EQUIVARIANT PRINCIPAL BUNDLES OVER SPHERES AND COHOMOGENEITY ONE MANIFOLDS

IAN HAMBLETON AND JEAN-CLAUDE HAUSMANN

Abstract. We classify SO(n)-equivariant principal bundles over S^{n} in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant (Π, G)-bundles over cohomogeneity one manifolds.

1. Introduction

Let Π and G be Lie groups. A principal (Π, G)-bundle is a locally trivial, principal G-bundle p: E → X such that E and X are left Π-spaces. The projection map p is Π-equivariant and γ(e · g) = (γe) · g, where γ ∈ Π and g ∈ G acts on e ∈ E by the principal action. Equivariant principal bundles, and their natural generalizations, were studied by T. E. Stewart [24], T. tom Dieck [5], [6, I (8.7)], R. Lashof [15], [16] together with P. May [17] and G. Segal [18].

These authors study equivariant principal bundles by homotopy theoretic methods. There exists a classifying space B(Π, G) for principal (Π, G)-bundles [5], so the classification of equivariant bundles in particular cases can be approached by studying the Π-equivariant homotopy type of B(Π, G). If the structural group G of the bundle is abelian, then the main result of [18] states that equivariant bundles over a Π-space X are classified by the ordinary homotopy classes of maps [X ×_{Π} EΠ, BG]. In practice, this program leads to an obstruction theory rather than a classification. See, however, the results of Lashof in the special cases where Π acts transitively [14] or semi-freely [16] on the base space X.

Another approach to equivariant principal bundles uses the “local” invariants arising from isotropy representations at singular points of (X, Π), together with equivariant gauge theory [1], [8], [9], [10]. By an isotropy representation at a Π-fixed point x_{0} ∈ X we mean the homomorphism α_{x_{0}} : Π → G defined by the formula

γ · e_{0} = e_{0} · α(γ)

where e_{0} ∈ p^{−1}(x_{0}). The homomorphism α is independent of the choice of e_{0} up to conjugation in G. The relationship between the local invariants and the homotopy classification (in the form of a Localization Theorem ?) deserves further study.

Date: Jan. 20, 2001.

The authors thank the Swiss National Fund for Scientific Research, the Université de Génève, and the Max Planck Institut für Mathematik in Bonn for hospitality and support.

In this paper, we use the second approach for Π = SO(n) acting in the standard way on X = S^{n}. In this concrete situation, we obtain a complete classification by relatively elementary geometric methods. It turns out that the local isotropy representations at the north and south poles of S^{n} explicitly determine the classification of (SO(n), G) principal bundles over S^{n} (for short (n, G)-bundles). One surprising consequence is that the set E(n, G) of (n, G)-bundles is finite for n ≥ 3. In contrast, the set of (non-equivariant) principal G-bundles over S^{n} is often infinite. A detailed statement of these results is given in the next section and their proofs, essentially self-contained, are explained in Sections 3 to 6. Several examples are given in Section 7. In Section 8 we show how these results fit into the more general setting of equivariant (Π, G)-bundles over certain Π-manifolds studied by K. Jänich [13] and E. Straume [25]. In particular, we obtain a classification of (Π, G)-bundles over cohomogeneity 1 manifolds.

The authors would like to thank P. de la Harpe for helpful discussions.

2. Statement of results

Let S^{n} be the n-dimensional sphere of radius 1 in R^{n+1}. We consider the action on S^{n} of the group SO(n), by orthogonal transformations fixing the poles (0, …, 0, ±1).

Let G be a Lie group. We denote by R(n, G) the set of smooth homomorphisms from SO(n) to G modulo the conjugations by elements of G. Unless specified, all maps between manifolds are smooth of class C^{∞}.

By a G-principal bundle η over S^{n}, we mean, as usual, a smooth map p: E → S^{n} from a manifold E = E(η) and a free right action E × G → E so that p(z · g) = p(z) with the standard local triviality condition. The isomorphism classes of G-bundles over S^{n} are in bijection with π_{i−1}(G)/π_{0}(G), the quotient of the homotopy group π_{n−1}(G) (based on the neutral element e of G) by the action of π_{0}(G) induced by the conjugation of G on itself. The bijection associates to a bundle η the class C(η) := [∂(id_{S^{n}})] ∈ π_{n−1}(G)/π_{0}(G), where ∂: π_{n}(S^{n}) → π_{n−1}(G) is the boundary operator in the homotopy exact sequence of η [23, Th. 18.5].

A SO(n)-equivariant principal G-bundle ξ over S^{n} (or an (n, G)-bundle for short) is a G-principal bundle ξ^{♭} over S^{n} together with a left action SO(n) × E(ξ) → E(ξ) commuting with the free right action of G and such that the projection to S^{n} is SO(n)-equivariant (we write E(ξ) for E(ξ^{♭})). Two (n, G)- bundles ξ_{1}, ξ_{2} are isomorphic if there exists a diffeomorphism h : E(ξ_{1}) → E(ξ_{2}) which is both SO(n) and G-equivariant and which commutes with the projections to S^{n}. We will compute the set E(n, G) of isomorphism classes of (n, G)-bundles.

Let ξ be a (n, G)-bundle. Choose points a, b ∈ E(ξ) such that p(a) = (0, …, −1) and p(b) = (0, …, 1). Let α, β be the maps from SO(n) to G determined by the formulae A · a = a · α(A) and A · b = b · β(A). We shall prove in Lemma 3.2 that α and β are smooth homomorphisms and that their class in R(n, G) depend only on [ξ] ∈ E(n, G). We call α and β the isotropy representations (associated to a and b). This defines a map J: E(n, G) → R(n, G) × R(n, G) by J(ξ) := ([α], [β]). When n = 2 and G is connected, J(ξ) is a complete invariant which, in particular, determines the (non-equivariant) isomorphism class of ξ^{♭}. More precisely, let ψ: R(2, G) × R(2, G) → π_{1}(G) be the map determined by ψ(α, β)(z) := [α(z)β(z)^{−1}] (one uses that SO(2) ≈ S^{1} and that ψ is well defined if G is connected). Theorem A. Suppose that G is connected Lie group. Then,

(i) the map J: E(2, G) → R(2, G) × R(2, G) is a bijection. (ii) if J(ξ) = ([α], [β]), then ψ(α, β) = C(ξ^{♭}). We shall now generalize Theorem A for n ≥ 2 or G any Lie group. In general,

J is then neither injective nor surjective and C(ξ^{♭}) is not determined by J(ξ). Consider SO(n − 1) as the subgroup of SO(n) fixing the last coordinate. The restriction [μ] ↦ [μ |_{SO(n−1)}] gives a map Res : R(n, G) → R(n − 1, G). Denote by R(n, G) ×_{(n−1)} R(n, G) the set of ([α], [β]) ∈ R(n, G) × R(n, G) such that Res [α] = Res [β]. If φ: H → G is a group homomorphism, we denote by Z_{φ} ⊂ G the centralizer of φ(H). Theorem B. Let G be any Lie group G. Then

(i) the image of J is R(n, G) ×_{(n−1)} R(n, G). (ii) let α, β: SO(n−1) → G be two smooth homomorphisms such that α |_{SO(n−1)} =

β |_{SO(n−1)} =: γ. Then J^{−1}([α], [β]) is in bijection with with the set of double cosets π_{0}(Z_{α})\π_{0}(Z_{γ})/π_{0}(Z_{β}).

Remark 2.1. The compatibility statement in Part (i) of Theorem B was also observed by K. Grove and W. Ziller [7, Prop. 1.6]. In § 8, Theorem B is extended to a more general setting, to include equivariant principal bundles over “special” Π-manifolds in the sense of Jänich [13]. In particular this provides a classification of the equivariant bundles considered by Grove and Ziller.

Since SO(1) is trivial, Theorem B reduces to Part (i) of Theorem A when n = 2. To determine C(ξ^{♭}) as in Part (ii) of Theorem A, we must choose particular representatives of [α] and [β] (in general, J(ξ) does not determine ξ^{♭}: see examples 7.2 and 7.5). An isotropic lifting for ξ is a smooth curve c̃: [−1, 1] → E(ξ) lifting the meridian arc c(t) = (0, …, cos(πt/2), sin(πt/2)) and such that B · c̃(t) = c̃(t)α(B) for all B ∈ SO(n − 1). Isotropic lifting always exist (see Lemma 3.5). Choosing a := c̃(−1) and b := c̃(−1) leads to isotropy representations α, β: SO(n) → G such that α |_{SO(n−1)} = β |_{SO(n−1)}. The map ψ(α, β): SO(n) → G constructed as in Theorem A then satisfies ψ(α, β)(AB) = ψ(α, β)(A) when B ∈ SO(n − 1). It thus induces a map

ψ̄(α, β): S^{n−1} ≅ SO(n)/SO(n − 1) → G.

Note that ψ̄ is well defined since α and β are actual homomorphisms and not conjugacy classes.

Proposition C. Let ξ be a (n, G)-bundle. Let α, β: SO(n) → G be the isotropy representation associated to the end points of an isotropic lifting. Then, [ψ̄(α, β)] = C(ξ^{♭}) in π_{n−1}(G)/π_{0}(G).

We shall prove two consequences of Theorem B and Proposition C which emphasize the constrast between the cases n = 2 and n ≥ 3.

Proposition D. Let η be a principal G-bundle over S^{2} with G a non-trivial Lie group. Then there exist infinitely many ξ ∈ E(2, G) such that ξ^{♭} ≅ η.

