The Orbit Space of the p-subgroup Complex is Contractible

Peter Symonds

The p-subgroup complex (or Brown complex or Quillen complex) was introduced by K.S. Brown [B]. It is defined for a group G and a prime p and will be denoted by S_{p}. It is a simplicial complex in which the n-simplices are chains of non-trivial finite p-groups (with strict inclusions):

Q_{0} < Q_{1} < Q_{2} < ⋯ < Q_{n},

with the face maps corresponding to inclusion of subchains. In other words, S_{p} is the geometric realisation of the poset of non-trivial p-subgroups of G.

This complex has played a prominent role in finite group theory since its introduction and the fundamental work of Quillen [Q]. For some more recent contributions see [ASe, ASm, KR, TW, W1, W2]. This paper consists of a proof of the following result.

Theorem. Let G be a finite group and p a prime which divides |G|. Let S_{p} denote the p-subgroup complex for G (considered as a topological space). Then S_{p}/G is contractible.

This was conjectured by Webb [W1, W2], who proved that S_{p}/G is mod-p acyclic. When G is a group of Lie type in characteristic p, then S_{p} is equivariantly homotopy equivalent to the Tits building of G, for which the orbit space consists of just one simplex, so the conjecture was known to be true. Various cases were also considered by Thévenaz [T], who showed that the conjecture held when G was p-solvable, or when the Sylow p-subgroup was either abelian, generalized quaternion or TI.

Instead of S_{p} we shall consider a subcomplex Δ, introduced by Robinson, in which the n-simplices are chains of p-groups (with strict inclusions), each one normal in the others:

Q_{0} ⊲ Q_{1} ⊲ Q_{2} ⋯ ⊲ Q_{n}, Q_{i} ⊲ Q_{n}, 0 ≤ i < n,

which we denote by (Q_{0}, ..., Q_{n}). This complex Δ does not arise from a partially ordered set, but it is equivariantly homotopy equivalent to S_{p} (and to various other subgroup complexes too) [TW], and we actually prove that Δ/G is contractible.

Now Δ is a simplicial complex, but Δ/G is naturally only a CW-complex. Each simplex of Δ is naturally oriented, because it is a chain. This orientation is preserved by G, and so induces an orientation on Δ/G.

Proof. We show that

a) π_{1}(Δ/G) = 1 and

b) H̃_{∗}(Δ/G; ZZ) = 0, and invoke Whitehead’s Theorem.

a) Let P be a Sylow p-subgroup of G. Any class x ∈ π_{1}(Δ/G, P) can be represented by a cellular loop s, i.e. a loop in the 1-skeleton which traverses each 1-cell at constant speed. This loop is determined by the sequence of directed 1-cells along which it travels.

Lift s to a cellular path s̃ in Δ starting at P and ending at some Sylow p-subgroup P′. Since Δ is a simplicial complex, s̃ is determined by the sequence of its vertices:

P → Q_{1} → Q_{2} → ⋯ → Q_{n} → P′.

There are two operations that we can perform on s̃ which do not change its image in π_{1}(Δ/G, P).

i) Homotopy. Change s̃ by a homotopy in Δ that fixes its endpoints.

ii) Change of Lift. If g ∈ N_{G}(Q_{j}) then we can replace

P → Q_{1} → Q_{2} → ⋯ → Q_{j} → ⋯ → Q_{n} → P′

by

P → Q_{1} → ⋯ → Q_{j−1} → Q_{j} → Q_{j+1}^{g} → ⋯ → Q_{n}^{g} → P^{′g}.

Define a height function h : Δ → IR by starting on the vertices with h(Q) = log_{p}|Q| and then extending linearly on each simplex. Define the depth of a path s̃ in Δ to be d(s̃) = min {h(Q)|Q a vertex of s̃}.

Now, for a given class c ∈ π_{1}(Δ/G, P), consider all the lifts starting at P of all the cellular paths representing c. Amongst these, restrict attention to those of maximal depth, and then choose one with the least possible number of vertices of minimal height. Call it s̃.

s̃: P → Q_{1} → ⋯ → Q_{n} → P′.

