ENDOTRIVIAL REPRESENTATIONS OF FINITE GROUPS AND EQUIVARIANT LINE BUNDLES ON THE BROWN COMPLEX

PAUL BALMER

Abstract. We relate endotrivial representations of a finite group in characteristic p to equivariant line bundles on the simplicial complex of non-trivial p-subgroups, by means of weak homomorphisms.

Dedicated to Serge Bouc on the occasion of his 60^{th} birthday

1. Introduction

Let G be a finite group, p a prime dividing the order of G and 𝕜 a field of characteristic p. For the whole paper, we fix a Sylow p-subgroup P of G.

Consider the endotrivial 𝕜G-modules M, i.e. those finite dimensional 𝕜-linear representations M of G which are ⊗-invertible in the stable category 𝕜G – stab = 𝕜G – mod / 𝕜G – proj; this means that the 𝕜G-module End_{𝕜}(M) is isomorphic to the trivial module 𝕜 plus projective summands. The stable isomorphism classes of these endotrivial modules form an abelian group, T_{𝕜}(G), under tensor product. This important invariant has been fully described for p-groups in celebrated work of Carlson and Thévenaz [CT04, CT05]. Therefore, for general finite groups G, the focus has moved towards studying the relative version:

T_{𝕜}(G, P) := Ker (T_{𝕜}(G) → T_{𝕜}(P)) .

We connect this piece of modular representation theory to the equivariant topology of the Brown complex S_{p}(G) of p-subgroups, see [Bro75]. This G-space S_{p}(G) is the simplicial complex associated to the poset of nontrivial p-subgroups of G, on which G acts by conjugation. The study of S_{p}(G) is a major topic in group theory, centered around Quillen’s conjecture [Qui78], which predicts that if S_{p}(G) is contractible then it is G-contractible, i.e. G admits a non-trivial normal p-subgroup. Here, we focus on the Picard group Pic^{G}(S_{p}(G)) of G-equivariant complex line bundles on S_{p}(G); see Segal [Seg68].

Our main result, Theorem 4.1, relates those two theories as follows (see Cor. 4.13):

1.1. Theorem. Suppose 𝕜 algebraically closed. Then there exists an isomorphism

T_{𝕜}(G, P) ≃ Tors_{p′} Pic^{G}(S_{p}(G))

where Tors_{p′} Pic^{G}(S_{p}(G)) is the prime-to-p torsion subgroup of Pic^{G}(S_{p}(G)).

Date: July 30, 2016.

2010 Mathematics Subject Classification. 20C20,55N91.

Key words and phrases. Endotrivial module, equivariant line bundle, Brown Quillen complex of p-subgroups, weak homomorphism.

Supported by NSF grant DMS-1303073 and Research Award of the Humboldt Foundation.

The left-hand abelian group T_{𝕜}(G, P) is always finite; see Remark 4.12. About the right-hand side, it is true for general finite G-CW-complex X that the group Pic^{G}(X) can be interpreted as an equivariant cohomology group, namely H_{G}^{2}(X, ℤ); in particular it is a finitely generated abelien group; see Remark 2.7. Some readers will consider Theorem 1.1 as the topological answer to the modular-representation-theoretic problem of computing T_{𝕜}(G, P).

Since its origin in [Bro75, Qui78], the space S_{p}(G) is related to the p-local study of G. Closer to our specific subject, Knörr and Robinson in [KR89] and Thévenaz in [Thé93] already exhibited interesting relations between modular representation theory and equivariant K-theory of S_{p}(G). The connection we propose here does not only relate invariants of both worlds but appears at a slightly deeper level, in that it connects actual objects. Indeed, in Construction 3.1, we build complex line bundles over S_{p}(G) from endotrivial representations of G. This construction then yields the isomorphism of Theorem 1.1. It would actually be interesting to see whether similar constructions exist for other classes of modular representations of G, beyond endotrivial ones.

The attentive reader will appreciate that modular representations of G live in positive characteristic whereas complex line bundles on the space S_{p}(G) are rather “characteristic zero” objects. This cross-characteristic connection is made possible thanks to the use of torsion elements and roots of unity. More precisely, we use in a crucial way the re-interpretation [Bal13] of the group T_{𝕜}(G, P) in terms of weak P -homomorphisms. Let us remind the reader.

1.2. Definition. Let K be a field – which will be either 𝕜 or ℂ in the sequel. A function u : G ⟶ K^{∗} = K−{0} is a (K-valued) weak P -homomorphism if (WH 1) u(g) = 1 when g ∈ P. (WH 2) u(g) = 1 if P ∩ P^{g} = 1. (WH 3) u(g_{2} g_{1}) = u(g_{2}) u(g_{1}) if P ∩ P^{g_{1}} ∩ P^{g_{2}g_{1}} ≠ 1. The name comes from (WH 3) which is a weakening of the usual homomorphism condition. We denote by A_{K}(G, P) the group of all weak P -homomorphisms from G to K^{∗}, equipped with pointwise multiplication: (uv)(g) = u(g) v(g).

The main result of [Bal13] is the existence of an explicit isomorphism

(1.3)  A_{𝕜}(G, P) ≃ T_{𝕜}(G, P) .

This result has already found interesting applications, for instance the computation of T_{𝕜}(G, P) for new classes of groups by Carlson-Mazza-Nakano [CMN14] and Carlson-Thévenaz [CT15]. Here, we will use the complex version A_{ℂ}(G, P) to build a homomorphism

𝕃 : A_{ℂ}(G, P) → Pic^{G}(S_{p}(G))

which will yield the isomorphism of Theorem 1.1 when suitably restricted to torsion. Injectivity of 𝕃 on torsion relies in an essential way on a result of Symonds [Sym98], namely the contractibility of the orbit space S_{p}(G)/G.