Proposition E. For G a compact Lie group, the set E(n, G) is finite when n ≥ 3.

These results are proved in § 5, while the former sections are devoted to preliminary material. In Section 6, we determine which (n, G)-bundles come from an SO(n + 1)-equivariant bundles. Examples are given in § 7.

3. Preliminary constructions

3.1. J is well defined. This follows from the following lemma.

Lemma 3.2. Let ξ be a (n, G)-bundle. Let a, b ∈ E(ξ) such that p(a) = (0, …, −1) and p(b) = (0, …, 1). Let β, α be the maps from SO(n) to G determined by the formulae A · a = a · α(A) and A · b = b · β(A). Then α and β are smooth homomorphisms and their class in R(n, G) depends only on [ξ] ∈ E(n, G).

Proof. Let A, B ∈ SO(n). One has

\begin{aligned}a · α(BA) & = & (BA) · a = B · (A · a) = B · (a · α(A)) = \\ & = & (B · a) · α(A) = a · (α(B)α(A)). \end{aligned}

Therefore, α and similarly, β, are homomorphism. They are smooth because the action of SO(n) is smooth. If a′ is another choice instead of a, there exists g ∈ G such that a′ = a · g and one has

a · (gα′(A)) = a′ · α′(A) = A · a′ = A · a · g = a · (α(A)g), (3.3)

whence α′(A) = g^{−1}α(A)g. This proves that the class of (α, β) in R(n, G) does not depend on the choice of a and b. Now, if h: E(ξ) ⟶^{≈} E(ξ′) is a (SO(n), G)- equivariant diffeomorphism over the identity of S^{n}, then, by choosing a′ := h(a) and b′ := h(b), one has (α′, β′) = (α, β). The proof of Lemma 3.2 is then complete.

3.4. Isotropic liftings. Let I := [−1, 1] and c: I → S^{n} be the parametrisation of the meridian arc c(t) = (0, …, cos(πt/2), sin(πt/2)). Let c̃: I → E = E(ξ) be a (smooth) lifting of c. As c(t) is fixed by SO(n−1), one has B·c̃(t) = c̃(t)·α_{t}(B), for B ∈ SO(n−1). As in the proof of Lemma 3.2, one checks that this gives a smooth path α_{t} (t ∈ I) of homomorphisms from SO(n − 1) to G, which depends on the lifting c̃. Call c̃ isotropic if α_{t} is constant: α_{t}(B) = α(B) for all B ∈ SO(n − 1).

Lemma 3.5. Any (n, G)-bundle admits an isotropic lifting.

We shall make use of connections on (n, G)-bundles which are SO(n)-invariant. These can be obtained by averaging any connection (see [1, p. 522]), since the space of connections is an affine space. If a curve u(t) in E(ξ) is horizontal for a SO(n)-invariant connection, then u(t) · g and A · u(t) are horizontal. Lemma 3.5 then follows from the following

Lemma 3.6. Let ξ be a (n, G)-bundle endowed with a SO(n)-invariant connection. Then, any lifting c̃ of c which is horizontal is isotropic.

Proof. If c̃ is an horizontal lifting , so are B · c̃ and c̃ · α(B) for B ∈ SO(n − 1). As B · c̃(−1) = c̃(−1) · α(B), one has B · c̃(t) = c̃(t) · α(B) for all t ∈ I.

3.7. The (n, G)-bundles ξ_{α,β}. If X is a topological space, the unreduced suspension ΣX is

ΣX := I × X/{(−1, x) ∼ (−1, x′) and (1, x) ∼ (1, x′), ∀ x, x′ ∈ X}.

We donote by C_{−}X the image of [−1, 1)×X in ΣX and by C_{+}X those of (−1, 1]× X.

Let (α, β) be a pair of smooth homomorphisms from SO(n) to G. Define the space Ê_{α,β} by

\begin{aligned}Ê_{α,β} & := & I × SO(n) × G / \{(−1, A′, g) ∼ (−1, A, α(A^{−1}A′) g) and \\ & & (1, A′, g) ∼ (1, A, β(A^{−1}A′) g), ∀A ∈ SO(n)\}. \end{aligned}

The space Ê_{α,β} admits an obvious free right action of G and a map p: Ê_{α,β} → ΣSO(n). This makes a principal G-bundle over ΣSO(n); indeed, trivializations on C_{±}SO(n) are given by

\begin{matrix}φ̂_{−}: & [t, A, g] & ↦ & ([t, A], α(A) g) & if & −1 ≤ t < 1 \\ φ̂_{+} : & [t, A, g] & ↦ & ([t, A], β(A) g) & if & −1 < t ≤ 1. \end{matrix} (3.8)

The change of trivializations is

φ̂_{−} ∘φ̂−1_{+}([t, A], g) = ([t, A], α(A)β(A)^{−1}g). (3.9)

Now, suppose that α |_{SO(n−1)} = β |_{SO(n−1)}. Form the space E_{α,β} as the quotient

E_{α,β} := Ê_{α,β}/{[t, AB, g] ∼ [t, A, α(B)g], ∀ B ∈ SO(n − 1)}.

Let ε: SO(n) → S^{n−1} be the map which associates to a matrix its last column; it is also the projection ε: SO(n) → SO(n)/SO(n − 1) ≅ S^{n−1}. There is an map p: E_{α,β} → ΣS^{n−1} given by p([t, A, g]) = [t, ε(A)] and a free G-action given by [t, A, g]·g_{1} := [t, A, gg_{1}] As above, we check that this defines a G-principal bundle over ΣS^{n−1}; the trivializations φ̂_{±} descend to trivializations φ̌_{±} over C_{±}S^{n−1}.

The map [t, A] ↦ A · c(t) descends to a homeomorphism f: ΣS^{n−1} ⟶^{≈} S^{n}. By replacing p by f ∘p, we obtain a (topological) principal G-bundle

ξ_{α,β}: E_{α,β} ⟶^{p} S^{n}.

Let S_{±}^{n} be the punctured spheres S_{±}^{n} := f(C_{±}S^{n−1}). The trivializations given by the compositions

φ_{±}: p^{−1}(S_{±}^{n}) ⟶^{φ̌_{±}} C_{±}S^{n−1} × G ⟶^{f× id} S_{±}^{n} × G (3.10)

are homeomorphisms from p^{−1}(S_{±}^{n}) onto manifolds. The change of trivialization is a diffeomorphism, being obtained by conjugating that of (3.9) by f. Therefore, φ_{±} produce a smooth manifold structure on E_{α,β}. The map p and the G action are smooth. One checks that the map

SO(n) × Ê_{α,β} → Ê_{α,β} given by C · [t, A, g] := [t, CA, g]

descends to a smooth SO(n)-action on E_{α,β} which makes ξ_{α,β} a (n, G)-bundle.

3.11. Proof of Part (i) of Theorem B. Let ξ be a (n, G)-bundle. By Lemma 3.5 there exists an isotropic lifting c̃: I → E(ξ) of c. Choosing a := c̃(−1) and b := c̃(1) produces a representative (α, β) of J(ξ) with α |_{SO(n−1)} = β |_{SO(n−1)}. Therefore, the image of J is contained in R(n, G) ×_{(n−1)} R(n, G).

Conversely, a class P ∈ R(n, G) ×_{(n−1)} R(n, G) is represented by a pair (α, β) with α |_{SO(n−1)} = β |_{SO(n−1)}. Let 1 be the identity matrix in SO(n) and e be the unit element of G. Computing J(ξ_{α,β}) with the points a := [−1, 1, e] and b := [1, 1, e] in E_{α,β} shows that J(ξ_{α,β}) = P.

4. The map J̃_{γ}

Let γ: SO(n − 1) → G be a smooth homomorphism. Define a set R_{γ}(n, G) as follows: an element of R_{γ}(n, G) is represented by a pair (α, β) of smooth homomorphisms from SO(n) to G such that α |_{SO(n−1)} = β |_{SO(n−1)} = γ. Two pairs (α_{1}, β_{1}) and (α_{2}, β_{2}) represent the same element of R_{γ}(n, G) if there is a smooth path {g_{t} | t ∈ [−1, 1]} in the centralizer Z_{γ} of γ(SO(n − 1)) such that α_{2}(A) = g_{−1}α_{1}(A)g_{−1}^{−1} and β_{2}(A) = g_{1}β_{1}(A)g_{1}^{−1}. There is an obvious map j: R_{γ}(n, G) → R(n, G) ×_{(n−1)} R(n, G).

Part (i) of Theorem B, already proven in (3.11), permits us to define a map J̄: E(n, G) → R(n − 1, G) by J̄(ξ) := Res [α] = Res [β]. We shall now compute the preimage J̄^{−1}([γ]).

Proposition 4.1. Let γ: SO(n − 1) → G be a smooth homomorphism. Then there exists a bijection J̃_{γ}: J̄^{−1}([γ]) ⟶^{≈} R_{γ}(n, G) such that j ∘J̃_{γ} = J.

The proof divides into several steps.

4.2. Definition of J̃_{γ}. Let ξ be a (n, G)-bundle with J̄(ξ) = [γ]. Choose, using Lemma 3.5, an isotropic lifting c̃_{0} : I → E(ξ) of c. As J̄(ξ) = [γ], the constant path α_{t}^{0} : SO(n − 1) → G associated to c̃_{0} is conjugated to γ: there exists g ∈ G such that α_{t}^{0}(B) = gγ(B)g^{−1}. Let c̃ := c̃_{0} · g. Like in Equation (3.3), one checks that c̃ is γ-isotropic, i.e. α_{t} = γ. Choosing a := c̃(−1) and b := c̃(1) then produces a pair (α, β) of smooth homomorphisms from SO(n) to G which represents a class J̃_{γ}(ξ) in R_{γ}(n, G).

