A topological consequence of Alperin’s fusion theorem

LAURENCE BARKER

Abstract: Some G-posets comprising Brauer pairs or local pointed groups belong to a class of G-posets which satisfy a version of Alperin’s Fusion Theorem, and as a consequence, have orbit spaces whose fundamental group is trivial.

Questions of fusion often arise in connection with the G-posets described in the next paragraph. A G-poset, recall, is a partially ordered set upon which G acts as automorphisms. We refer to Benson [1, Chapter 6] for an introduction to the theory of G-posets (pioneered by K. S. Brown, S. Bouc, D. Quillen, Thévenaz, Webb, and others). We refer to Thévenaz [6, Chapter 7] for an introduction to the theory of G-fusion within the G-posets awaiting description (this theory pioneered by J. Alperin, M. Broué, L. Puig, and others). However, these two references are more by way of motivation than prerequisite for the material herein. Aside from some comments intended to locate the material within its cultural context, we shall work in an abstract scenario, and our arguments will largely be self-contained.

Throughout, we work with a finite group G. Given a prime p, and a p-block B of G, we write B(B) for the G-poset of Brauer pairs associated with B, and write L(B) for the G-poset of local pointed groups associated with B. If B is projective, then B(B) and L(B) are trivial, and we have nothing else to say about them. So let us henceforth assume that B is non-projective, whereupon p must divide |G|. We shall also consider the upwardly closed G-subposets of B(B) and L(B); a Gsubposet Y of a G-poset X is said to be upwardly closed provided, given y ≤ x in X with y ∈ Y, then x ∈ Y. In particular, writing b for the unity of the block algebra of B, the G-poset S(B) := B(B)−{(1, b)} seems to be of interest (for reasons indicated below). Note that if B is the principal block of G, then Brauer’s Third Main Theorem tells us that S(B) is isomorphic to the G-poset S_{p}(G) consisting of the non-trivial p-subgroups of G.

For an upwardly closed G-subposet X of either B(B) or L(B), our purpose is to relate G-fusion in X to a property of the topological space |X |/G (defined below). In order to give a unified treatment to the two cases X ⊆ B(B) and X ⊆ L(B), and also to entertain the possibility of application to other suitable G-posets, we have interpreted a few properties of B(B) and L(B) as axioms from which all our deductions are to be made.

Let us begin by reviewing, for any finite G-poset X, the topological constructions involved. Write sd(X) for the G-poset consisting of the chains in X (partially ordered by the subchain relation). The G-sets X and sd(X) comprise the vertices and the simplexes, respectively, of a G-simplicial complex Δ(X) whose associated polyhedron |X| is a G-space. Since Δ(sd(X)) is the barycentric subdivision of Δ(X), there is a G-equivariant homeomorphism

ϕ_{X} : |sd(X)| → |X|

linearly extending the function sending each element of sd(X) to its centroid in |X|. We have a canonical projection to the orbit space

θ_{X} : |X| → |X|/G

and a homeomorphism

\overline{ϕ}_{X} : |sd(X)|/G → |X|/G

such that \overline{ϕ}_{X}θ_{sd(X)} = θ_{X}ϕ_{X}. We also have a homeomorphism

ψ_{X} : |sd(X)/G| → |sd(X)|/G

linearly extending the function on the orbit poset sd(X)/G such that the geometric realisation of the G-orbit of an element ζ ∈ sd(X) is sent to the G-orbit of the geometric realisation of ζ (see Thévenaz [4, Section 1]). Let

ρ_{X} : |sd(X)| → |sd(X)/G|

be the projection linearly extending the canonical surjection sd(X) → sd(X)/G. It is easy to check that ψ_{X}ρ_{X} = θ_{sd(X)}, hence

θ_{X}ϕ_{X} = \overline{ϕ}_{X}ψ_{X}ρ_{X}.

In the local theory of finite groups, particularly of their representations and cohomology, one frequently encounters problems concerning fusion. The Brown complex, defined to be Δ(S_{p}(G)), has been used to tackle such problems; see a discussion and citations in Thévenaz [4]. Many of the applications of the Brown complex are based on the Lefschetz invariant of S_{p}(G), which is an isomorphism invariant of the G-space |S_{p}(G)|; see, for instance, Benson [1]. Results in Knörr– Robinson [2], Thévenaz[5], and Webb [8] involve sums over the elements of the poset sd(S_{p}(G))/G. In a letter to A. Watanabe, dated 23 November 1992, L. Puig paraphrases [2, 3.8], reformulating Alperin’s Weight Conjecture in terms of a sum over the elements of sd(S(G, b))/G. These developments are behind our impulse to investigate the interplay between fusion and topological invariants.

Though our methods are different, we follow in the wake of Symonds [3] proof of Webb’s conjecture that |S_{p}(G)| is contractible. Earlier, Thévenaz [4] had verified Symonds’ Theorem in cases where a control of fusion condition was available. He had suggested to me that the general case might succumb to Alperin’s Fusion Theorem. We now introduce a class of G-posets which will be shown to satisfy a version of Alperin’s Fusion Theorem, and in consequence, to satisfy part of Symonds’ Theorem.

