟶ [BZ/pZ, BG]

is a bijection, where ∼ is equivalence up to G - conjugation.

The previous proofs relied on Quillen’s description of H^{∗}(BG, F_{p}) as an inverse limit of certain algebras indexed over the elementary p-groups of G. The present proof does not use this description, but has in common with the other proofs the use of Lannes’ identification [Lannes_{1}]

Hom_{A−alg}(H^{∗}(X), H^{∗}(BZ/pZ)) ≈ [BZ/pZ, X]

for suitable X.

∗The research of both authors was partially supported by the NSF, and that of the second author by sabbatical funds from Wayne State University.

In spite of its limitation to rank one elementary p-groups, Theorem I is exactly the input needed by the machinery of [Dwyer-Zabrodsky] in the general p-group case:

Theorem II: If π is a finite p-group and G is a connected Lie group, then

α : Hom_{grp}(π, G)/(∼) ⟶ [Bπ, BG]

is a bijection, where ∼ denotes equivalence up to G-conjugation.

Finally, in the last section, counterexamples are presented to α being an isomorphism for general finite source groups :

Theorem III: If the target group G is SU(2), then α is not surjective if π is the symmetric group of degree 3, and not injective if π is the cyclic group of order 15.

That is, the behavior observed for SL(2, 5) by Milnor and Adams is generic for non-p-groups.

Although the proof of Theorem I avoids Quillen’s main result, the study of the action of Z/pZ on the flag manifold G/T is very much in the spirit of [Quillen].

1. Obtaining Homomorphisms from Maps

Fix a prime p, let π be a cyclic group of order p, and let G be a compact connected Lie group. The purpose of this paragraph is to prove the following proposition:

\underline{\underline{1.1 Proposition}} Given a map f : Bπ ⟶ BG, there exists a homomorphism φ : π ⟶ G such that f is homotopic to Bφ.

\underline{\underline{Notation}} The symbol H^{∗}( −) will stand for H^{∗}( −; Z/p), A - alg will denote the category of unstable algebras over the mod p Steenrod algebra A, and [U, V] will stand for the set of homotopy classes of unpointed maps from U to V.

The proof of 1.1 depends mainly on the following result:

\underline{\underline{1.2 Proposition}} Let X be a p-complete space with mod p cohomology of finite type, and let q : E ⟶ Bπ be a fibration over Bπ with fiber X. Then the space of sections Γ(q) of q is non-empty iff the map q^{∗} : H^{∗}(Bπ) ⟶ H^{∗}(E) is a monomorphism.

\underline{\underline{Proof}} It is obvious that q^{∗} is a monomorphism if there is a section. Suppose now that q^{∗} is a monomorphism. The space E is p - complete (since the monodromy action of π on H^{∗}(X) is necessarily nilpotent) so by [Lannes_{1}] the map

q_{♯} : [Bπ, E] ⟶ [Bπ, Bπ]

is naturally isomorphic to the induced map

Hom_{A−alg}(H^{∗}E, H^{∗}(Bπ)) ⟶ Hom_{A−alg}(H^{∗}(Bπ), H^{∗}(Bπ)).

Since q^{∗} is a monomorphism and H^{∗}(Bπ) is an injective object of A-alg, it is possible to find a dotted arrow in A-alg to make the following diagram commute:

[figure: H^{∗}(Bπ); \underrightarrow{q^{∗}}; H^{∗}E; id; H^{∗}(Bπ)]

This dotted arrow corresponds to a map Bπ ⟶ E which projects under q_{♯} to the identity element of [Bπ, Bπ]; such a map immediately gives a section of the fibration q.

\underline{\underline{1.3 Corollary}} If X is a simply-connected space of finite type and q : E ⟶ Bπ is a fibration over Bπ with fiber X, then the space of sections Γ(q) of q is nonempty iff q^{∗} : H^{∗}(Bπ) ⟶ H^{∗}E is a monomorphism.