To see that J̃_{γ} is thus well defined, let c̃′ be another γ-isotropic lifting of c. The smooth path t ↦ g_{t} ∈ G defined by c̃′(t) = c̃(t) · g_{t} then satisfies, for all B ∈ SO(n − 1):

γ(B) = α′_{t}(B) = g_{t}^{−1}α_{t}(B)g_{t} = g_{t}^{−1}γ(B)g_{t}.

Therefore, g_{t} ∈ Z_{γ}. One has α′(A) = g_{−1}^{−1}α_{t}(A)g_{−1} and β′(A) = g_{1}^{−1}β_{t}(A)g_{1}, for all A ∈ SO(n), which proves that J̃_{γ}(ξ) does not depend on the choice of a γ-isotropic lifting.

Now, if h: E(ξ) ⟶^{≈} E(ξ′) is a (SO(n), G)-equivariant diffeomorphism over the identity of S^{n} and c̃: I → E(ξ) is a γ-isotropic lifting for ξ, then c′ := h∘c̃ is a γ-isotropic lifting for ξ′ giving (α′, β′) = (α, β). This proves that J̃_{γ} is well defined.

4.3. Surjectivity of J̃_{γ}. Let (α, β) represent a class P in R_{γ}(n, G). One checks that J̃_{γ}(ξ_{α,β}) = P, using that the path t ↦ [t, 1, e] is a γ-isotropic lifting for ξ_{α,β}.

4.4. Injectivity of J̃_{γ}. Let γ: SO(n − 1) → G be a smooth homomorphism. and let ξ be a (n, G)-bundle with J̄(ξ) = [γ]. There exists a ∈ E(ξ) with p(a) = (0, …, −1) and B · a = a · γ(B) for all B ∈ SO(n − 1). Choose a SO(n)- invariant connection on ξ and let c̃ an horizontal lifting of c with c̃(−1) = a. By Lemma 3.6, c̃ is γ-isotropic. If β: SO(n) → G is defined by A · c̃(1) = c̃(1) · β(A), then (α, β) represents J̃_{γ}(ξ).

Consider the map λ̂: Ê_{α,β} → E(ξ) given by

λ̂([t, A, g]) := A · c̃(t) · g.

The map λ̂ descends to a continuous map λ: E_{α,β} → E(ξ) which is both SO(n) and G-equivariant and which covers the identity of S^{n}. Therefore ξ and ξ_{α,β} are isomorphic as topological (n, G)-bundles. What remains to prove is that λ is a diffeomorphism, which is only non-trivial around the fibers E_{±} above the north and south poles.

The connection on ξ provides a smooth trivialization of ξ restricted to the punctured sphere Sn_{−} (see 3.7) in the following way. Consider the map s_{−}: p^{−1}(Sn_{−}) → E_{−} assigning to z the end point in E_{−} of the horizontal path through z above the meridian arc through p(z). Define the G-equivariant map σ_{−}: p^{−1}(Sn_{−}) → G by s_{−}(z) = a · σ_{−}(z). The required trivialization τ_{−}: p^{−1}(Sn_{−}) → Sn_{−} × G is τ_{−}(z) := (p(z), σ_{−}(z)).

Take the trivialization φ_{−} for ξ_{α,β} defined in Equations (3.10) of (3.7). As c̃ is horizontal, one has

τ_{−} ∘λ∘φ−1_{−}(x, g) = (x, g).

This, and the same for E_{+}, prove that λ is a diffeomorphism. We have thus established that if J̃_{γ}(ξ) is represented by (α, β) then the (n, G)-bundle ξ is isomorphic to ξ_{α,β}, which proves the injectivity of J̃_{γ}.

The proof of Proposition 4.1 is now complete.

5. Proof of the main results

This section contains the proofs of the results stated in § 2.

5.1. Proof of Theorem B. Part (i) has already been proven in (3.11). We shall now prove Part (ii). Let α, β: SO(n − 1) → G be two smooth homomorphisms such that α |_{SO(n−1)} = β |_{SO(n−1)} = γ. The pair (α, β) defines a class [α, β] ∈ R_{γ}(n, G). The group Z_{γ} × Z_{γ} acts on R_{γ}(n, G) by

(g, h) · [α, β] := [gαg^{−1}, hβh^{−1}].

The set J^{−1}([α], [β]) ⊂ J̄^{−1}([γ]) is, via the bijection J̃_{γ}: J̄^{−1}([γ]) ⟶^{≈} R_{γ}(n, G) of Proposition 4.1, in bijection with an orbit of the above action. Let “∼” be the equivalence relation on Z_{γ} × Z_{γ} defined by (g_{1}, h_{1}) ∼ (g_{2}, h_{2}) iff (g_{1}, h_{1})[α, β] = (g_{2}, h_{2})[α, β]. Let

ϕ: Z_{γ} × Z_{γ} → π_{0}(Z_{α})\π_{0}(Z_{γ})/π_{0}(Z_{β})

be the map defined by ϕ(g, h) := [g^{−1}h]. Part (ii) of Theorem B then follows from the following lemma.

Lemma 5.2. (g_{1}, h_{1}) ∼ (g_{2}, h_{2}) iff ϕ(g_{1}, h_{1}) = ϕ(g_{2}, h_{2}).

Proof. Suppose that (g_{1}, h_{1}) ∼ (g_{2}, h_{2}). This means that there exist s_{−}, s_{+} ∈ Z_{γ}, with [s_{−}] = [s_{+}] in π_{0}(Z_{γ}), such that the following equality

(g_{1}αg_{1}^{−1}, h_{1}βh_{1}^{−1}) = (s_{−}g_{2}αg_{2}^{−1}s−1_{−}, s_{+}h_{2}βh_{2}^{−1}s−1_{+})

holds in Z_{γ} × Z_{γ}. This implies that

g_{1} = s_{−} g_{2} A and h_{1} = s_{+} h_{2} B

with A ∈ Z_{α} and B ∈ Z_{β} (the centralizers of the images of α and β). Therefore

g_{1}^{−1}h_{1} = A^{−1}g_{2}^{−1}s−1_{−}s_{+} h_{2} B, which implies ϕ(g_{1}, h_{1}) = ϕ(g_{2}, h_{2}).

To prove the converse, observe that

• (g, h) ∼ (C g, C h) for C ∈ Z_{γ}.

• (g, h) ∼ (gA, hB) for A ∈ Z_{α} and B ∈ Z_{β}.

• (g, h) ∼ (g, uh) for u in the identity component of Z_{γ}.

Suppose that ϕ(g_{1}, h_{1}) = ϕ(g_{2}, h_{2}). This means that there are A ∈ Z_{α}, B ∈ Z_{β} and u in the identity component of Z_{γ} auch that g_{1}^{−1}h_{1} = A^{−1}g_{2}^{−1}uh_{2}B (u can be put in the middle since the identity component of Z_{γ} is a normal subgroup of Z_{γ}). One then has

(g_{1}, h_{1}) ∼ (e, g_{1}^{−1}h_{1}) = (e, A^{−1}g_{2}^{−1}uh_{2}B) ∼ (g_{2}A, u h_{2}B) ∼ (g_{2}, h_{2}) .

Proof of Proposition C. Let ξ, α and β as in the statement of Proposition C. Let γ := α_{|SO(n−1)} = β_{|SO(n−1)}. Then, ξ ∈ J̄^{−1}([γ]) and, by 4.2, one has J̃(ξ) = [α, β] in R_{γ}(n, G). By 4.4, ξ = ξ_{α,β}. Therefore, C(ξ^{♭}) = C(ξ_{α,β}^{♭}) = [ψ̄(α, β)], the last equality coming from Equation (3.9) of 3.7 and that C(ξ^{♭}) can be represented by the transition function [23, Th. 18.4].

Proof of Theorem A. Since SO(1) is trivial, Z_{γ} = G which is suppose to be connected. Therefore, Part (i) is a particular case of Part (i) of Theorem B. Let c: I → S^{n} parametrizing the meridian arc, as in (3.4). Let α, β: SO(2) → G be two homomorphisms representing J(ξ). One can find a and b so that α and β are the isotropy representations associated to a and b. As G is connected, the submanifold P_{0} := p^{−1}(c(I)) of E(ξ) is connected and there is a smooth lifting c̃ of c such that c̃(−1) = a and c̃(1) = b. As SO(1) is trivial, c̃ is isotropic. Part (ii) of Theorem A then follows from Proposition C.

Proof of Proposition D. Recall that any element of π_{1}(G, e) can be represented by a homomorphism (a geodesic in a maximal compact subgroup K of G, with a Kbi-invariant Riemannian metric, being a 1-parameter subgroup [11, Ch. IV, § 6]). Therefore, if η is a G-bundle over S^{2}, there exists a homomorphism α: S^{1} → G such that C(η) = [α]. For q ∈ N, let α_{q}: S^{1} → G given by α_{q}(z) := α(z)^{q}. If α is not trivial, the classes [α_{q}] are all distinct in R(2, G). Indeed, the set R(2, G) is in bijection with lattice points in a Weyl chamber of the Lie algebra of a maximal torus of G and the point representing α_{q} is q times those representing α.

Suppose first that η is not trivial. Hence, α is not trivial and [α_{q+1}, α_{q}] are all different classes in R_{γ}(2, G) with [ψ̄(α_{q+1}, α_{q}]) = C(η). The result then follows from Propositions C and 4.1.

