G-HOMOTOPY EQUIVALENCES OF SUBGROUP POSETS

JOHN MARTINO AND STEWART PRIDDY

Let G be a compact Lie group. We study the G-homotopy properties of posets of subgroups of G partially ordered by inclusion:

C, the closed subgroups of G;

P ⊆ C, the non-trivial p-toral subgroups;

B ⊆ P, the p-reduced subgroups, i.e., the Q ∈ P such that the Weyl groups W_{G}(Q = N_{G}(Q)/Q) contain no non-trivial normal p-toral subgroup.

Φ ⊆ P, the non-trivial p-toral subgroups with finite Weyl groups.

S = Φ ∩ B, the p-stubborn subgroups.

The Lie group G acts on each poset by conjugation and hence upon the geometric realizations of their nerves.

Theorem 0.1. The inclusions (1) i : B → P

(2) i : S → Φ are G-homotopy equivalences.

We note for G = SU(2) the inclusion S → P is not a G-homotopy equivalence. This can be seen by examining the induced map on fixed points ϕ → Z(G) which is not an equivalence.

We begin with some preliminaries on the topology of the subgroups of G, [1]. Under inclusion C is a poset which is a category with Obj(C) the elements of C and

Mor(C) = ⋃_{H,K∈Obj(C)} mor(H, K)

where mor(H, K) = {H ≤ K} is the singleton set if H ≤ K and empty otherwise. Moreover, C is a topological category, i.e., the object and morphism sets are topological spaces and the structure maps are continuous maps. In fact G has a bi-invariant metric which can be used to give the object set the structure of a metric space with the Hausdorff metric [1] which we denote by d. Let δ be a bi-invariant metric on G. For H and K closed subgroups of G, define δ(x, K) = inf{δ(x, y) | y ∈ K} and r(H, K) = sup{δ(x, K) | x ∈ H}. Finally define

Date: March 23, 2009.

d(H, K) = max{r(H, K), r(K, H)}.

The morphism sets are also metric spaces under

D(H ≤ K, H′ ≤ K′) = d(H, H′) + d(K, K′).

Both d and D are bi-invariant. It is clear from this definition that the structure maps are continuous and G acts continuously. The poset C is a compact topological category and the orbit space C/G is a countable, compact, totally disconnected, Hausdorff space [1].

Proposition 0.2 ([1, IV, Prop. 3.3]). Let (H) = lim(H_{i}) in C/G. Then there exists an n_{0} and K_{n} ∈ C for n ≥ n_{0} such that (K_{n}) = (H_{n}), K_{n} ≤ H, and lim K_{n} = H.

Proposition 0.3. The orbit space B/G is finite.

Proof: This follows from the proof of Theorem 2.4 in [2] where Jackowski, McClure, and Oliver show that S has only finitely many conjugacy classes of objects. Their proof does not depend on the Weyl groups of the objects being finite. □

Corollary 0.4. B is a closed subset of C.

Proof: This is immediate since B is the preimage of a closed subset of C/G. □

We will need

Lemma 0.5. For G a compact Lie group and Q a p-toral subgroup of G, then a spherical neighborhood of radius ϵ > 0 can be chosen small enough so that if S ∈ N_{ϵ}(Q), then there is a subgroup R such that (R) = (S), R ∈ N_{ϵ}(Q), R ≤ Q, and N_{G}(R) ≤ N_{G}(Q).

Proof: If not then by compactness (specifically that any sequence has a convergent subsequence), this will contradict Proposition 0.2. Also, from the proof of [2, Prop. 1.6] we may assume N_{G}(R) ≤ N_{G}(Q_{n,i}). □

To prove Theorem 0.1 we will need the following proposition to contract P to a G-poset on a finite number of G-conjugacy classes. More precisely, we will cover P with a finite number of disjoint Gneighborhoods each within a tube about a subgroup and then contract to the G-orbit of that subgroup by the functor ϕ defined in the proof of Lemma 0.8.

We begin by fixing a maximal p-toral subgroup P of G. We choose a complete set R of conjugacy-class representatives of P so that each representative is a subgroup of P.