As often in such matters, it is difficult to predict which way traffic will go on the new bridge opened by Theorem 1.1. Computations of T_{𝕜}(G, P) have already been performed for many classes of finite groups and it seems quite possible that these examples will produce new equivariant line bundles for people interested in the G-homotopy type of S_{p}(G). Conversely, Theorem 1.1 might prove useful to modular representation theorists in endotrivial need. Only future work will tell.

ENDOTRIVIAL REPRESENTATIONS AND LINE BUNDLES ON S_{p}(G) 3

Finally, we emphasize that the G-space S_{p}(G) can of course be replaced by any G-homotopically equivalent G-space, like Quillen’s version [Qui78] via elementary abelian p-subgroups, Bouc’s variant [Bou84], or Robinson’s, see Webb [Web87].

2. The Brown complex and roots of functions

In this preparatory section, we gather some background and notation.

2.1. Notation. For an integer m ≥ 1 and a field K (which will be 𝕜 or ℂ), we denote by μ_{m}(K) = { ζ ∈ K | ζ^{m} = 1 } the group of m^{th} roots of unity in K.

2.2. Notation. The Brown complex S_{p}(G) is (the geometric realization of) the simplical complex with one non-degenerate n-simplex [Q_{0} < Q_{1} < ⋯ < Q_{n}] for each sequence of n proper inclusions of nontrivial p-subgroups, with the usual faceoperations “dropping Q_{i}”. For n = 0, we thus have a point [Q] in S_{p}(G) for each non-trivial p-subgroup Q ≤ G. The space S_{p}(G) admits an obvious right G-action given by conjugation on the p-subgroups, that is Q · g := Q^{g} = g^{−1}Qg. This G-action on S_{p}(G) is compatible with the cell structure.

Since we have fixed a Sylow p-subgroup P ≤ G, we can consider the subcomplex

Y := S_{p}(P) ⊆ S_{p}(G)

on those subgroups contained in P, i.e. we keep in Y those n-cells [Q_{0} < ⋯ < Q_{n}] of S_{p}(G) corresponding to non-trivial subgroups of P. This closed subspace Y of S_{p}(G) is contractible, for instance towards the point [P]. But more than that, Y is an N-subspace of S_{p}(G) for N = N_{G}(P) the normalizer of P. As such, Y is even N-contractible. See [TW91] if necessary. A fortiori, Y is P -contractible. The translates Y g = S_{p}(P^{g}) of the closed subspace Y cover the space S_{p}(G):

S_{p}(G) = ∪_{g∈G} S_{p}(P^{g}) = ∪_{g∈G} Y g.

We shall perform several “G-equivariant constructions” over S_{p}(G) by first performing a basic construction over Y and then showing that the translates of this basic construction on Y g_{1} and on Y g_{2} agree on the intersection Y g_{1} ∩ Y g_{2} for all g_{1}, g_{2}.

2.3. Remark. We will be tacitly using the following fact. For g_{1}, …, g_{n} ∈ G (typically with n ≤ 3), we have P^{g_{1}} ∩ ⋯ ∩ P^{g_{n}} ≠ 1 if and only if Y g_{1} ∩ ⋯ ∩ Y g_{n} is not empty. Clearly a nontrivial P^{g_{1}} ∩ ⋯ ∩ P^{g_{n}} gives a point in Y g_{1} ∩ ⋯ ∩ Y g_{n}. Conversely, as G acts simplicially on S_{p}(G), a non-empty intersection Y g_{1} ∩ ⋯ ∩ Y g_{n} must contain some 0-simplex [Q], i.e. some nontrivial p-subgroup Q ≤ P^{g_{i}} for all i.

We shall also often use the following standard notation:

2.4. Notation. When λ : L_{1} → L_{2} is a map of complex line bundles on a space X and ϵ : X → ℂ^{∗} is a continuous function, we denote by λ · ϵ the map λ composed with the automorphism (of L_{1} or L_{2}) which scales by ϵ(x) the fiber over x.

2.5. Remark. A G-equivariant complex line bundle L over a (right) G-space X consists of a complex line bundle π : L → X such that L is also equipped with a G-action making π equivariant and such that the action of every g ∈ G on fibers L_{x} → L_{xg} is ℂ-linear. More generally, see [Seg68] for G-equivariant vector bundles. We denote by Pic^{G}(X) the group of G-equivariant isomorphism classes of such L, equipped with tensor product. The contravariant functor Pic^{G}(−) is invariant under G-homotopy. In particular, if X is G-equivariantly contractible, the map Hom_{gps}(G, ℂ^{∗}) ≅ Pic^{G}(∗) ⟶ Pic^{G}(X) is an isomorphism.

In the case of X = S_{p}(G), restriction to the P -subspace Y = S_{p}(P) yields a group homomorphism from Pic^{G}(S_{p}(G)) to the one-dimensional complex representations of P, that we shall simply denote by Res_{P}^{G}

(2.6)  Res_{P}^{G} : Pic^{G}(S_{p}(G)) → Pic^{P} (S_{p}(P)) ≅ Hom_{gps}(P, ℂ^{∗}).