When η is trivial, one takes any non-trivial homomorphism α: SO(2) → G. The classes [α_{q}, α_{q}] in R_{γ}(2, G) represent infinitely many distinct SO(2)- equivariant G-bundles ξ_{q} with trivial ξ_{q}^{♭}.

Proof of Proposition E. If n ≥ 3, the group SO(n) is semi-simple and the set R(n, G) is finite. The latter follows from the following known results:

• a homomorphism is determined by its tangent map at the identity (as a homomorphism of Lie algebras).

• the Lie algebra of G contains only finitely many semi-simple Lie subalgebras, up to inner automorphism [21, Prop. 12.1].

• there are only finitely many homomorphisms between two semisimple Lie algebras.

Also, if G is compact, the group Z_{γ} is compact and then π_{0}(Z_{γ}) is finite. Proposition E then follows from Theorem B.

Remark 5.3. To remove the hypothesis “G compact” from Proposition E, it is enough to consider the case G connected. Indeed, R_{γ}(n, G) is a quotient of R_{γ}(n, G_{e}), where G_{e} is the connected component of e. One would then need the following kind of result: if H is a compact Lie subgroup of a connected Lie group G, then π_{0}(Z(H)) is finite. We do not know whether this true.

6. SO(n + 1)-equivariant bundles

In this section, we describe the (n, G)-bundles which are SO(n+1)-equivariant G-bundles, for the natural action of SO(n + 1) on S^{n}. Let δ: SO(n) ⟶^{≈} SO(n) be the conjugation by the diagonal (n × n)-matrix Diag (1, …, 1, −1) (or, equivalently, Diag (−1, …, −1, 1)). If α: SO(n) → G is a smooth homomorphism, observe that Res [α] = Res [α∘δ] in R(n − 1, G).

Theorem 6.1. Let ξ be a (n, G)-bundle. If ξ comes from an SO(n+1)-equivariant G-bundle then J(ξ) is of the form ([α], [α∘δ]).

For any [α] ∈ R(n − 1, G) there is a unique ξ ∈ R(n, G) which comes from a SO(n + 1)-equivariant G-bundle and such that J(ξ) = ([α], [α∘δ]).

Proof. For θ ∈ [0, π], let R_{θ} ∈ SO(n + 1) be the rotation of angle θ in the plane of the last 2 coordinates. Let R := R_{π}, the diagonal matrix with entries (1, …, 1, −1, −1).

Let ξ be an SO(n + 1)-equivariant bundle. Choose a, b ∈ E(ξ), with p(a) = (0, …, −1) and let b := R · a. For A ∈ SO(n), one has R^{−1}AR = δ(A) and

\begin{aligned}b β(A) & = & A · b = A · (R · a) = R · (R^{−1}AR) · a \\ & = & R · a α(δ(A)) = b α(δ(A)), \end{aligned} (6.2)

whence β = α∘δ, which proves Part (i).

Let α: SO(n) → G be a smooth homomorphism and set γ := α_{|SO(n−1)}. Suppose that ξ is an SO(n + 1)-equivariant G-bundle with a ∈ E(ξ) such that p(a) = (0, …, −1) and A · a = a α(A) for A ∈ SO(n). Then ξ ∈ J̄^{−1}([γ]). let c: I → E(ξ) be the curve c(t) := R_{θ(t)} · a, where θ(t) := \frac{π}{2}(t − 1). Using that R_{θ}^{−1}BR_{θ} = B for B ∈ SO(n − 1), on checks, as in equation (6.2), that c is a γ-isotropic lifting. By (4.2) and equation (6.2), one has J̃_{γ}(ξ) = ([α], [α∘δ]) in R_{γ}(n, G). By Proposition 4.1, this proves that ξ is determined by α, which proves the uniqueness statment of Part (ii).

It remains to construct, for a smooth homomorphism α: SO(n) → G, an SO(n + 1)-equivariant G-bundle with J(ξ) = ([α], [α∘δ]). Consider the map p: SO(n+1) → S^{n} sending a matrix to its last column. This makes an SO(n+1)- equivariant SO(n)-bundle (the principal SO(n)-bundle associated to the tangent bundle of S^{n}). Let ξ be the G-bundle obtained by the Borel construction, using the homomorphism α∘δ: SO(n) → G

E(ξ) := SO(n + 1) × G/{(BA, g) = (B, α(δ(A))g)}. (6.3)

This is an SO(n + 1)-equivariant G-bundle. Choosing a := (R, e) and b := (I, e), one sees that

A · a = (AR, e) = (Rδ(A), e) = (R, α(A)e) = a α(A)

and

A · b = (A, e) = (I, α(δ(A))e) = b α(δ(A)).

Therefore, ξ is an SO(n + 1)-equivariant G-bundle with J(ξ) = ([α], [α∘δ]).

Remark 6.4. Theorem 6.1 and its proof show that any SO(n + 1)-equivariant G bundle is derived from the tangent bundle to S^{n} by the Borel construction (formula (6.3)). This can be compared with [10, § 6].

7. Examples and applications

Notation: if X is a set, we denote by ΔX the diagonal in X × X.

7.1. n = 2 and G = U(m). A homomorphism α: S^{1} → U(m) has, up to conjugacy, a unique diagonal form α(z) = Diag (z^{p_{1}}, …, z^{p_{m}}), with p_{1} ≥ ⋯ ≥ p_{m}. The same holds for β. By Theorem A, E(2, U(m)) is then in bijection with the set of pairs (p, q) of m-tuples of non-decreasing integers. In π_{1}(U(m)) = Z, one has [α] = ∑_{i=1}^{m} p_{i} and [β] = ∑_{i=1}^{m} q_{i}, so, by Proposition C:

C(ξ^{♭}) = ∑_{i=1}^{m}(p_{i} − q_{i}).

If one wishes instead to characterize ξ^{♭} by its first Chern number c(ξ^{♭}) ∈ H^{2}(S^{2}) = Z, then c(ξ^{♭}) = −C(ξ^{♭}) [19, p. 445].

For instance, if τ is the unit tangent bundle over S^{2} with the natural action, then α(z) = z, β(z) = z^{−1}, so C(τ) = −2 and c(τ) = 2 = χ(S^{2}).

Note that, if ξ comes from a SO(3)-bundle, then, by Theorem 6.1, one has q_{i} = −p_{i} (like for τ above). In particular c(ξ^{♭}) must be even (see also [10, (6.3)]).

7.2. n = 2 and G = O(2). For q ∈ Z, let α_{q}: SO(2) → O(2) be the homomorphism A → A^{q}. The set R(2, O(2)) is in bijection with N given by α_{q} ↦ |q|. The same recipe produces a bijection π_{1}(O(2), e)/π_{0}(O(2)) ≅ N. Let ξ_{p,q} := ξ_{α_{p},α_{q}} be the (2, O(2))-bundles constructed in (3.7). By Proposition 4.1 and (4.3), each E(2, O(2)) is represented by some ξ_{p,q}, with the only relation ξ_{p,q} = ξ_{−p,−q}. One has J(ξ_{p,q}) = (|p|, |q|) and C(ξ_{p,q}) = |p − q|. Therefore, J^{−1}(r, s) contains 1 element if rs = 0 and 2 otherwise.

7.3. n = 2 and G = SO(m) with m ≥ 3. A maximal torus of SO(m) is formed by matrices containing 2-blocks concentrated around the diagonal, so isomorphic to SO(2)^{k} where k = [m/2]. As in 7.1, by Theorem A, E(2, SO(m)) is then in bijection with the set of pairs (p, q) of k-tuples of non-decreasing integers. The bundle ξ^{♭} is determined by its second Stiefel-Whitney number w(ξ^{♭}) ∈ Z_{2} which is then given by

w(ξ^{♭}) = ∑_{i=1}^{[ \frac{m}{2} ]}(p_{i} − q_{i}) (mod 2).

Again, ξ comes from a SO(3)-equivariant bundle iff q_{i} = −p_{i} and then ξ^{♭} is trivial.

7.4. n = 2k + 1 ≥ 3 and G is a compact classical group other than SO(2m). The important thing is that SO(n − 1) contains a maximal torus of SO(n). Therefore, by [3, Ch. 6, Corollary 2.8], for any embedding ψ: G ↪ U(m), the representations ψ ∘α and ψ ∘β are conjugate in in U(M). For G a compact classical group other than SO(2m), this implies that α and β are conjugate in G [20, Pro. 8, p.56]. Therefore, the image of J is the diagonal ΔR(n, G). We do not know whether this true for G = SO(2m) (see [20, Rm. p.57] for a possible source of counter-examples).

7.5. n = 2k + 1 ≥ 3 and G = SO(n). The set R(n, SO(n)) has just two elements, represented by the trivial homomorphism and the identity id of SO(n). If ι: SO(n−1) ⊂ SO(n) denotes the inclusion, then Z_{[ι]} contains 2 elements, represented by the identity matrix and the diagonal matrix D with entries (−1, …, −1, 1). Let δ be the inner automorphism of SO(n) given by the conjugation by D. By Theorem B, the set E(n, SO(n)) for n = 2k + 1 ≥ 3 then contains 3 elements:

(i) the trivial bundle S^{n} × SO(n) with the action A · (z, B) = (A · z, B). The isotropy representations are both trivial.

(ii) the trivial bundle S^{n} × SO(n) with the action A · (z, B) = (A · z, AB). It is characterized by J̄(ξ) = [ι] and J_{ι}(ξ) = ([id], [id]). This does not come from an SO(n + 1)-equivariant bundle.