Assume that c ≠ 1 so that there are at least three vertices. Let Q_{j} be a vertex of minimal height and let R be a Sylow p-subgroup of N_{G}(Q_{j}) containing Q_{j−1} (clearly Q_{j} ⊲ Q_{j−1} since Q_{j} is of minimal height). Then for some g ∈ N_{G}(Q_{j}), ^{g}R contains Q_{j+1}, and we can change the lift to obtain

s′ : P → Q_{1} → ⋯ → Q_{j−1} → Q_{j} → Q_{j+1}^{g} → ⋯ → Q_{n}^{g} → P^{′g}.

figure = orbit1.ps, height = 2in, width = 1.7in, angle = 270

We now have 1-simplices: where Q_{j−1}, Q_{j+1}^{g} ≤ R but they need not be normal. However there are sequences

Q_{j−1} ⊲ S_{1} ⊲ ⋯ ⊲ S_{s} ⊲ R

and

Q_{j+1}^{g} ⊲ T_{1} ⊲ ⋯ ⊲ T_{t} ⊲ R,

so we have 2-simplices:

figure = orbit2.ps, height = 3in, width = 4in, angle = 270

We can now change the path s′ by a homotopy to s′′:

P → Q_{1} → ⋯ → Q_{j−1} → S_{1} → ⋯ → S_{s} → R → T_{t} → ⋯ → T_{1} → Q_{j+1}^{g} → ⋯ → Q_{n}^{g} → P^{′g}.

But s′′ has fewer vertices of minimal height, a contradiction.

b) The case of homology is similar but a little more complicated. Clearly H̃_{0}(Δ/G; ZZ) = 0, i.e. Δ/G is connected, because for every p-subgroup Q there is a sequence Q ⊲ Q_{1} ⊲ ⋯ ⊲ Q_{n}, where Q_{n} is a Sylow p-subgroup of G. This yields a path from Q to Q_{n}, and all Sylow p-subgroups are conjugate. From now on we assume that n ≥ 1.

Each n-cycle in the CW-homology of Δ/G can be regarded as a linear combination s of oriented n-cells. This can be lifted to a linear combination s̃ of n-simplices of Δ, s̃ = ∑ n_{σ}σ. We do not assume that this lifting is necessarily done in such a way that only one σ appears from each G-orbit.

There are two operations that we can perform on s̃ which do not change its image in H_{n}(Δ/G, ZZ).

i) Homology. Add a boundary (i.e. something homologous to zero).

ii) Change of Lift. Any of the simplices can be replaced by another in the same G-orbit.

Define the height h(σ) of a simplex to be the height of its barycentre (i.e. the average height of its vertices) and its depth to be the minimum height of its codimension 1 faces. The depth of a chain is defined by d(∑ n_{σ}σ) = {min d(σ)|n_{σ} ≠ 0}.

Given a class c ∈ H_{n}(Δ/G; ZZ) consider all the liftings s̃ to Δ of all cycles s representing c. Amongst these consider only those of maximal depth d, and write s̃ = ∑ n_{σ}σ. Now pick an s̃ that minimizes the multiplicity,

m(s̃) = ∑_{d(σ)=d} |n_{σ}|.

Assume that c ≠ 0, so there must be a simplex ρ_{1} with n_{ρ_{1}} ≠ 0 and d(ρ_{1}) = d. Now ρ_{1} has a face μ = (Q_{0} ⊲ ⋯ ⊲ Q_{n−1}) with h(μ) = d. Let R_{1} be the vertex of ρ_{1} not in μ. Then h(R_{1}) > h(Q_{i}) for any i, otherwise ρ_{1} would have a face of depth less than d, so ρ_{1} = (μ, R_{1}).

Since the image of s̃ in Δ/G is a cycle, there must be another simplex ρ′ with n_{ρ′} ≠ 0 such that some conjugate ρ_{2} = hρ′ (h ∈ G) also has a face μ, and n_{ρ_{1}} and n_{ρ′} have opposite signs (but not necessarily the same absolute value). Again, ρ_{2} = (μ, R_{2}) by minimality, and by changing our attention to −c if necessary, we can assume that n_{ρ_{1}} > 0 and n_{ρ′} < 0. Note that minimality under change of lift implies that the coefficient function n_{σ} can not take both positive and negative values on the same orbit, so ρ′ ≠ ρ_{1} ≠ ρ_{2} and also n_{ρ_{2}} ≤ 0. A change of lift alters s̃ to

s′ = s̃ + ρ′ − ρ_{2} = ∑ n_{σ}σ + ρ′ − ρ_{2} = ∑ nσ^{′}σ,

where it is easy to check that d(s′) = d(s̃), m(s′) = m(s̃), n′_{ρ_{1}} > 0 and n_{ρ_{2}} < 0.