Proposition 0.6. There exists a finite set X = {Q_{k,i}} ⊂ R of p-toral subgroups of G and numbers ϵ_{k} > 0 depending only on the rank k of Q_{k,i} such that the spherical neighborhoods {N_{ϵ_{k}}(Q_{k,i})} are mutually disjoint, and their union meets all the conjugacy classes of P. Furthermore,

(1) If S ∈ Obj(P(G)) then there is a Q_{k,i} ∈ X and a subgroup R ≤ Q_{k,i} such that (S) = (R), R ∈ N_{ϵ_{k}}(Q_{k,i}), and N_{G}(R) ≤ N_{G}(Q_{k,i}). If S ∈ B, Q_{k,i} can be taken in X ∩ B.

(2) For Q_{i}, Q_{j} ∈ X, R_{i} ∈ N_{ϵ_{i}}(Q_{i}) ∩ R, and R_{j} ∈ N_{ϵ_{j}}(Q_{j}) ∩ R, if R_{i} ≤ R_{j}, then Q_{i} ≤ Q_{j}.

Proof: For 0 ≤ k ≤ n, let B(k) consist of all the subgroups in R ∩ B of rank k.

Let n = rank(G), the dimension of a maximal torus. We note there are only a finite number of conjugacy-class representatives of p-toral subgroups of rank n since the Weyl group is finite. Let C(n) consist of all the subgroups of rank n in R, that are not in B. Let X(n) = B(n)∪C(n) = {Q_{n,i}}. Let Δ = min{d(B, B′) | B, B′ ∈ R∩B, B ≠ B′} and choose ϵ < Δ/2 so that the neighborhoods N_{ϵ}(Q_{n,i}) are disjoint. Let ϵ_{n} = ϵ.

We claim all but a finite number of conjugacy classes of the rank (n − 1) subgroups of P meet U_{n} = ⋃_{Q_{n,i}∈X(n)} N_{ϵ_{n}}(Q_{n,i}). To see this, suppose {(R_{j})} is an infinite sequence of distinct classes lying outside U_{n}. By compactness of C/G there is a (R) ∈ C/G such that (R_{i}) → (R) and (R) ∩ U_{n} = ∅. Furthermore for large i we may assume R_{i} ≤ R, by Proposition 0.2. R contains a maximal p-toral subgroup S and thus there is a conjugate convergent subsequence R′_{j} ≤ S. By biinvariance of the metric, lim R_{j} = R, S = R, and so R is p-toral. The rank(R) = n − 1 or n since in the limit rank cannot decrease. If rank(R) = n − 1 then R_{i} = R for large i contradicting the distinctness of the R_{i}. Thus rank(R) = n, hence (R)∩U_{n} ≠ ∅, also a contradiction.

Let C(n − 1) be the subgroups in R, not in B ∪ U_{n}, of rank n − 1. Let X(n − 1) = B(n − 1) ∪ C(n − 1) = {Q_{n−1,j}}, a finite set.

Choose an ϵ > 0 at least as small as ϵ_{n} so that the N_{ϵ}(Q_{n−1,j}) are mutually disjoint among themselves and with the N_{ϵ_{n}}(Q_{n,i}). As before we may also require that the conclusions of Lemma 0.5 hold. Set ϵ_{n−1} = ϵ.

Continuing in this manner by a finite downward induction on the rank of the p-toral subgroups constructs for each k, 0 ≤ k ≤ n the finite set X(k) = B(k) ∪ C(k) = {Q_{k,j}} ⊂ R of subgroups with neighborhoods N_{ϵ_{k}}(Q_{k,j}). Since each X(k) is finite we can argue as above that ϵ_{k} can be chosen so the conclusions of Lemma 0.5 hold.

Define X = ⋃_{k=0}^{n} X(k). By construction the collection {N_{ϵ_{k}}(Q_{k,i})} meets every conjugacy class of the objects of P. Thus (1) holds by construction. Furthermore by Lemma 0.5 for each Q_{k,i}, the conjugacyclass representatives in R ∩ N_{ϵ_{k}}(Q_{k,i}) may be replaced with subgroups R < Q_{k,i} such that N_{G}(R) ≤ N_{G}(Q_{k,i}).