(iii) the principal SO(n)-bundle T S^{n} associated with the tangent bundle of S^{n}. It is characterized by J̄(T S^{n}) = [ι] and J_{ι}(T S^{n}) = ([id], [δ]). This comes from an SO(n + 1)-equivariant bundle.

Observe that ([id], [δ]) = ([δ], [id]) in R_{ι}(n, SO(n)). By Proposition C, this implies that C(T S^{n}) = −C(T S^{n}). This proves again the classical fact that C(T S^{2k+1}) ∈ π_{2k}(SO(2k + 1)) is of order 2 [4, Cor. IV.1.11].

7.6. n = 2k ≥ 6 and G = SO(n). The set R(n, SO(n)) contains 3 elements, represented by the trivial homomorphism, the identity id of SO(n) and the conjugation δ by the diagonal matrix with entries (−1, …, −1, 1). The non-trivial homomorphisms restrict to the inclusion ι: SO(n − 1) ⊂ SO(n). The group Z_{ι} is trivial. Therefore, J is injective and the set E(n, SO(n)) for n = 2k then contains 5 elements:

(i) the trivial bundles S^{n} × SO(n) with the actions A · (z, B) = (A · z, B), A · (z, B) = (A · z, AB) and A · (z, B) = (A · z, δ(A)B). Their images by J is the diagonal ΔR(n, SO(n)).

(ii) the principal SO(n)-bundle T S^{n} associated with the tangent bundle of S^{n}.

One has J(T S^{n}) = ([id], [δ]). (iii) the (n, G)-bundle −T S^{n} with J(−T S^{n}) = ([δ], [id]). Its underlying SO(n)-

principal bundle is stably trival with Euler number −2. The trivial bundle with action A · (z, B) = (A · z, B), as well as the bundles in

(ii) and (iii) are the ones coming from SO(n + 1)-equivariant bundles.

7.7. n = 4 and G = SO(3). The groups SO(4) and SO(3) are built up out of the unit quaternions S^{3} by SO(4) ≅ (S^{3} × S^{3})/{(1, 1), (−1, −1)} and SO(3) ≅ S^{3}/{±1}. Recall that these isomorphisms are constructed as follows: the orthogonal transformation A_{p,q} ∈ SO(4) associated to (p, q) ∈ S^{3} × S^{3} is A_{p,q}(x) := pxq̄, where x ∈ H is a quaternion and H is made isomorpohic to R^{4} by choosing (i, j, k, 1) as a basis. The correspondence p → A_{p,p} then induces the inclusion ι: SO(3) ⊂ SO(4). As for the automorphism δ of SO(4) of § 6, the conjugation by D := Diag (−1, …, −1, 1), as Dx = x̄, one checks easily that δ(A_{p,q}) = A_{q,p}.

The non-equivariant isomorphism class of a SO(3)-principal bundle η is characterized by C(η) ∈ π_{3}(SO(3)) = π_{3}(S^{3}) = Z. It is also determined by its first Pontrjagin number p(η) ∈ 4 Z, with the relation p(η) = 4 C(η).

The set R(4, SO(3)) contains 3 elements represented by the trivial homomorphism and those induced by the projections S^{3} × S^{3} → S^{3} given by σ_{1}(p, q) := p, σ_{2}(p, q) := q. The last two restrict over SO(3) to the identity id of SO(3). The group Z_{ι} being trivial, the map J is injective. This shows that the set E(4, SO(3)) contains 5 elements:

(i) the trivial bundles S^{4} × SO(3) with the actions A · (z, B) = (A · z, B) and A · (z, B) = (A · z, σ_{i}(A)B) for i = 1, 2. Their images by J is the diagonal ΔR(4, SO(3)).

(ii) the principal SO(3)-bundle H: RP^{7} → S^{4} coming from the quaternionic Hopf bundle S^{7} → S^{4}; the SO(4)-action comes from the SU(2) × SU(2)- action on S^{7} given by (p, q)·(z_{1}, z_{2}) = (pz_{1}, qz_{2}). One has J(H) = ([σ_{1}], [σ_{2}]) and p(H^{♭}) = 4.

(iii) the (n, G)-bundle −H with J(−H) = ([σ_{2}], [σ_{1}]) and p(H^{♭}) = −4.

The trivial bundle with action A · (z, B) = (A · z, B), as well as the bundles in (ii) and (iii) are the ones coming from SO(5)-equivariant bundles.

7.8. n = 4 and G = SO(4). Taking the notations of 7.7, the set R(4, SO(4)) contains 5 elements represented by:

• the trivial homomorphism.

• those induced by σ_{1}(p, q) := (p, p) and σ_{2}(p, q) := (q, q).

• the identity id of SO(4).

• the homomorphism δ(p, q) := (q, p).

The non-trivial homomorphisms all restrict to to ι over SO(3). The group Z_{ι} is trivial and then J is injective. The non-equivariant isomorphism class of a SO(4)-principal bundle η is characterized by C(η) ∈ π_{3}(SO(4)) = π_{3}(S^{3} × S^{3}) = Z⊕Z. More usually, one takes the pair of integers (p(η), e(η)) formed by the first Pontrjagin number (∈ 2Z) and the Euler number of η. The linear map which sends C(η) to (p(η), e(η)) has matrix (_{1}^{2} _{−1}^{2}) (as can be checked on the examples below).

One sees that the set E(4, SO(3)) then contains 17 elements:

(i) the trivial bundles S^{4} × SO(3) with the 5 actions using the above representations. Their images by J are just the diagonal elements ΔR(4, SO(4)).

(ii) the principal SO(4)-bundle Ĥ whose total space is E(Ĥ) := RP^{7} ×_{SO(3)} SO(4). One has J(Ĥ) = ([σ_{1}], [σ_{2}]) and C(Ĥ^{♭}) = (1, 1) therefore its characteristic classes are (p(H^{♭}), e(H^{♭})) = (4, 0). Also its “inverse” −Ĥ, with J(−Ĥ) = ([σ_{2}], [σ_{1}]) and (p(H^{♭}), e(H^{♭})) = (−4, 0).

(iii) the principal SO(4)-bundle T S^{4} associated with the tangent bundle of S^{4}. One has J(T S^{4}) = ([id], [δ]). Therefore, C(T S^{4}) = (1, −1) and (p(T^{♭}), e(T^{♭})) = (0, 2). Again, one can consider its inverse.

(iv) the (n, G)-bundles ξ_{i} (i = 1, 2) with J(ξ_{i}) = ([id], [σ_{i}]) and their inverses −ξ_{i}. They satisfy C(ξ_{1}) = (0, −1) and C(ξ_{2}) = (1, 0), or, equivalently:

(p(ξ_{1}^{♭}), e(ξ_{1}^{♭})) = (−2, 1) and (p(ξ_{2}^{♭}), e(ξ_{2}^{♭})) = (2, 1).

(v) the (n, G)-bundle ξ_{i,δ} (i = 1, 2) with J(ξ_{i}) = ([δ], [σ_{i}]) and their inverses. They satisfy C(ξ_{1,δ}) = (−1, 0) and C(ξ_{1,δ}) = (0, 1), or, equivalently:

(p(ξ_{1,δ}^{♭}), e(ξ_{1,δ}^{♭})) = (−2, −1) and (p(ξ_{2,δ}^{♭}), e(ξ_{2,δ}^{♭})) = (2, −1).

Only the trivial bundle with action A · (z, B) = (A · z, B) and the bundles in (ii) and (iii) come from SO(5)-equivariant bundles.

7.9. G = U(m). In order to have non-trivial (n, U(m))-bundles, one must have dim U(m) > dim SO(n). We check that we are then in the stable range, where, by Bott periodicity,

π_{n−1}(U(m)) ≈ π_{n−1}(U(m + k)) ≈ \{ \begin{matrix}0 & if n is odd \\ Z & if n is even. \end{matrix}

Problem: which integers occur as C(ξ^{♭}) for a (n, U(m))-bundle ξ ?

8. A more general setting

The orthogonal action of SO(n) on S^{n} is an example of the special Π-manifolds defined by Jänich [13, 1.2]. Other examples include the “cohomogeneity one” actions studied by E. Straume [25] (see [7] for a recent application and other references). In this section we give the classification of equivariant (Π, G)-bundles over special Π-manifolds. We will assume in this section that Π and G are both compact Lie groups.

Let X be a smooth, connected, closed n-dimensional manifold with a smooth Π-action. Choose a Π-invariant Riemannian metric on X, and then each tangent space T_{x}X contains a Π_{x} invariant subspace V_{x} perpendicular to the orbit Π · x. Then X is called a special Π-manifold if for each x ∈ X the representation of Π_{x} on the normal space V_{x} is the direct sum of a trivial representation and a transitive representation.

It follows that the orbit space M = X/Π admits a natural structure as a topological manifold with boundary [13, 1.3], with dimension equal to n − dim(Π/H) where H is the principal isotropy type. Under the “functional” smooth structure [2, VI.6], the orbit space M is a smooth manifold with boundary. The pair (X, π: X → M) is called a special Π-manifold over M.

Special Π manifolds over M were classified by Jänich [13, 3.2], and independently by W.-C. Hsiang and W.-Y. Hsiang [12] (see also [2, V.5, VI.6]).