\underline{\underline{Proof}}: Consider the commutative diagram

[figure: [Bπ, E]; [Bπ, Bπ]; q_{♯}; (q′)_{♯}; [Bπ, E′]]

where E′ \underrightarrow{q′} Bπ is the fibration with fiber X_{p}^{∧} obtained from E by fiberwise p-completion. The fiber of the map E ⟶ E′ is the same as the fiber of the map X ⟶ X_{p}^{∧}, and is therefore a connected simple space with uniquely p-divisible homotopy groups. Obstruction theory shows that [Bπ, E] ⟶ [Bπ, E′] is an isomorphism, so the corollary follows immediately from 1.2.

\underline{\underline{1.4 Lemma}} If T is a compact torus and f : Bπ ⟶ BT a map, then there exists a homomorphism φ : π ⟶ T such that f is homotopic to Bφ.

\underline{Proof} This is clear.

\underline{\underline{Proof of 1.1}} Let T be the maximal torus of G. By 1.4, it is enough to find a lifting up to the homotopy in the diagram

[figure: Bπ; BG; BT; f]

The homotopy fiber of BT ⟶ BG is the flag manifold G/T, so finding the indicated dotted arrow is equivalent to finding a section of the fibration E ⟶ Bπ with fiber G/T induced by f. This fibration is simple because BG is simply connected. Since H^{∗}(G/T; Z) is torsion-free and both H^{∗}(G/T; Z) and H^{∗}(Bπ; Z) are concentrated in even dimensions, the integral cohomology Serre spectral sequence of E ⟶ Bπ collapses. Reducing mod p shows that the mod p cohomology Serre spectral sequence of E ⟶ Bπ also collapses; in effect, the E_{2} term of the spectral sequence is generated over H^{∗}(Bπ) by the reduction of integral classes. It follows that H^{∗}(Bπ) ⟶ H^{∗}(E) is a monomorphism, so the proof can be completed by an application of 1.3.

2. Homotopy Implies Conjugacy

Let p, π and G be as in §1, but drop the assumption that G is connected.

\underline{\underline{2.1 Proposition}} If φ_{1}, φ_{2} : π ⟶ G are homomorphisms such that Bφ_{1} is homotopic to Bφ_{2}, then φ_{1} is conjugate to φ_{2}.

\underline{\underline{2.2 Lemma}} If X is a finite dimensional π-complex, then the fixed-point set X^{π} is non-empty iff the homotopy fixed-point set X^{hπ} is non-empty.

\underline{\underline{Proof of 2.2}} If X^{π} is empty, then X ×_{π} Eπ is a homotopy equivalent to the finite-dimensional space X/π, so that the projection X ×_{π} Eπ ⟶ Bπ has no section, i.e., X^{hπ} = ϕ. If X^{π} ≠ ϕ, then X^{hπ} ≠ ϕ by virtue of the natural map X^{π} ⟶ X^{hπ}.

\underline{\underline{Proof of 2.1}} Consider the solid arrow diagram

[figure: Bπ; BG; Bπ; Bφ_{1}; Bφ_{2}]

in which Bφ_{2} is thought of as having been replaced by a fibration. If Bφ_{1} is homotopic to Bφ_{2}, then the indicated dotted arrow exists. The fiber of the map from Bπ to BG is, up to homotopy, G/φ_{2}(π), so the existence of the dotted arrow indicates that the bundle over Bπ associates to the left translation action of π on G/φ_{2}(π) via φ_{1} has a section; i.e., that this action of π on G/φ_{2}(π) possesses a homotopy fixed point. By 2.2, the action has a genuine fixed point. This fixed point corresponds exactly to an element of G which conjugates φ_{1}(π) to a subgroup of φ_{2}(π). By symmetry it is possible to find another element of G which conjugates φ_{2}(π) to a subgroup of φ_{1}(π); this leads easily to the desired conclusion.

3. The Counterexamples D_{2q} and Z/15Z

If π is not a p-group, then it is quite possible that α is not onto or monic. Milnor observed that for π = SL(2, 5) and G = SU(2), α is not onto. A discussion of this case appears in [Adams_{2}, Adams_{3}]. In fact, in this section it is shown that α is not onto for the dihedral groups {D_{2q}, q an odd prime} and that α is not monic for {Z/pqZ, p and q distinct odd primes}, with target group SU(2). Thus Σ_{3} ≈ D_{6} is the smallest group for which α is not onto. On the other hand while α is not monic for Z/15Z, this may not be a minimal example.