2.7. Remark (Totaro). For a compact Lie group G acting on a manifold M, there is an isomorphism Pic^{G}(M) ≃ H_{G}^{2}(M, ℤ) = H^{2}(M ×_{G}EG, ℤ), where EG → BG is the universal G-principal bundle on the classifying space BG; see [GGK02, Thm. C.47], where the similar result for a finite group acting on a finite CW-complex is attributed to [HY76]. Alternatively, one can see the latter by reducing to the case of manifolds, since every finite G-CW-complex is G-homotopy equivalent to a (noncompact) G-manifold. Then the group H^{2}(X ×_{G} EG, ℤ) can be approached via a Serre spectral sequence for the fibration X → X ×_{G} EG → BG. In particular, using that G is finite, the spectral sequence collapses rationally to an isomorphism H^{2}(X ×_{G} EG, ℚ) ≃ H^{0}(BG, H^{2}(X, ℚ)) showing that Pic^{G}(X)⊗ℚ ≃ (Pic(X)⊗ℚ)^{G}.

2.8. Notation. For a subspace Y of a G-space X, like our Y = S_{p}(P) ⊆ S_{p}(G) = X, every element g ∈ G yields a homeomorphism ·g : Y →^{∼} Y g. We can transport things from Y to Y g via this homeomorphism, and we use g_{∗}(−) to denote this idea. For instance, if f : Y → ℂ is a function, then g_{∗}f : Y g → ℂ is g_{∗}f(x) := f(xg^{−1}).

Another situation will be that of G-equivariant line bundles L →^{π} X and L′ →^{π′} X and a morphism λ : L_{|Y} → L′_{|Y} of bundles over Y, in which case the morphism g_{∗}λ : L_{|Y g} → L′_{|Y g} is defined by the commutativity of the following top face:

(2.9)  [figure: L_{|Y}; λ; π; ·g; ≃; L_{|Y g}; =: g_{∗}(λ); π; L′_{|Y}; ·g; ≃; π′; L′_{|Y g}; π′; Y; ·g; ≃; Y g .]

As we use right actions (that is (·g_{2}g_{1}) = (·g_{1})∘(·g_{2})) we have (g_{2}g_{1})_{∗} = (g_{1})_{∗}∘(g_{2})_{∗}.

Let us now say a word of roots of complex functions.

2.10. Remark. Throughout the paper, ℂ is given the trivial G-action. Hence a Gmap f : X → ℂ from a (right) G-space X to ℂ is simply a continuous function such that f(xg) = f(x) for all x ∈ X and all g ∈ G, that is essentially a continuous function f̄: X/G → ℂ on the orbit space.

2.11. Proposition. Let m ≥ 1 be an integer, X a G-space and f : X → ℂ^{∗} a G-map. Suppose that f is G-homotopic to the constant map 1. Then f admits an m^{th} root in Cont_{G}(X, ℂ^{∗}), i.e. a G-map f^{1/m} : X → ℂ^{∗} such that (f^{1/m})^{m} = f.

Proof. By assumption, the induced map f̄: X/G → ℂ^{∗} is homotopic to 1. Then it suffices to observe that f̄ has an m^{th} root by a standard determination-of-the-logarithm argument. (Let X̄ = X/G and let H : X̄ × [0, 1] → ℂ^{∗} be a homotopy between H(x, 0) = f̄(x) and H(x, 1) = 1. Lifting each t ↦ H(x, t)/|H(x, t)| ∈ 𝕊^{1} along the fibration ℝ ↠ 𝕊^{1}, we find a map θ : X̄ × [0, 1] → ℝ such that H(x, t) = |H(x, t)| · e^{iθ(x,t)} and θ(x, 1) = 0. One can then define the m^{th} root of f̄ via f̄^{1/m}(x) = |f̄(x)|^{1/m} · e^{iθ(x,0)/m} for all x ∈ X̄.) □

2.12. Corollary. If X/G is contractible (e.g. if X is G-contractible) then for every integer m ≥ 1, every G-map f : X → ℂ^{∗} admits an m^{th} root f^{1/m} ∈ Cont_{G}(X, ℂ^{∗}).

Proof. As such a map f factors via X ↠ X/G, the contractibility of X/G implies that f is G-homotopically trivial and we conclude by Proposition 2.11. □

2.13. Corollary. For every integer m ≥ 1, every G-map f : S_{p}(G) → ℂ^{∗} on the Brown complex admits an m^{th} root f^{1/m} ∈ Cont_{G}(S_{p}(G), ℂ^{∗}).

Proof. The orbit space S_{p}(G)/G is contractible by Symonds [Sym98]. □

3. Constructing line bundles from weak homomorphisms

We now want to associate a G-equivariant complex line bundle L_{u} on S_{p}(G) to each complex-valued weak homomorphism u ∈ A_{ℂ}(G, P) as in Definition 1.2. In essence, this is a very standard gluing procedure, familiar to every geometer. We spell out some details for the sake of clarity and to see where the “weak homomorphism” conditions (WH 1-3) show up.

3.1. Construction. Let u : G → ℂ^{∗} be a weak P -homomorphism and Y = S_{p}(P) ⊆ S_{p}(G) as in Notation 2.2. Define L_{u} as the following topological space:

L_{u} := ( ⨆_{s∈G} Y s × ℂ )/ ∼

where ∼ is the equivalence relation defined in (3.2) below. We use the notation (y, a)_{s} to indicate a point (y, a) in the space Y s × ℂ with index s ∈ G; and we shall write [y, a]_{s} ∈ L_{u} for its class modulo ∼. (As the subsets Y s do intersect in S_{p}(G), the lighter notation (y, a) would be ambiguous.) Note that the weak P -homomorphism u does not appear so far; it is used in the equivalence relation:

(3.2)  (y, a)_{s} ∼ (z, b)_{t}  iff  \{ \begin{matrix}y = z \\ and \\ a · u(s t^{−1}) = b. \end{matrix}

Direct inspection shows that ∼ is an equivalence relation: Reflexivity uses (WH 1); symmetry uses that u(g^{−1}) = u(g)^{−1}, see [Bal13, Rem. 4.2(1)]; transitivity relies on (WH 3) and Remark 2.3. Of course, L_{u} is equipped with the quotient topology.