Let ∂M_{A} = {B_{α}}_{α∈A} denote the set of boundary components of M. An admissible isotropy group system (H, U_{A}) over M consists of a closed subgroup H of Π and a set U_{A} = {U_{α}}_{α∈A} of closed subgroups in Π containing H, with the property that for each α ∈ A there exists a transitive representation in which H appears as the isotropy group of a non-zero vector. Let Γ = N(H)/H and Ω_{α} = N(U_{α}) ∩ N(H)/H for each α ∈ A. The idea of the classification is to re-construct X from the unique principal Γ-bundle P over M such that P_{|M_{0}} = {x ∈ X | Π_{x} = H}, and a reduction of the structural group of P |B_{α} to Ω_{α} over each of the boundary components B_{α}.

Let B_{α} × [0, 1] be a collar neighbourhood of some component B_{α} in M, and let Y_{α} = π^{−1}(B_{α} × [0, 1]) denote its pre-image in X. The key fact is the following identification of Y_{α} as a Π-space.

Theorem 8.1. (Tube Theorem [2, V.4.2]) Let Y = Y_{α}, B = B_{α} and Ω = Ω_{α}. There exists a right Ω-principal bundle Q = Q_{α} over B, and a Π-equivariant diffeomorphism

M_{ψ} ×_{Ω} Q ⟶^{≈} Y

commuting with the projection to [0, 1], where M_{ψ} denotes the mapping cylinder of the canonical projection ψ: Π/H → Π/U.

Let X_{0} = X − ⋃_{α} π^{−1}(B_{α} × [0, 1/2)) and M_{0} = X_{0}/Π. The Tube Theorem shows that X is Π-diffeomorphic to the union

X = X_{0} ∪ ⋃_{α} Y_{α} = Π/H ×_{Γ} P ∪ ⋃_{α} M_{ψ_{α}} ×_{Ω_{α}} Q_{α}

with the identification on the overlaps B_{α} × [1/2, 1] induced by a reduction of structural groups P |B_{α} ≅ Γ ×_{Ω_{α}} Q_{α}.

Two special Π-manifolds (X_{1}, π_{1}) and (X_{2}, π_{2}) over M are called isomorphic when there exists a Π-equivariant diffeomorphism f: X_{1} → X_{2} so that the induced diffeomorphism f̄: M → M is the identity. By the the smooth isotopy lifting theorem of G. Schwarz [22, Corollary 2.4] this is equivalent to Jänich’s original definition where f was assumed to be only isotopic to the identity, by an isotopy fixing ∂M pointwise. Let S[H, U_{A}] denote the set of isomorphism classes of special Π-manifolds over M, with isotropy group system fine equivalent to (H, U_{A}) (cf. [13, §2]).

Theorem 8.2. ([13, 3.2]) Let (H, U_{A}) be an admissible isotropy group system over M, where M is a smooth, compact connected manifold with boundary. Then

S[H, U_{A}] ≅ [M, ∂M_{A}; BΓ, BΩ_{A}] .

In order to classify equivariant (Π, G)-bundles (E, p) over a special Π-manifold X, called special (Π, G)-bundles for short, we will generalize the results of Lashof [16] to describe the bundles over Π-spaces X_{0} and Y_{α} with one orbit, and then follow Jänich’s method to glue the pieces together.

First some general definitions: if ρ: H → G is a (smooth) homomorphism, [ρ] denotes the set of homomorphisms ρ′: H → G such that ρ′(h) = gρ(h)g^{−1} for some g ∈ G and all h ∈ H. We will say that the fibre over x belongs to [ρ] if for each z ∈ p^{−1}(x) there exists ρ′ ∈ [ρ] such that hz = z · ρ′(h) for all h ∈ H. Then let

X^{[ρ]} = {x ∈ X^{H} | the fibre over x belongs to [ρ]}

Let X_{0}^{[ρ]} = X_{0} ∩ X^{[ρ]} and notice that X_{0}^{[ρ]} ⊂ P = {x ∈ X | Π_{x} = H}. By [16, Lemma 1.2], the space

E^{ρ} = {z ∈ E | hz = z · ρ(h), ∀h ∈ H}

is an Z_{ρ}-bundle over X^{[ρ]}, where Z_{ρ} is the centralizer of ρ in G. The group Ĝ = Π×G has a left action on the total space E given by the formula (γ, g)·z = γz·g^{−1} for any (γ, g) ∈ Ĝ and any z ∈ E. Let us set

H⟨ρ⟩ := {(h, ρ(h)) | h ∈ H} ⊂ Π × G

and

Γ⟨ρ⟩ := N(H⟨ρ⟩)/H⟨ρ⟩ .

Then E^{ρ} is just the fixed set of H⟨ρ⟩ in E under this action.

Two special (Π, G)-bundles (E_{1}, p_{1}) and (E_{2}, p_{2}) over M are called equivalent when there exists a Π-equivariant G-bundle isomorphism ϕ: E_{1} → E_{2}, covering a Π-equivariant diffeomorphism f: X_{1} → X_{2}, so that the induced diffeomorphism f̄: M → M is the identity.

We now give the classification of the part of (E, p) lying over M_{0}. After the remarks above, we see that it follows directly from the slice theorem [2, II.5.8].

Theorem 8.3. ([16, 1.9]) Let ρ: H → G be a smooth homomorphism. The equivalence classes of (Π, G)-equivariant bundles (E_{0}, p_{0}) over M_{0} with all fibres belonging to [ρ] are in bijection with the homotopy classes of maps [M_{0}, BΓ⟨ρ⟩].

We next define an admissible (Π, G)-isotropy group system over M to be a set (H, U_{A}; ρ, ρ_{A}), where H and U_{A} are as above, ρ: H → G is a homomorphism, and ρ_{A} = {ρ_{α}}_{α∈A} is a set of homomorphisms ρ_{α}: U_{α} → G such that ρ_{α}|H = ρ. We define

Ω_{α}⟨ρ_{α}⟩ = N(U_{α}⟨ρ_{α}⟩) ∩ N(H⟨ρ⟩)/H⟨ρ⟩

and

Ω_{A}⟨ρ_{A}⟩ = {Ω_{α}⟨ρ_{α}⟩}_{α∈A} .

A special (Π, G)-bundles (E, p) realizes an admissible (Π, G)-isotropy group system (H, U_{A}; ρ, ρ_{A}) over M if (i) there exists points {y_{α} ∈ Y_{α}} such that Π_{y_{α}} = U_{α}, (ii) for each y_{α}, the normal space V_{y_{α}} to the orbit Π · y_{α} has a point z_{α} ∈ V_{y_{α}}

with Π_{z_{α}} = H, (iii) the images c_{α}(t) of rays in V_{y_{α}} joining z_{α} to y_{α}, have isotropic liftings c̃_{α}(t)

to E such that c̃_{α}(t) ⊂ E^{ρ} for 0 ≤ t < 1 and c̃_{α}(1) ⊂ E^{ρ_{α}}. It is not difficult to check (following [13, §2]) that every special (Π, G)-bundle (E, p) over M realizes some admissible (Π, G)-isotropy group system (H, U_{A}; ρ, ρ_{A}). This isotropy group system is unique up to a natural notion of equivalence, extending the “fine-orbit structure” of Jänich.

We say that two isotropy group systems (H, U_{A}; ρ, ρ_{A}) and (H′, U′_{A}; ρ′, ρ′_{A}) are fine-equivalent if the following conditions hold:

(i) there exists an element γ ∈ Π × G such that H′⟨ρ′⟩ = γH⟨ρ⟩γ^{−1}, and (ii) there exist n_{α} ∈ NH⟨ρ⟩ such that U′_{α}⟨ρ′_{α}⟩ = (γn_{α})U_{α}⟨ρ_{α}⟩(γn_{α})^{−1}. Let S[H, U_{A}; ρ, ρ_{A}] denote the set of equivalence classes of special (Π, G)-bundles over M realizing the given (Π, G)-isotropy group system, up to fine equivalence.

Theorem 8.4. Let (H, U_{A}; ρ, ρ_{A}) be an admissible (Π, G)-isotropy group system over M, where M is a smooth, compact connected manifold with boundary. Then

S[H, U_{A}; ρ, ρ_{A}] ≅ [M, ∂M_{A}; BΓ⟨ρ⟩, BΩ_{A}⟨ρ_{A}⟩] .

Proof. Suppose that we are given an admissible (Π, G)-isotropy group system over M. Let (E, p) be a special (Π, G)-bundle over M realizing the given (Π, G)- isotropy group system. By restricting the bundle to M_{0}, we get a map ω_{0} : M_{0} → BΓ⟨ρ⟩ classifying the principal Γ⟨ρ⟩-bundle P ⟨ρ⟩ (which completely determines (E_{0}, p_{0})) by Theorem 8.3. We can apply the Tube Theorem 8.1 to the Ĝ = Π×G action on p^{−1}(Y_{α}) ⊂ E, since z ∈ E^{ρ} means that Ĝ_{z} = H⟨ρ⟩ and similarly for z ∈ E^{ρ_{α}}. This identifies the restriction of our bundle to the part over B_{α} × [0, 1] as M_{φ_{α}} ×_{Ω_{α}⟨ρ_{α}⟩} Q⟨ρ_{α}⟩, where

φ_{α}: Ĝ/H⟨ρ⟩ → Ĝ/U_{α}⟨ρ_{α}⟩

is the Ĝ-equivariant projection, and Q⟨ρ_{α}⟩ is a principal right Ω_{α}⟨ρ_{α}⟩-bundle over B_{α}. The classifying map ω_{0} for P ⟨ρ⟩ therefore extends to a map

ω: (M, ∂M_{A}) → (BΓ⟨ρ⟩, BΩ_{A}⟨ρ_{A}⟩)

where the notation means that each boundary component B_{α} is mapped into the α-component of BΩ_{A}⟨ρ_{A}⟩. The restriction of ω to B_{α} classifies Q⟨ρ_{α}⟩. This shows that (E, p) is determined up to equivalence by ω.