The main theoretical tool is Proposition 3.1 below, which states that there is no “coherence” for the mapping sets at various primes. The proof of Proposition 3.1 and Proposition 3.2 are deferred to the end of the section.

\underline{\underline{3.1 Proposition}} If π is a finite group, and G is a compact connected Lie group, then

[Bπ, BG] = ∏_{p∈P}[(Bπ)_{p}^{∧}, BG_{p}^{∧}]

where ( )_{p}^{∧} denotes the Bousfield-Kan completion functor.

Proposition 3.2 below allows us to compute (Bπ)_{p}^{∧} in some favorable cases.

\underline{\underline{3.2 Proposition}} If the p-Sylow subgroup S_{p}(π) is abelian, then

(Bπ)_{p}^{∧} ≃ (BN_{π}(S_{p}(π)))_{p}^{∧} ≃ (BS_{p}(π) ×_{W} EW)_{p}^{∧}

where W = N_{π}(S_{p}(π)/π.

\underline{\underline{3.3 Example}} For D_{2q} = \{Z/qZ ><Z/2Z.\}, for q prime ≠ 2, 1) ♯(Hom(D_{2q}, SU(2))/ ∼) = 2

2) ♯[BD_{2q}, BSU(2)] = 2 · (\frac{q + 1}{2}) = q + 1

3) ♯[(BD_{2q})_{2}^{∧}, BSU(2)_{2}^{∧}] = ♯[BZ/2Z, BSU(2)] = 2

Thus for D_{6}, α is not onto.

\underline{\underline{3.4 Example}} Let G = SU(2) and π = Z/pqZ, where p and q are distinct odd primes.

1) ♯(Hom(Z/pqZ)/ ∼) = 1 + (pq − 1)/2 = (pq + 1)/2

2) [BZ/pqZ, BSU(2)] = [BZ/pZ, BSU(2)] × [BZ/qZ, BSU(2)]

so [BZ/pqZ, BSU(2)] = ((1 + p)/2)(1 + q)/2. Thus for Z/pq, α is not monic.

A purely formal argument allows examples 3.3 and 3.4 to be combined into one common counterexample.

\underline{\underline{3.5 Proposition}} For a fixed G, if α is not monic for π, it is not monic for π × π′. If α is not onto for π′, then it is not onto for π × π′.

\underline{\underline{3.6 Example}} : For G = SU(3), and π = Σ_{3} × Z/15Z, α is neither onto nor monic.

Demonstration of Example 3.3

Since the center of SU(2) is Z/2Z, and contains all elements of order 2, any homomorphism φ : D_{2q} ⟶ SU(2) must factor through Z/2Z. Thus ♯(Hom(D_{2q}, SU(2))/ ∼) = 2. On the other hand,

(BD_{2q})_{2}^{∧} ≈ BZ/qZ by Prop. 3.2
and (BD_{2q})_{q}^{∧} ≈ (BZ/2Z) ×_{Z/2Z} EZ/2Z

so

[BD_{2q}, BSU(2)] = [BZ/2Z, BSU(2)_{2}^{∧}] × π_{0}(Map(BZ/qZ, BSU(2)_{2}^{∧})^{hZ/2Z}).

This latter set of components is by obstruction theory

[BZ/qZ, BSU(2)_{q}^{∧}]^{Z/2Z}] = [BZ/qZ, BSU(2)]^{Z/2Z}

which has (q+1)/2 elements. Demonstration of Example 3.4

Hom(Z/pqZ, SU(2))/ ∼= (Z/pqZ)/(action of ± 1)

so

♯(Hom(Z/pqZ, SU(2))/ ∼) = 1 + ((pq − 1)/2)
= (pq + 1)/2

On the other hand ♯[BZ/pZ, BSU(2)_{p}^{∧}] = ♯(Hom(Z/pZ, SU(2))/ ∼) = 1+(p−1)/2 = (p+1)/2

so by 3.1, ♯[BZ/pqZ, BSU(2)] = \frac{(p + 1)}{2} · \frac{(q + 1)}{2}

\underline{\underline{Proof of 3.1}} Apply the “arithmetic square” of Sullivan, Bousfield-Kan to BG.