3.3. Remark. A good way to keep track of what happens is to think of the class [y, a]_{s} as a fictional element “a·s ∈ ℂ living in a fiber over y ∈ S_{p}(G)”, which is not defined since we do not know how s ∈ G should act on ℂ. Still, equality between “a · s over y” and “b · t over z” should nonetheless mean that they live in the same fiber, i.e. y = z, and that “a · (st^{−1}) = b”. So we decide that the action of st^{−1}, i.e. the difference of the two actions over the point y = z in Y s ∩ Y t, is given via the weak homomorphism u. This can be compared to [Bal13, Eq. (2.7)].

The space L_{u} admits a continuous projection to the Brown complex

π_{u} : L_{u} → S_{p}(G)

simply given by [y, a]_{s} ↦ y and whose fibers are isomorphic to ℂ. More precisely, for every s ∈ G, we have a homeomorphism

(3.4)  [figure: α_{s} :; 𝟙_{Y s} := Y s × ℂ; ≃; π_{u}^{−1}(Y s) ⊆ L_{u}; (y, a); [y, a]_{s}]

(We denote trivial line bundles by 𝟙.) These are trivializations of L_{u} over Y s. For all s, t ∈ G, the transition maps α_{t}^{−1}α_{s} on the intersection

\xymatrix{(Y s ∩ Y t) × ℂ \ar[r]^{α_{s}}_{≃} & π_{u}^{−1}(Y s ∩ Y t) & (Y s ∩ Y t) × ℂ \ar[l]_{α_{t}}^{≃} \\ (y, a) \ar@{|->}[r] & [y, a]_{s} =^{(3.2)} [y, a · u(s t^{−1})]_{t} \ar@{|->}[r] & (y, a · u(s t^{−1}))}

is given by the (constant) linear isomorphism, multiplication by the unit u(s t^{−1}). In other words, L_{u} ⟶^{π_{u}} S_{p}(G) is a complex line bundle on S_{p}(G). We record the above computation in compact form : for all s, t ∈ G we have an equality

(3.5)  α_{s} = α_{t} · u(s t^{−1})  over Y s ∩ Y t

as isomorphisms 𝟙_{Y s∩Y t} →^{∼} (L_{u})_{|Y s∩Y t}. Here we used Notation 2.4.

The right G-action on the space L_{u} is defined, in the spirit of Remark 3.3, by

[y, a]_{s} · g := [yg, a]_{sg}.

This action clearly makes π_{u} : L_{u} → S_{p}(G) into a G-map. In view of the above, G acts linearly on the fibers of π_{u} and thus makes L_{u} into a G-equivariant complex line bundle over S_{p}(G). We can also observe that the collection of local trivializations α_{s} : 𝟙_{Y s} ⟶^{∼} (L_{u})_{|Y s} given in (3.4) is “G-coherent” (^{1}) by which we mean that for all s, g ∈ G we have

(3.6)  g_{∗}(α_{s}) = α_{sg}

as isomorphisms 𝟙_{Y sg} →^{∼} (L_{u})_{|Y sg}. This fact results directly from the definitions, see (2.9) and (3.4). Combining this with (3.5) we note for later use the formula:

(3.7)  g_{∗}(α_{1}) = α_{1} · u(g)  over Y ∩ Y g

as isomorphisms 𝟙_{Y ∩Y g} →^{∼} (L_{u})_{|Y ∩Y g}, for all g ∈ G such that P ∩ P^{g} ≠ 1.

3.8. Proposition. For any two weak P -homomorphisms u, v ∈ A_{ℂ}(G, P) we have a G-equivariant isomorphism L_{uv} ≃ L_{u} ⊗ L_{v} of complex line bundles over S_{p}(G).

Proof. Note that the trivializations (3.4) of L_{u} are performed on the closed cover of S_{p}(G) given by (Y s)_{s∈G}, which is independent of u. So, it is the same cover for L_{u}, L_{v} and L_{uv}. The statement then follows from the observation that the following obvious isomorphisms over Y s (where we temporarily decorate the three morphisms α as α^{(u)}, α^{(v)} and α^{(uv)} to distinguish the respective line bundles)

\xymatrix{(L_{u} ⊗ L_{v})_{|Y s} ≅ (L_{u})_{|Y s} ⊗ (L_{v})_{|Y s} & 𝟙_{Y s} ⊗ 𝟙_{Y s} ≅ 𝟙_{Y s} \ar[l]^{≃}_{α_{s}^{(u)}⊗α_{s}^{(v)}} \ar[r]^{α_{s}^{(uv)}} & (L_{uv})_{|Y s}}

1 We do not say “G-equivariant” to avoid confusion.

patch together into a G-equivariant isomorphism L_{u} ⊗ L_{v} →^{∼} L_{uv} on S_{p}(G). Verification of this patching is immediate from (3.5) and the following agreement:

[figure: 𝟙_{Y s∩Y t} ⊗ 𝟙_{Y s∩Y t} ≅ 𝟙_{Y s∩Y t}; (·u(st^{−1}))⊗(·v(st^{−1})); ·uv(st^{−1}); 𝟙_{Y s∩Y t} ⊗ 𝟙_{Y s∩Y t} ≅ 𝟙_{Y s∩Y t}]

on the trivial bundle. Finally, the map L_{u} ⊗ L_{v} →^{∼} L_{uv} is G-equivariant because each {α_{s}^{(...)}}_{s∈G} is a G-coherent collection of maps, as we saw in (3.6). □

3.9. Notation. As in the Introduction, we denote by 𝕃 : A_{ℂ}(G, P) ⟶ Pic^{G}(S_{p}(G)) the homomorphism u ↦ [L_{u}] defined by Construction 3.1 and Proposition 3.8.