Conversely, if we are given a map ω as above we can reconstruct a special (Π, G)-equivariant bundle over M realizing the isotropy group system, up to fine equivalence. It can be checked that this bundle is unique up to equivalence.

Remark 8.5. Note that a special Π × G-manifold over M is a special (Π, G)- bundle over M precisely when the subgroup 1 × G acts freely on the total space.

The isotropy group system for the bundle is just the collection of isotropy groups for the Π × G-action. This observation shows that Theorem 8.4 follows from Jänich’s results.

Corollary 8.6. Let Π and G be compact Lie groups. The set of special (Π, G)- equivariant bundles over M is finite, provided that dim M ≤ 1 and the isotropy groups are semi-simple.

Proof. This is proved using [21], as for the finiteness of E(n, G).

To conclude this section, we discuss the connection between these results and Theorem B. Let X be a special Π-manifold over M, and let E(X, F) denote the set of bundle isomorphism classes of principal (Π, G)-bundles over X with isotropy group system fine equivalent to F := (H, U_{A}; ρ, ρ_{A}). Here a bundle isomorphism is a Π-equivariant G-bundle isomorphism ϕ: E_{1} → E_{2} covering the identity on X.

Lemma 8.7. There is a commutative diagram:

\xymatrix{E(X, F) \ar[r] \ar[d]^{v} & S[H, U_{A}; ρ, ρ_{A}] \ar[r]^{λ} \ar[d]^{≈} & S[H, U_{A}] \ar[d]^{≈} \\ [M, ∂M_{A}; BZ_{ρ}, BZ_{ρ_{A}}] \ar[r]^{u} & [M, ∂M_{A}; BΓ⟨ρ⟩, BΩ_{A}⟨ρ_{A}⟩] \ar[r] & [M, ∂M_{A}; BΓ, BΩ_{A}]}

where the horizontal composites have images represented by the element [X].

Proof. The exact sequences 1 → Z_{ρ} → Γ⟨ρ⟩ → Γ and 1 → Z_{ρ_{α}} → Ω_{α}⟨ρ⟩ → Ω_{α} of groups induce a map u: [M, ∂M_{A}; BZ_{ρ}, BZ_{ρ_{A}}] → [M, ∂M_{A}; BΓ⟨ρ⟩, BΩ_{A}⟨ρ_{A}⟩]. By Theorem 8.4 and Theorem 8.2, we also have a surjective map

v: E(X, F) → [M, ∂M_{A}; BZ_{ρ}, BZ_{ρ_{A}}]

induced by u and our construction of equivariant bundles.

Theorem 8.8. Let X be a special Π-manifold over M, and F = (H, U_{A}; ρ, ρ_{A}) an admissible isotropy group system. Then

v: E(X, F) ≅ [M, ∂M_{A}; BZ_{ρ}, BZ_{ρ_{A}}] .

Proof. The map v is given in the diagram above, and we have already observed that it is surjective. Suppose that ξ_{1}, ξ_{2} ∈ E(X, F) with v(ξ_{1}) = v(ξ_{2}). Then we have a continuous map

(M × I, ∂(M × I)) → (BZ_{ρ}, BZ_{ρ_{A}})

realizing the homotopy between the classifying maps for ξ_{1} and ξ_{2}. By the surjectivity of v for (Π, G)-bundles over M ×I, we get a bundle (E, p) over X ×I which restricts to ξ_{1} and ξ_{2} at the ends X × ∂I. Since E ≅ E_{0} × I, we get ξ_{1} ≅ ξ_{2}.

Let Aut(X) be the group of Π-equivariant isotopy classes of Π-equivariant diffeomorphisms of X over the identity of M. The group Aut(X) preserving acts on E(X, F) by pulling back: f · ξ := (f^{−1})^{∗}ξ. This action is well defined by the equivariant Covering Homotopy Theorem of Palais [2, II.7.3]. If

λ: S[H, U_{A}; ρ, ρ_{A}] → S[H, U_{A}]

denote the natural forgetful map, then there is an induced map

ψ: E(X, F) → λ^{−1}(X)

given by applying our stronger equivalence relation on bundles (which allows ϕ to cover a self-diffeomorphism of X).

Proposition 8.9. The map ψ induces a bijection between λ^{−1}(X) and the quotient of E(X, F) by the action of Aut(X).

Proof. The map from one set of bundles to the other is defined by regarding a (Π, G)-bundle over X as an element of λ^{−1}(X), and this is well-defined since the equivalent relation in S[H, U_{A}; ρ, ρ_{A}] is stronger. Moreover, two bundles with isotropy group system F over X are equivalent if and only if they are in the same orbit of the action of Aut(X), hence our correspondence is injective. On the other hand, if (E′, p′) is a bundle with base space X′ in λ^{−1}(X), then there exists a Π-equivariant diffeomorphism h: X → X′ covering the identity on M. Then E := h^{∗}(E′) is an equivalent element in λ^{−1}(X), and is a bundle over X, so our correspondence is surjective.

These results and Theorem 8.4 can sometimes be used for explicit classification of equivariant (Π, G)-bundles over special Π-manifolds. Notice that Bredon in [2, V.7] together with [2, Theorem V.6.4] has determined Aut(X) in many cases of interest.

We shall now specialize to special Π-manifolds over I := [−1, 1], and extend Theorem B to this setting. Examples include cohomogeneity 1 actions on spheres, classified in [25, Thm. C, Table II]. The two components {±1} of ∂I are denoted {±}, and the notation F = (H, U_{±}; ρ, ρ_{±}) will be used for the admissible (Π, G)- isotropy group systems, as well as Γ, Ω_{±}, etc. The classification of special Πmanifolds over I takes the following form.

Theorem 8.10. Let (H, U_{±}) be an admissible isotropy group system over I. Then S[H, U_{±}] is in bijection with the double cosets π_{0}(Ω_{−})\π_{0}(Γ)/π_{0}(Ω_{+}).

Proof. This follows directly from Theorem 8.2, since

[I, ∂I; BΓ, BΩ_{±}] ≅ π_{1}(BΩ_{−})\π_{1}(BΓ)/π_{1}(BΩ_{+}) ≅ π_{0}(Ω_{−})\π_{0}(Γ)/π_{0}(Ω_{+}).

Remark 8.11. The bijection of Theorem 8.10 can be seen in a more constructive way. Given a special Π-manifold X over I, we can choose an H-meridian c: I → X, i.e. a smooth section of X → I so that Π_{c(t)} = H for t in the interior of I; this can be obtained from a smooth section of the (trivial) principal Γ-bundle P → I.

Let U_{±} := Π_{c(±1)}. We say that (H, U_{±}) is an H-meridian isotropy group system for X. Choosing another smooth section of P gives isotropy groups conjugate to U_{±} by elements in the same connected component of N(H).

As in (3.7), the special Π-manifold X can be reconstructed as a quotient of I × Π:

X = (I × Π/H) /{(±1, g) ∼ (±1, gu_{±}), ∀ u_{±} ∈ U_{±}}

Therefore, X is determined by the subgroups U_{±} (compare [7, Prop. 1.6]). Moreover, any set (H, U ′_{±}), where the U ′_{±} are conjugate to U_{±}, occurs as an H-meridian isotropy group system for some special Π-manifold X′ over I (proved as in (4.3)). In this way, S[H, U_{±}] is a quotient of π_{0}(N(H)) × π_{0}(N(H)). The diagonal group acts trivially, and by carefully examining the relevant equivalence relation one sees that S[H, U_{±}] is in bijection with π_{0}(Ω_{−})\π_{0}(Γ)/π_{0}(Ω_{+}).

Let X be a special Π-manifold over I. We will now compute the set E(X, G) of isomorphism classes of Π-equivariant principal G-bundles over X. Fix an Hmeridian isotropy group system (H, U_{±}) for X induced by a smooth H-meridian c: I → X. If ξ = (E, p) is a (Π, G)-bundle over X, then we can choose an isotropic lift c̃: I → E, and obtain an admissible isotropy group system F = (H, U_{±}; ρ, ρ_{±}) for the bundle.

Since the composition of an isotropic lift with a bundle isomorphism is again isotropic, the conjugacy classes [ρ_{±}] ∈ R(U_{±}, G) depend only on [ξ] ∈ E(X, G). This defines a map

J: E(X, G) → R(U_{−}, G) × R(U_{+}, G) .

We write R(U_{−}, G) ×_{H} R(U_{+}, G) for the set of pairs ([ρ_{−}], [ρ_{+}]) ∈ R(U_{−}, G) × R(U_{+}, G) such that Res [ρ_{−}] = Res [ρ_{+}] in R(H, G).

Theorem B generalizes to special manifolds over I as follows:

Theorem 8.12. Let X be a special Π-manifold over I realizing the isotropy group system (H, U_{±}). The set E(X, G) of isomorphism classes of (Π, G)-bundles over X is determined by the following properties:

(i) the image of J is R(U_{−}, G) ×_{H} R(U_{+}, G). (ii) Let ρ_{±}: U_{±} → G be two smooth homomorphisms such that ρ_{−} |_{H} = ρ_{+} |_{H} =: ρ.