BG ⟶ Π(BG)_{p}^{∧}
↓  ↓
BG_{0} ⟶ (Π(BG)_{p}^{∧})_{0}

After taking the unpointed maps from Bπ into this fiber square, one obtains the fiber square below :

BG^{Bπ} → ∏(BG_{p}^{∧})^{Bπ}
↓  ↓
BG_{0} ≃ BG_{0}^{Bπ} → ((∏ BG_{p}^{∧})_{0})^{Bπ} ≃ (∏ BG_{p}^{∧})_{0}

The Vietoris homotopy sequence of this fiber square has π_{1}((∏(BG_{p}^{∧}))_{0}) → π_{0}(BG^{Bπ}) → π_{0}((∏ BG_{p}^{∧})^{Bπ} × BG_{0}) → π_{0}((∏ BG_{p}^{∧})_{0}) But ∏(BG_{p}^{∧})_{0} is connected and simply connected and BG_{0} is connected, so 3.1 follows from exactness.

\underline{\underline{Proof of 3.2}}

A slightly nontrivial calculation of the stable elements in the sense of [Cartan-Eilenberg, p.257] in these cases with S_{p}(π) abelian shows that

H^{∗}(Bπ, Z/pZ) ≈ H^{∗}(BS_{p}(π), Z/pZ)^{N_{π}(S_{p}(π))/S_{p}(π)}
≈ H^{∗}(BN_{π}(S_{p}), Z/pZ).

\underline{\underline{Proof of 3.5}}

Since π is a retract of π×π′, and Bπ is a retract of Bπ×Bπ′, Hom(π, G)/ ∼ is a retract of Hom(π ×π′, G)/ ∼ and [Bπ, BG] is a retract of [Bπ ×Bπ′, BG]. Thus

Hom(π, G)/ ∼ \underrightarrow{α_{1}}  [Bπ, BG]
↑ i^{∗}  ↑ Bi^{∗}
Hom(π × π′, G)/ ∼ \underrightarrow{α_{2}} [Bπ × Bπ′, BG]
↑ p^{∗}  ↑ Bp^{∗}
Hom(π, G)/ ∼ \underrightarrow{α_{1}}  [Bπ, BG]

Now, if α_{1}φ = α_{1}φ′, then α_{2}p^{∗}φ = α_{2}p^{∗}φ′. But p^{∗}φ ≠ p^{∗}φ′ so α_{2} is not monic if α_{1} is not. If f : Bπ → BG is not in the image of α_{1}, then if Bp^{∗}f ∈ im α_{2}, then Bi^{∗}Bp^{∗}f = f ∈ im α_{1}, which is a contradiction.

References

[Adams 1] J.F. Adams, private communication, Fall 1985.

[Adams 2] J.F. Adams, Maps between classifying spaces, Enseigs. Math.(2), 24(1978), no. 1-2, pp.79-85.

[Adams 3] J.F. Adams, Maps between classifying spaces II, Invent. Math. 49(1978), no.1, pp 1-65.

[Cartan-Eilenberg] \underline{Homological Algebra}, Princeton University Press, Princeton, 1956.

[Dwyer-Zabrodsky] W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, 1986 Barcelona Conference Proceedings, Springer Lecture Notes in Mathamatics, to appear.

[Lannes 1] J. Lannes, Sur la cohomologie modulo p des p-groups abeliens elementaires, to appear.

[Lannes 2] J. Lannes, Cohomology of groups and function spaces, to appear.

[Miller-Wilkerson] H.R. Miller and C.W. Wilkerson, Maps of elementary p-groups into compact Lie groups, talk by C.W.Wilkerson at 1985 Yale Conference on Algebraic Topology.