We define a Sylow G-poset to be a finite G-set X equipped with G-stable relations ≤ and ≤ such that • ≤ is a partial ordering, and is the transitive closure of ≤,

• G acts transitively on the maximal elements of X, and for any x ∈ X, the stabiliser N_{G}(x) acts transitively on the elements y ∈ X which are maximal subject to x ≤ y.

Though Sylow G-posets have more structure than (abstract) G-posets, we point out that the conclusion of Theorem 3 below holds for any G-poset which can be made into a Sylow G-poset. Any upwardly closed G-subposet of a Sylow G-poset is a Sylow G-poset. By results of J. Alperin, M. Broué, and L. Puig in Thévenaz [6, 40.10, 40.15, 48.1], the G-posets B(B) and L(B) are Sylow.

For a Sylow G-poset X, we write < and < for the anti-reflexive relations corresponding in the usual way to ≤ and ≤. When x≤ y in X, we say that y normalises x. If y is maximal subject to normalising x, we call y a maximal normaliser of x. Given x ≤ z in X, we say that x is fully normalised in z provided z contains a maximal normaliser of x.

Stronger versions of the following result are already known in the case of the G-posets B(B) (due to J. Alperin and M. Broué), and in the case of the G-poset L(B) (due to L. Puig); see Thévenaz [6, Section 48]. The presentation in [6] inspired our axiomatic approach here.

THEOREM 1 (Alperin’s Fusion Theorem for Sylow G-posets): Let X be a Sylow G-poset, let x ≤ s in X with s maximal, and let g ∈ G such that x^{g} ≤ s. Then there exist some n, and for 1 ≤ i ≤ n, elements s_{i}, t_{i} ∈ X with s_{i} maximal, and elements g_{i} ∈ N_{G}(t_{i}) such that

• s ≥ t_{i} ≤ s_{i}, and t_{i} is fully normalised in both s and s_{i},

• x^{g_{1}...g_{i}} ≤ t_{i}, and g = g_{1}...g_{n}.

Proof: We say that the elements g_{1}, ..., g_{n} (when they exist) accomplish fusion from x to x^{g} in s. We argue by induction on the depth of x (the maximal length r of a chain x = x_{0} < ... < x_{r} starting at x). Supposing the depth to be positive, choose y ∈ X with x < y ≤ s, and let z be a maximal normaliser of x containing y. We have z^{h} ≤ s for some h ∈ G. Since y^{h} ≤ s, we may, by induction, write h as a product of elements h_{1}, h_{2}, ... accomplishing fusion from y to y^{h} in s. Perforce, h_{1}, h_{2}, ... accomplish fusion from x to x^{h} in s. By replacing x, z, g with x^{h}, z^{h}, h^{−1}g, respectively, we may assume that z ≤ s. A similar argument allows us to assume that s^{g^{−1}} contains a maximal normaliser z′ of x. We have z′ = z^{f} for some f ∈ N_{G}(x). By induction, we can write gf as a product of elements f_{1}, f_{2}, ... accomplishing fusion from z^{g^{−1}} to z^{f} in s^{g^{−1}}. Noting that x is (by our assumptions) fully normalised in s and s^{g^{−1}}, we see that f, f_{1}^{g}, f_{2}^{g}, ... accomplish fusion from x to x^{g} in s. ⊓ ⊔

We shall need a lemma explaining how suitable paths in the orbit space |X|/G of a finite G-poset X may, up to homotopy, be lifted to |X|. First, let us discuss homotopy classes of paths in a finite simplicial complex K. It is well-known (and easily proved) that for vertices u, v ∈ K, any path from u to v in the polyhedron |K| is homotopic to a path α in the 1-skeleton of |K| such that the preimage under α of the 0-skeleton of |K| is finite. (Homotopies of paths are to preserve end-points.) Given a simplex {x, y} in K (allowing the possibility that x = y), we choose a path σ(x, y) from x to y in |K| whose image is confined to the geometric simplex |{x, y}| (which has dimension 0 or 1). The homotopy class of σ(x, y) is independent of any choice. We shall sometimes write expressions of the form σ(x_{0}, x_{1})σ(x_{1}, x_{2})...σ(x_{m−1}, x_{m}) where each {x_{i−1}, x_{i}} is a simplex in K. Such an expression denotes a concaternation of the paths σ(x_{0}, x_{1}), σ(x_{1}, x_{2}), ..., σ(x_{m−1}, x_{m}), and is well-defined up to homotopy. Observe that, given a simplex {x, y, z} in K (allowing the possibility of repetitions), we have a homotopy σ(x, y)σ(y, z) ≃ σ(x, z).

LEMMA 2: Given a finite G-poset X, and elements x, y ∈ X, then any path in |X|/G from θ_{X}(x) to θ_{X}(y) is homotopic to a path of the form

θ_{X}(σ(z_{0}, z_{1})σ(z_{1}, z_{2})...σ(z_{2m−1}, z_{2m}))

where x = z_{0} ≥ z_{1} ≤ z_{2} ≥ ... ≤ z_{2m−2} ≥ z_{2m−1} ≤ z_{2m} and θ_{X}(z_{2m}) = θ_{X}(y). Furthermore, we may insist that the elements z_{1}, z_{3}, ..., z_{2m−1} are minimal, and that the elements z_{2}, z_{4}, ..., z_{2m−2} are maximal.