This homomorphism is easily seen to be natural in the following sense:

3.10. Proposition. Let G′ ≤ G be a subgroup containing P and consider the G′- subspace S_{p}(G′) ⊆ S_{p}(G). Then the following diagram

\xymatrix{A_{ℂ}(G, P) \ar[r]^{𝕃} \ar[d]_{Res} & Pic^{G}(S_{p}(G)) \ar[d]^{Res} \\ A_{ℂ}(G′, P) \ar[r]^{𝕃} & Pic^{G′}(S_{p}(G′))}

is commutative. □

3.11. Example. Let u : G → ℂ^{∗} be a group homomorphism, i.e. a one-dimensional representation. Assume that u is trivial on P. One associates to u a weak P - homomorphism ũ ∈ A_{ℂ}(G, P) by forcing (WH 2), i.e. by setting for every g ∈ G

(3.12)  ũ(g) := \{ \begin{matrix}u(g) & if P ∩ P^{g} ≠ 1 \\ 1 & if P ∩ P^{g} = 1. \end{matrix}

Then Lũ is isomorphic to the “constant” line bundle (in the sense of [Seg68]), that is, the line bundle 𝟙_{u} := S_{p}(G) × ℂ with action (y, a) · g = (yg, au(g)). Indeed, inspired by Remark 3.3, one easily guesses the G-equivariant isomorphism Lũ →^{∼} 𝟙_{u} by sending the class [y, a]_{s} in Lũ (see Construction 3.1) to the point (y, a · u(s)) in S_{p}(G) × ℂ = 𝟙_{u}. Verifications are left to the reader.

The modification (3.12) of u into a weak homomorphism ũ is irrelevant for the construction of Lũ since (3.2) only uses values ũ(g) over the subset Y ∩Y g. Indeed, either P ∩ P^{g} = 1 and this subset is empty, or P ∩ P^{g} ≠ 1 and ũ(g) = u(g) anyway. Furthermore, the homomorphism u ↦ ũ is often injective, even after (post-) composition with 𝕃. We do not use the latter but state it for peace of mind:

3.13. Proposition. Suppose that S_{p}(G) is connected. Let u : G → ℂ^{∗} be a group homomorphism which is trivial on P and such that the G-equivariant line bundle 𝟙_{u} ≃ 𝕃(ũ) is G-equivariantly trivial on S_{p}(G) (for instance if ũ = 1). Then u = 1.

Proof. A G-equivariant isomorphism 𝟙 →^{∼} 𝟙_{u} is given by multiplication by a map f : S_{p}(G) → ℂ^{∗} such that f(xg) = f(x) · u(g) for all g ∈ G and x ∈ S_{p}(G). Choose an integer m ≥ 1 such that u(g)^{m} = 1. Then f^{m} : S_{p}(G) → ℂ^{∗} is a G-map. By Corollary 2.13, this f^{m} admits an m^{th} root in Cont_{G}(S_{p}(G), ℂ^{∗}), i.e. there exists a G-map f̂: S_{p}(G) → ℂ^{∗} such that f̂^{m} = f^{m}. Since S_{p}(G) is assumed connected, we have f̂ = f · ρ for some constant ρ ∈ μ_{m}(ℂ); see Notation 2.1. Then f is also a G-map and the above relation f(xg) = f(x) · u(g) forces u(g) = 1 for all g ∈ G. □

Assuming S_{p}(G) connected is a mild condition. According to [Qui78, Prop. 5.2], if S_{p}(G) is disconnected then the stabilizer H ≤ G of a component is a strongly p-embedded subgroup, and our discussion can be safely reduced from G to H.

4. The results

We now prove our main result, from which we will deduce Theorem 1.1 stated in the Introduction. Recall from Notation 3.9 the homomorphism 𝕃 : A_{ℂ}(G, P) → Pic^{G}(S_{p}(G)), u ↦ [L_{u}], from the group of complex-valued weak P -homomorphisms (Def. 1.2) to the G-equivariant Picard group (Rem. 2.5) of the Brown complex S_{p}(G).

4.1. Theorem. The homomorphism 𝕃 : A_{ℂ}(G, P) → Pic^{G}(S_{p}(G)) is injective on torsion subgroups (denoted Tors) and its image is the kernel of restriction to onedimensional representations of P, see (2.6). In other words, the following sequence

(4.2)  \xymatrix{0 \ar[r] & Tors A_{ℂ}(G, P) \ar[r]^{𝕃} & Tors Pic^{G}(S_{p}(G)) \ar[r]^{^{Res{G \atop P}}} & Hom_{gps}(P, ℂ^{∗})}

is exact. Consequently, for every integer m ≥ 1 prime to p, our 𝕃 restricts to an isomorphism on the m-torsion subgroups (^{2})

𝕃 : Tors_{m} A_{ℂ}(G, P) →^{∼} Tors_{m} Pic^{G}(S_{p}(G)).

Proof. The proof will occupy the next couple of pages. First note that by naturality of 𝕃 (Prop. 3.10 applied to G′ = P), the following square commutes:

[figure: A_{ℂ}(G, P); 𝕃; Res; Pic^{G}(S_{p}(G)); Res; 0 = A_{ℂ}(P, P); 𝕃; Pic^{P} (S_{p}(P)) ≅ Hom_{gps}(P, ℂ^{∗}) .]