We now consider (2). We identify P/G with R. Extend the partial order by inclusion of subgroups to R < Q iff d(R, P) > d(Q, P). Then extend this partial order to a linear order (reference?) and consider the interval (R, Q] where R < Q. If ϵ is small enough the subgroups of N_{ϵ}(Q) are subgroups of Q by Lemma 0.5. If ϵ < d(R, Q) then N_{ϵ}(Q) ⊆ (R, Q]. Similarly for S ∈ (R, Q) and ϵ small enough N_{ϵ}(S) ⊆ (R, Q]. Therefore the intervals (R, Q] are open in the metric topology on R. Thus we can assume the neighborhoods N_{ϵ_{k}}(Q_{k,i}) belong to disjoint intervals and so (2) follows directly.

□

Definition 0.7. Choose ϵ_{k} > 0, X = {Q_{k,i}} as in the proof of the Proposition. Let I = ⟨X⟩ be the G-set generated by X. Then I ⊆ P is a topological poset.

Note, that since X contains a representative of every conjugacy class in B, it follows that B ⊆ I.

Lemma 0.8. The inclusion i : I → P is a G-homotopy equivalence.

Proof: We define a functor

ϕ : P → I.

For R ∈ R∩N_{ϵ_{k}}(Q_{k,i}), define ϕ(R) = Q_{k,i}. Then extend ϕ equivariantly to the conjugacy class of R, i.e., ϕ(gR) = gQ_{k,i}. Since ⋃_{Q_{k,i}∈X} N_{ϵ_{k}}(Q_{k,i}) meets all the conjugacy classes of P this defines ϕ on all of P. Recall from the proof of Proposition 0.6 that R ≤ Q_{k,i} and N_{G}(R) ≤ N_{G}(Q_{k,i}) so that ϕ is a well-defined, surjective function.

To show the functor ϕ : P → I is continuous on objects fix Q = Q_{k,i} ∈ Obj(P), ϵ = ϵ_{k}. Consider R ∈ R ∩ N_{ϵ}(Q) and let V be an open set about ϕ(R) = Q. Choose an open ball B ⊆ G about 1 small enough so that B·Q ⊆ V and B·R ⊆ N_{ϵ}(Q). Then ϕ(B·R) = B·Q ⊆ V. Hence ϕ is continuous at R. Since the action of G on Obj(I) is continuous and by definition ϕ is extended equivariantly to subgroups in the conjugacy class of R, i.e., ϕ(gR) = gϕ(R), it follows that ϕ is continuous on Obj(I).

The function ϕ is also continuous on morphisms, since the map on morphisms is a restriction of (Obj(P))^{2} ⟶^{ϕ^{2}} (Obj(I))^{2}. The map ϕ is a functor by conclusion (2) of Proposition 0.6. Furthermore, Q ≤ ϕ(Q) for all Q; hence by an argument analogous to [3, Cor. 1.2], ϕ : P → I is G-homotopy equivalence having i : I → P as a G-homotopy inverse. □

Lemma 0.9. P_{>Q} is G_{Q}-contractible if and only if Q ∈ P − B.

Proof: This is the same argument as in Lemma 2.1 of [3]. □

Lemma 0.10. The inclusion j : B → I is a G-homotopy equivalence.

Proof: Since I has only finitely many conjugacy classes we may apply an argument analogous to [3, Prop. 1.7] and Lemma 0.9 to conclude j : B → I is a G-homotopy equivalence. □

Proof of Theorem 0.1: (1) By Lemmas 0.8, 0.10, the inclusion B → P factors into a composite of G-homotopy equivalences i, j.

(2) A similar argument for S → I ∩Φ → Φ shows that this inclusion too is a G-homotopy equivalence. □

References

[1] T. tom Dieck, Transformation Groups, Walter de Gruyer, Berlin, 1987.

[2] S. Jackowski, J. McClure, and B. Oliver, “Homotopy classifications of self maps of BG via G-actions,” Ann. of Math., 135, (1992), 183–224.

[3] J. Thévenaz and P. Webb, “Homotopy equivalences of posets with a group action,” J. of Combin. Th., Ser. A, 56, (1991), 173–181.

Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008

  E-mail address: john.martino@wmich.edu

Department of Mathematics, Northwestern University, Evanston, IL 60208

  E-mail address: priddy@math.northwestern.edu