Proof: We have (x) ∈ sd(X), and ρ_{X}((x)) ∈ sd(X)/G. Consider a path in |sd(X)/G| of the form

\overline{μ} = σ(ρ_{X}(ζ_{0}), ρ_{X}(ζ_{1}))σ(ρ_{X}(ζ_{1}), ρ_{X}(ζ_{2}))...σ(ρ_{X}(ζ_{2m−1}), ρ_{X}(ζ_{2m}))

where ρ_{X}((x)) = ρ_{X}(ζ_{0}) ≥ ρ_{X}(ζ_{1}) ≤ ρ_{X}(ζ_{2}) ≥ ... ≤ ρ_{X}(ζ_{2m}) = ρ_{X}((y)), and each ζ_{i} ∈ sd(X). In view of the identity θ_{X}ϕ_{X} = \overline{ϕ}_{X}ψ_{X}ρ_{X}, together with our above comments about paths, the main assertion will follow when we have shown that \overline{μ} lifts via ρ_{X} to a path in |sd(X)|. We can write (x) = η_{0} ≥ η_{1} ≤ η_{2} ≥ ... ≤ η_{2m} where each η_{i} is a G-conjugate of ζ_{i}. Putting

μ = σ(η_{0}, η_{1})σ(η_{1}, η_{2})...σ(η_{2m−1}, η_{2m})

then \overline{μ} = ρ_{X}μ, and the main assertion is proved.

For 1 ≤ j ≤ m − 1, let z′_{2j} be a maximal element of X containing z_{2j}. Then

σ(z_{2j−1}, z_{2j})σ(z_{2j}, z_{2j+1})
≃ σ(z_{2j−1}, z′_{2j})σ(z′_{2j}, z_{2j})σ(z_{2j}, z′_{2j})σ(z′_{2j}, z_{2j+1})
≃ σ(z_{2j−1}, z′_{2j})σ(z′_{2j}, z_{2j+1})

so we may insist that the elements z_{2j} are maximal. Similarly, we may insist that z_{i} is minimal for odd i. ⊓ ⊔

THEOREM 3: Given a Sylow G-poset X, then the orbit space |X|/G is simply connected, and the fundamental group π_{1}(|X|/G) is trivial.

Proof: The transitivity of G on the maximal elements of X implies that |X|/G is simply connected. Letting s be a maximal element, we take θ_{X}(s) to be the basepoint of |X|/G. Lemma 2 tells us that any element of π_{1}(|X|/G) is the homotopy class of a path of the form

θ_{X}(σ(x_{0}, x_{1})σ(x_{1}, x_{2})...σ(x_{2m−1}, x_{2m}))

where s = x_{0} ≥ x_{1} ≤ x_{2} ≥ ... ≤ x_{2m}, each x_{i} ∈ X, and each x_{2j} is maximal. It suffices to show that θ(σ(x_{2j−2}, x_{2j−1})σ(x_{2j−1}, x_{2j})) is null-homotopic. This is equivalent to the assertion that, given x ≤ s ≥ x^{g} with g ∈ G, and writing σ := σ(x, s)σ(s, x^{g}), then the closed path θ_{X}(σ) (based at θ_{X}(x) ) is null-homotopic.

Let n, and the elements s_{i}, t_{i}, g_{i} be as in Theorem 1. Since σ is homotopic to a path passing consecutively through the points x, s, x^{g_{1}}, s, ..., x^{g_{1}...g_{n−1}}, s, x^{g}, we may assume that n = 1. The element t := t_{1} is fixed by G. We have homotopic paths

σ ≃ σ(x, t)σ(t, s)σ(s, t)σ(t, x^{g}) ≃ σ(x, t)(σ(t, x))^{g}

whose composites with θ_{X} are manifestly null-homotopic. ⊓ ⊔

Must the orbit space of a Sylow G-poset be contractible? An adaptation of the argument in Symonds [3] would seem to give an affirmative answer given appropriate generalisations of results in Thévenaz–Webb [7]. However, the relevant proofs in [7] do not extend easily.

Acknowledgement: I would like to thank Jacques Thévenaz for some very helpful comments.

REFERENCES

[1] D. J. Benson, Representations and cohomology II: cohomology of groups and modules, Cambridge Univ. Press, Cambridge 1991.

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

[3] P. Symonds, The orbit space of the p-subgroup complex is contractible, (preprint).

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

[5] J. Thévenaz, Locally determined functions and Alperin’s conjecture, J. London Math. Soc. (2) 45 (1992), 446-468.

[6] J. Thévenaz, G-Algebras and modular representation theory, Clarendon press, Oxford 1995.

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

[8] P. J. Webb, A local method in group cohomology, Comment. Math. Helvetici 62 (1987), 135-167.

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey. e-mail: barker@fen.bilkent.edu.tr

Feburary 1997.