This proves that Res_{P}^{G} ∘𝕃 is trivial (even outside torsion).

We now prove injectivity of 𝕃 on the torsion of A_{ℂ}(G, P). Let u ∈ A_{ℂ}(G, P) be an element of m-torsion for some m ≥ 1, meaning that u(g)^{m} = 1 for all g ∈ G. Suppose that we have a G-equivariant trivialization ψ : 𝟙_{S_{p}(G)} →^{∼} L_{u} of the line bundle 𝕃(u) = L_{u} (see Constr. 3.1). Comparing the restriction ψ_{|Y} to the trivialization α_{1} : 𝟙_{Y} →^{∼} (L_{u})_{|Y} given in (3.4), we find a P -map δ : Y → ℂ^{∗} with

ψ_{|Y} = α_{1} · δ

as isomorphisms 𝟙_{Y} →^{∼} (L_{u})_{|Y} . Combining the G-equivariance of ψ with the relation g_{∗}(α_{1}) = α_{1} · u(g) on Y ∩ Y g from (3.7), we see that for every g ∈ G such that P ∩ P^{g} ≠ 1, we have for every y ∈ Y ∩ Y g

(4.3)  u(g) = \frac{δ(y)}{g_{∗}δ(y)} = \frac{δ(y)}{δ(yg^{−1})}.

As the left-hand side belongs to μ_{m}(ℂ), we deduce that δ^{m} and g_{∗}(δ^{m}) agree on the intersection Y ∩ Y g. Consequently the family of functions (g_{∗}(δ^{m}))_{g∈G} patch together into a G-map f : S_{p}(G) → ℂ^{∗} by setting f(x) = δ(xg^{−1})^{m} whenever x ∈ Y g. By Corollary 2.13, f admits an m^{th} root, i.e. there exists a G-map f^{1/m} : S_{p}(G) → ℂ^{∗} such that (f^{1/m})^{m} = f. On Y, the two roots f^{1/m} and δ of the same map f must differ by an m^{th} root ρ ∈ μ_{m}(ℂ) which must be constant since Y is connected, say δ = ρ · f^{1/m}. But then for every g ∈ G such that P ∩ P^{g} ≠ 1 and for any y ∈ Y ∩ Y g ≠ ∅ (for which yg^{−1} ∈ Y too), relation (4.3) becomes

2 By “m-torsion” we mean exactly the annihilator of m itself, not of powers of m.

u(g) = \frac{δ(y)}{δ(yg^{−1})} = \frac{ρ · f^{1/m}(y)}{ρ · f^{1/m}(yg^{−1})} = 1

by G-equivariance of f^{1/m}. In the other case where P ∩ P^{g} = 1, we have u(g) = 1 by (WH 2). In short, u = 1 is trivial. This proof uses the contractibility of S_{p}(G)/G, since Corollary 2.13 relies on Symonds [Sym98].

We now prove exactness of (4.2) in the middle via another construction.

4.4. Construction. Let L be a G-equivariant complex line bundle on S_{p}(G), which is torsion and such that Res_{P}^{G}(L) = 1, i.e. L restricts to the trivial P -bundle on S_{p}(P). Choose for some m ≥ 1 a G-equivariant isomorphism

ω : 𝟙_{S_{p}(G)} ⟶^{∼} L^{⊗m}

over S_{p}(G) and choose a P -equivariant isomorphism over Y = S_{p}(P)

β : 𝟙_{Y} ⟶^{∼} L_{|Y}

between the trivial bundle 𝟙_{Y} = Y × ℂ and the restriction of L to Y. The P - equivariance of β means that, for every h ∈ P, we have

(4.5)  h_{∗}(β) = β

as isomorphisms 𝟙_{Y} →^{∼} L_{|Y} . There is a choice in the isomorphism β, and we can replace β by β · δ for any P -map δ : Y → ℂ^{∗}. We shall use this flexibility shortly.

Observe that β^{⊗m} yields another trivialization of L^{⊗m} on Y, that we can compare to the initial ω, restricted to Y. It follows that we have ω_{|Y} = β^{⊗m} ·ϵ for some P -map ϵ : Y → ℂ^{∗}. Since the space Y is P -contractible, Corollary 2.12 produces an m^{th}-root of ϵ, say ϵ^{1/m} ∈ Cont_{P} (Y, ℂ^{∗}) with (ϵ^{1/m})^{m} = ϵ. Using this unit to replace β by β ·ϵ^{1/m}, we can and shall assume that β : 𝟙_{Y} →^{∼} L_{|Y} moreover satisfies

(4.6)  β^{⊗m} = ω_{|Y} .

Then, for each g ∈ G, consider as before the translate Y g = S_{p}(P^{g}) ⊆ S_{p}(G) and transport β into an isomorphism g_{∗}(β) : 𝟙_{Y g} →^{∼} L_{|Y g}; see (2.9). If the isomorphisms β and g_{∗}(β) were to agree on the intersection of their domains of definition Y ∩ Y g for all g ∈ G, the collection of isomorphisms (g_{∗}(β))_{g∈G} would patch together into a global isomorphism 𝟙_{S_{p}(G)} →^{∼} L, automatically G-equivariant by construction. Since this cannot happen for nontrivial L, there is an obstruction, and this happens to be a weak P -homomorphism. Indeed, for every g ∈ G such that P ∩ P^{g} ≠ 1, define what is a priori a function u_{L}(g) ∈ Cont(Y ∩ Y g, ℂ^{∗}) by

(4.7)  g_{∗}(β) = β · u_{L}(g)  over Y ∩ Y g

i.e. by the commutativity of the following diagram of line bundles on Y ∩ Y g:

(4.8)  [figure: 𝟙_{Y ∩Y g}; (g_{∗}(β))_{|Y ∩Y g}; ≃; ·u_{L}(g):=; ≃; (L_{|Y g})_{|Y ∩Y g} = L_{|Y ∩Y g}; 𝟙_{Y ∩Y g}; β_{|Y ∩Y g}; ≃; (L_{|Y} )_{|Y ∩Y g} = L_{|Y ∩Y g} .]