Now write

s′ = ∑ m_{σ}σ + ρ_{1} − ρ_{2} = t + ρ_{1} − ρ_{2},

so m_{σ} = n′_{σ} unless σ is ρ_{1} or ρ_{2}, and m_{ρ_{1}} = nρ_{1}^{′} − 1 ≥ 0, m_{ρ_{2}} = nρ_{2}^{′} + 1 ≤ 0. Thus m(t) = m(s̃)−2. Let R be a Sylow p-subgroup of stab_{G}(μ) containing R_{1}. Then R_{2} ≤ R^{g} for some g ∈ stab_{G}(μ), so a change of lift alters s′ to s′′ = t+ρ_{1}−gρ_{2}, where gρ_{2} = (μ,^{g}R_{2}).

Suppose, for the moment, that R_{1} ≠ R ≠ R_{2}^{g}. Then we can find sequences of subgroups

R_{1} ⊲ S_{1} ⊲ ⋯ ⊲ S_{s} ⊲ R

and

gR_{2} ⊲ T_{1} ⊲ ⋯ ⊲ T_{t} ⊲ R.

Let

v_{1} = (μ, R_{1}, S_{1}) + (μ, S_{1}, S_{2}) + ⋯ + (μ, S_{s}, R),

and

v_{2} = (μ,^{g}R_{2}, T_{1}) + (μ, T_{1}, T_{2}) + ⋯ + (μ, T_{t}, R).

Then for i = 1, 2,

(−1)^{n}∂v_{i} = g^{i−1}ρ_{i} − (μ, R) + X_{i},

where X_{i} is a sum of cells which do not contain μ, but their vertices which are not in μ contain (as groups) all the vertices of μ. It follows that X_{i} involves only cells of depth strictly greater than d, and therefore that

s′′ = t + ρ_{1} − gρ_{2} ≡ t + (−1)^{n}∂(v_{1} − v_{2}), modulo cells of depth greater than d,

and a change by homology alters s′′′ to s′′′ = s′′ − (−1)^{n}∂(v_{1} − v_{2}) and yields

s′′′ ≡ t, modulo cells of depth greater than d.

But m(s′′′) = m(t) = m(s̃) − 2, a contradiction.

As for the remaining cases, if R_{1} = R = ^{g}R_{2} then s′ = t. If R_{1} ≠ R = ^{g}R_{2}, then (−1)^{n}∂v_{1} ≡ ρ_{1} − gρ_{2}, modulo cells of depth greater than d. The case R_{1} = R ≠ ^{g}R_{2} is similar.

Remark. A relative version of this theorem also holds. Let Y be a set of subgroups of G that is closed under subgroups and conjugation. Let Δ_{Y} be the subcomplex of Δ in which we only allow chains of p-subgroups not in Y. Then if Δ_{Y} is not empty, Δ_{Y} /G is contractible.

References

[ASe] A. Aschbacher and Y. Segev, Extending morphisms of groups and graphs, Ann. Math. (2) 137 (1992) 297-323.

[ASm] A. Aschbacher and S.D. Smith, On Quillen’s Conjecture for the psubgroup complex, Ann. Math. (2) 137 (1993) 473-529.

[B] K.S. Brown, Euler characteristics of groups: the p-fractional part, Invent. Math. 29 (1975) 1-5.

[KR] R. Knörr and G.S. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1989) 48-60.

[Q] D. Quillen, Homotopy properties of the poset of non-trivial p-subgroups of a group, Advances in Math, 28 (1978) 101-128.

[T] J. Thévenaz, On a conjecture of Webb, Arch. Math. 58 (1992) 105-109.

[TW] J. Thévenaz and P.J. Webb, Homotopy equivalence of posets with a group action, J. Combinatorial Theory Ser. A 56 (1991) 173-181.

[W1] P.J. Webb, Subgroup complexes, in the Arcata Conference on Representations of Finite Groups, part 1. Proc. Symp. Pure Math. 47 (1987) 349-365.

[W2] P.J. Webb, A split exact sequence of Mackey functors, Comm. Math. Helv. 66 (1991) 34-69.

Address: Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027. e-mail: symonds@ms.uky.edu