There is no choice at this step. By convention, we set

(4.9)  u_{L}(g) = 1  if  P ∩ P^{g} = 1.

In the case P ∩ P^{g} ≠ 1, we are going to prove that u_{L}(g) : Y ∩ Y g → ℂ^{∗} is a constant function. Taking (4.8) to the m^{th} tensor power, replacing both instances of β^{⊗m} by ω thanks to (4.6) and using that ω is G-equivariant, we deduce that (u_{L}(g))^{m} = 1 on Y ∩ Y g. Since this space is non-empty and connected (even contractible), this implies that the function u_{L}(g) is actually constant, with value equal to some complex m^{th} root of unity u_{L}(g) ∈ μ_{m}(ℂ). In other words, the function

u_{L} : G → μ_{m}(ℂ) , g ↦ u_{L}(g)

is a candidate to be a complex-valued weak P -homomorphism. It satisfies (WH 1) by P -equivariance of β, see (4.5) and (4.8) for g = h ∈ P; and u_{L} satisfies (WH 2) by definition (4.9). To verify the last property (WH 3), consider g_{1}, g_{2} ∈ G such that P ∩ P^{g_{1}} ∩ P^{g_{2}g_{1}} ≠ 1, i.e. such that the subset Z := Y ∩ Y g_{1} ∩ Y g_{2}g_{1} is nonempty. Then juxtaposing the defining diagram (4.8) for u_{L}(g_{1}) and the one for u_{L}(g_{2}) transported by (g_{1})_{∗}, both suitably restricted to this triple intersection Z, we obtain the following commutative diagram over Z:

(4.10)  [figure: 𝟙_{Z}; (g_{1}_{∗}g_{2}_{∗}(β))_{|Z}; ≃; g_{1}_{∗}(·u_{L}(g_{2}))=·u_{L}(g_{2}); ≃; L_{|Z}; 𝟙_{Z}; (g_{1}_{∗}β)_{|Z}; ≃; ·u_{L}(g_{1}); ≃; L_{|Z}; 𝟙_{Z}; β_{|Z}; ≃; L_{|Z} .]

We used at the top left that g_{1}_{∗}(−) is ℂ-linear. Using now that g_{1}_{∗}g_{2}_{∗} = (g_{2}g_{1})_{∗}, the left-hand vertical composite satisfies the commutativity expected of u_{L}(g_{2}g_{1}), i.e. fits in place of u_{L}(g_{2}g_{1}) in (4.8) for g = g_{2}g_{1}, after restriction of the latter to Z. This is where we use that Z ≠ ∅ to deduce that u_{L}(g_{2}g_{1}) = u_{L}(g_{2}) · u_{L}(g_{1}).

It is interesting to see the parallel of these arguments with those of [Bal13], where the non-emptiness of Z is replaced by the non-vanishing of a suitable stable category. Both properties are equivalent, namely they both are avatars of the fact that the Sylow P and its conjugates P^{g_{1}} and P^{g_{2}g_{1}} intersect non-trivially.

At this stage, we have associated a weak P -homomorphism u_{L} ∈ Tors_{m} A_{ℂ}(G, P) to an m-torsion G-equivariant line bundle L on S_{p}(G) and choices of isomorphisms ω : 𝟙_{S_{p}(G)} →^{∼} L^{⊗m} and β : 𝟙_{Y} →^{∼} L_{|Y} satisfying (4.6). We now claim that 𝕃(u_{L}) ≃ L. For this, recall the line bundle L_{u_{L}} of Construction 3.1, which describes 𝕃(u_{L}). It comes with an isomorphism α_{1} : 𝟙_{Y} →^{∼} (L_{u_{L}})_{|Y} satisfying

g_{∗}(α_{1}) = α_{1} · u_{L}(g) over Y ∩ Y g

by (3.7). Comparing this formula to the similar one for β in (4.7), we see that the following isomorphism φ := β ∘ α_{1}^{−1} over Y

\xymatrix{φ : (L_{u_{L}})_{|Y} \ar[r]_{≃}^{α_{1}^{−1}} & 𝟙_{Y} \ar[r]_{≃}^{β} & L_{|Y}}

satisfies g_{∗}(φ) = φ on Y ∩Y g for all g ∈ G. Therefore, the (g_{∗}φ)_{g∈G} patch together into a morphism φ : L_{u_{L}} → L which is G-equivariant and an isomorphism by construction. This finishes the proof of the exactness of the sequence (4.2).

It is immediate that 𝕃 restricts to an isomorphism on prime-to-p torsion since Hom_{gps}(P, ℂ^{∗}) is p^{r}-torsion, where |P | = p^{r}, hence every L ∈ Tors_{m} Pic^{G}(S_{p}(G)) with m prime to p maps to zero under Res_{P}^{G}.

This finishes the proof of Theorem 4.1. □

4.11. Remark. Construction 4.4 describes the inverse of 𝕃 on prime-to-p torsion.

Let us now connect these results over ℂ to positive characteristic objects. We recall some well-known facts, to facilitate cognition.

4.12. Remark. The group T_{𝕜}(G, P) is always finite. (Indeed, every endotrivial module in T_{𝕜}(G, P) is a direct summand of 𝕜(G/P) – an explicit projector depending on u ∈ A_{𝕜}(G, P) is given in [Bal13]. By Krull-Schmidt it follows that T_{𝕜}(G, P) has at most dim_{𝕜}(𝕜(G/P)) = [G : P] elements.) Also, the order of T_{𝕜}(G, P) is prime to p; see [Bal13, Cor. 5.3]. For an algebraic closure 𝕜̄ of 𝕜, one can easily identify the image of T_{𝕜}(G, P) ↪ T𝕜̄(G, P); see [Bal13, Cor. 5.5].

In fact, the group T_{𝕜}(G, P) “stabilizes” once 𝕜 contains all roots of unity by which we mean it contains all m^{th} roots of unity for all integers m ≥ 1 prime to p. Here, “stabilization” means that T_{𝕜}(G, P) → T_{𝕜′}(G, P) is an isomorphism for every further extension 𝕜 → 𝕜′; see [Bal13, Cor. 5.5]. This condition is of course fulfilled if the field 𝕜 = 𝕜̄ is algebraically closed, or simply if 𝕜 contains 𝔽̄_{p}, the algebraic closure of the prime field. Our Theorem 1.1 is another way of seeing why T_{𝕜}(G, P) stabilizes once 𝕜 contains all roots of unity, by giving it a topological interpretation:

4.13. Corollary. The prime-to-p torsion Tors_{p′} Pic^{G}(S_{p}(G)) is a finite subgroup of Pic^{G}(S_{p}(G)). For any field 𝕜 of characteristic p which contains all roots of unity (see Remark 4.12), we have an isomorphism as announced in Theorem 1.1

T_{𝕜}(G, P) ≃ Tors_{p′} Pic^{G}(S_{p}(G))

where Tors_{p′} denotes the prime-to-p torsion subgroup.

Proof. Let 𝕜 contain all roots of unity (or just the [G:P]^{th}-roots) and let e be the exponent of T_{𝕜}(G, P). Let m ≥ 1 be an integer, prime to p and divisible by e.

By (1.3), the integer e is also the exponent of A_{𝕜}(G, P) ≃ T_{𝕜}(G, P) hence u^{m} = 1 for all u ∈ A_{𝕜}(G, P). Thus every u : G → 𝕜^{∗} in A_{𝕜}(G, P) takes values in μ_{m}(𝕜). In other words, we can identify the group of 𝕜-valued weak P -homomorphisms A_{𝕜}(G, P) with the set of functions u : G → μ_{m}(𝕜) satisfying (WH 1-3).

Consider now inside the group A_{ℂ}(G, P) of complex-valued weak P -homomor-phisms, the subgroup Tors_{m} A_{ℂ}(G, P) of elements of order dividing m. Again, this is just the subset of those functions u : G → μ_{m}(ℂ) satisfying (WH 1-3).

Choose now an isomorphism μ_{m}(𝕜) ≃ ℤ/m ≃ μ_{m}(ℂ). This uses that 𝕜 contains all m^{th} roots of unity. Combining the above we obtain an isomorphism

(4.14)  A_{𝕜}(G, P) ≃ Tors_{m} A_{ℂ}(G, P).

Since the left-hand side is independent of such m (prime to p and divisible by e), we get Tors_{p′} A_{ℂ}(G, P) = Tors_{e} A_{ℂ}(G, P). Using now Theorem 4.1, it follows that Tors_{p′} Pic^{G}(S_{p}(G)) = Tors_{e} Pic^{G}(S_{p}(G)) ≃ Tors_{e} A_{ℂ}(G, P) via 𝕃. The latter is itself isomorphic to A_{𝕜}(G, P) ≃ T_{𝕜}(G, P) by a last instance of (4.14) and (1.3). □

4.15. Remark. The isomorphism of Corollary 4.13 is essentially induced by the canonical homomorphism 𝕃 : A_{ℂ}(G, P) → Pic^{G}(S_{p}(G)) of Section 3, up to the choice of an identification between e^{th} roots of unity in 𝕜 and e^{th} roots of unity in ℂ, for e the exponent of T_{𝕜}(G, P). Another choice of an isomorphism μ_{e}(𝕜) ≃ μ_{e}(ℂ) simply changes the isomorphism (4.14) by multiplication with some integer prime to e, a rather harmless operation which is of course invertible.

Combining the above with Example 3.11, we obtain:

4.16. Corollary. The following properties of G and p are equivalent:

(i) For 𝕜 = 𝔽̄_{p} the group T_{𝕜}(G, P) consists only of one-dimensional representations G → 𝕜^{∗}.

(i’) For every field 𝕜 containing all roots of unity, the group T_{𝕜}(G, P) consists only of one-dimensional representations G → 𝕜^{∗}.

(ii) Every G-equivariant complex line bundle on S_{p}(G) which is torsion of order prime to p is constant, i.e. Tors_{p′} Pic^{G}(∗) → Tors_{p′} Pic^{G}(S_{p}(G)) is onto. □

Acknowledgments : This paper is a satellite of a project with Serge Bouc, based on his insight on how “sipp cohomology” in the sense of [Bal15] might be connected to the Brown complex. I am extremely grateful to him for numerous detailed discussions, both at philosophical and technical levels. The idea for the present article originated in this interaction. I thank Burt Totaro for several useful discussions and for suggesting Remark 2.7. I also thank Ivo Dell’Ambrogio and Jacques Thévenaz for valuable comments. Finally, I thank Henning Krause and Bielefeld University for their hospitality during the preparation of this work.

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Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

E-mail address: balmer@math.ucla.edu
URL: http://www.math.ucla.edu/∼balmer