Then there is a bijection between J^{−1}([ρ_{−}], [ρ_{+}]) and the set of double cosets π_{0}(Z_{ρ_{−}})\π_{0}(Z_{ρ})/π_{0}(Z_{ρ_{+}}).

Proof. Since ρ_{±} comes from an admissible system, the image of J is contained in R(U_{−}, G) ×_{H} R(U_{+}, G). On the other hand, let ρ_{±}: U_{±} → G be two smooth homomorphisms such that ρ_{−} |_{H} = ρ_{+} |_{H} =: ρ. The special (Π × G)-manifold constructed as in Remark 8.11, with isotropy group system (H⟨ρ⟩, U_{±}⟨ρ_{±}⟩) (associated to an H⟨ρ⟩-meridian) is a (Π, G)-bundle ξ over X (the special Π-manifold with H-meridian isotropy group system (H, U_{±})). One has J(ξ) = ([ρ_{−}], [ρ_{+}]), which proves Part (i).

If (E, p) ∈ J^{−1}([ρ_{−}], [ρ_{+}]) then its isotropy group system is fine equivalent to F = (H, U_{±}; ρ, ρ_{±}). In the notation introduced earlier, we have

E(X, F) = J^{−1}([ρ_{−}], [ρ_{+}])

and the result now follows from Theorem 8.8.

We also have a more explicit version of Proposition 8.9. Note that J(f · ξ) = J(ξ), for f ∈ Aut(X), so we must investigate the action of Aut(X) on a preimage E(X, F) ≅ J^{−1}([ρ_{−}], [ρ_{+}]). The group Aut(X) has a homotopy description: choose a base point (•) ∈ Ω_{−}\Γ/Ω_{+} which corresponds to the class of X in π_{0}(Ω_{−})\π_{0}(Γ)/π_{0}(Ω_{+}).

Proposition 8.13. (Bredon [2, V.7.3, VI.6.4]) There is a group anti-isomorphism

Aut(X) ≅ [I, ∂I; Γ, Ω_{±}]_{•}

Proof. The pointed maps (I, ∂I) → (Γ, Ω_{±}) send ∂I into the component of Ω_{−}\Γ/Ω_{+} containing the base point (•). Let f ∈ Aut(X). Using our H-meridian c, one defines a smooth path d: I → Π/H by the formula

f(c(t)) = d(t) · c(t).

The map f being Π-equivariant, one has, for all h ∈ H

hd(t) · c(t) = h · f(c(t)) = f(hc(t)) = f(c(t)) = d(t)c(t)

This implies, for all t ∈ I, that d(t) ∈ N(H)/H = Γ. For t = ±1, one gets in addition that d(±1) ∈ Ω_{±}. We check that this defines an anti-homomorphism from Aut(X) to the group [I, ∂_{−}I, ∂_{+}I; Γ, Ω_{−}, Ω_{+}]. Now, if d(t) represents a class in the latter, the formula

f_{d}(αc(t)) := αd(t) · c(t) , α ∈ Π, t ∈ I

defines an element f_{d} of Aut(X) and constitutes an inverse to the above antihomomorphism.

Suppose that the homomorphisms Γ⟨ρ⟩ → Γ and Ω_{±}⟨ρ⟩ → Ω_{±} are surjective. Then we have a fibre bundle

Z_{ρ_{−}}\Z_{ρ}/Z_{ρ_{+}} → Ω_{−}⟨ρ⟩\Γ⟨ρ⟩/Ω_{+}⟨ρ⟩ → Ω_{−}\Γ/Ω_{+}. (8.14)

Therefore, π_{1}(Ω_{−}\Γ/Ω_{+}) acts on π_{0}(Z_{ρ_{−}}\Z_{ρ}/Z_{ρ_{+}}) = π_{0}(Z_{ρ_{−}})\π_{0}(Z_{ρ})/π_{0}(Z_{ρ_{+}}).

Theorem 8.15. Suppose that the homomorphisms Γ⟨ρ⟩ → Γ and Ω_{±}⟨ρ⟩ → Ω_{±} are surjective. Let ρ_{±}: U_{±} → G be two smooth homomorphisms such that ρ_{−} |_{H} = ρ_{+} |_{H} =: ρ. Then the quotient of J^{−1}([ρ_{−}], [ρ_{+}]) by the action of Aut(X) is in bijection with with the quotient of π_{0}(Z_{ρ_{−}})\π_{0}(Z_{ρ})/π_{0}(Z_{ρ_{+}}) by the action of π_{1}(Ω_{−}\Γ/Ω_{+}).

Proof. By Theorem 8.8, there is a surjective map ψ: J^{−1}([ρ_{−}], [ρ_{+}]) → λ^{−1}([X]). Since Γ⟨ρ⟩ → Γ and Ω_{±}⟨ρ⟩ → Ω_{±} are surjective, we have the fibration (8.14). From the homotopy exact sequence of this fibration, we see that λ^{−1}([X]) is in bijection with the quotient of π_{0}(Z_{ρ_{−}}\Z_{ρ}/Z_{ρ_{+}}) by the action of π_{1}(Ω_{−}\Γ/Ω_{+}). But this is just the action of Aut(X) by Proposition 8.13.

Example 8.16. Consider the standard action of Π = SO(n) on S^{n}, with H = SO(n − 1). As U_{±} = Π, the homomorphisms Γ⟨ρ⟩ → Γ and Ω_{±}⟨ρ⟩ → Ω_{±} are surjective.

When n = 2, H is the trivial group. If G is connected, then Z_{ρ} = G is connected and the map J is a bijection, as seen in Theorem A. Also, the group of Π-automorphism of S^{2} is equal to Π, so Aut(S^{2}) is trivial.

If n ≥ 3, the group Γ = Ω_{±} has 2 elements, the non-trivial one represented by the diagonal matrix D := Diag (1, … 1, −1, −1). The space Ω_{−}\Γ/Ω_{+} being reduced to a point, it follows from Theorem 8.15 that the group Aut(S^{n}) acts trivially on E(S^{n}, G). This has the following consequence:

Proposition 8.17. Let ρ_{±}: SO(n) → G be two representations into a compact Lie group G such that ρ_{±}_{|SO(n−1)} = ρ. Then, the element ρ_{−}(D)ρ_{+}(D)^{−1} belongs to Z_{ρ} and represent the trivial element in π_{0}(Z_{ρ_{−}})\π_{0}(Z_{ρ})/π_{0}(Z_{ρ_{+}}).

Proof. As ρ_{±}_{|H} = ρ, one has, for all h ∈ H,

ρ_{+}(D)^{−1}ρ(h)ρ_{+}(D) = ρ_{−}(D)^{−1}ρ(h)ρ_{−}(D).

Therefore, ρ_{−}(D)ρ_{+}(D)^{−1} ∈ Z_{ρ}.

Recall that our H-meridian for S^{n} is c(t) = (0, …, cos(πt/2), sin(πt/2)). By Proposition 4.1 and its proof there exists a (Π, G)-bundle ξ over S^{n}, with a ρisotropic lifting c̃: I → E(ξ) of c such that the isotropy representations associated to c̃(±1) are ρ_{±}. The curve c̃ is the horizontal lifting of c for a Π-invariant connection. There is no problem to extend the definitions of c(t) and c̃(t) for t ∈ R. This defines μ ∈ G by

c̃(3) = c̃(−1) · μ and c̃(5) = c̃(1) · μ.

For t ∈ [−1, 1], one has D · c̃(t) = c̃(2 − t)ρ_{+}(D) (since this is true for t = 1 and both side are horizontal). For t = −1, this gives

c̃(−1)μρ_{+}(D) = c̃(3)ρ_{+}(D) = D · c̃(−1) = c̃(−1)ρ_{−}(D)

Therefore, μ = ρ_{−}(D)ρ_{+}(D)^{−1}.

Now, let ξ_{1} := δ^{∗}ξ; one has

E(ξ_{1}) = {(x, u) ∈ S^{n} × E(ξ) | δ(x) = π(u)}

and the Π-action on E(ξ_{1}) comes from the diagonal action. A ρ-isotropic lifting c̃_{1} : I → E(ξ_{1}) of c is then given by

c̃_{1}(t) := (c(t), c̃(t − 2)).

By Proposition 4.1, one has ξ = ξ_{1} in E(S^{n}, G) iff J̃_{ρ}(ξ) = J̃_{ρ}(ξ_{1}) in R_{ρ}(n, G). By 4.2, using the curves c̃ and c̃_{1}, one has in R_{ρ}(n, G)

J̃_{ρ}(ξ) = [ρ_{−}, ρ_{+}] and J̃_{ρ}(ξ_{1}) = [μ^{−1}ρ_{−}μ, ρ_{+}].

This proves that μ represents the unit element in π_{0}(Z_{ρ_{−}})\π_{0}(Z_{ρ})/π_{0}(Z_{ρ_{+}}).

Example 8.18. Consider the action of SO(3) on the traceless (3×3)-symmetric matrixes by conjugation. Restricting this action to the unit sphere (for the invariant scalar product tr (AB)) makes S^{4} a special SO(3)-manifold over I (see [7, §3]) for more details on this classical cohomogeneity one SO(3)-action on S^{4}). Here H = S(O(1)×O(1)×O(1)), U_{−} = S(O(1)×O(2)) and U_{+} = S(O(2)×O(1)). Observe that any homomorphism ρ_{±}: U_{±} → G is trivial unless its restriction to H is injective.

Take G = SO(3) and use the standard inclusions of U_{±} into SO(3). Any nontrivial homomorphism ρ_{−} is conjugate to ρ̃_{−}: (ε, r_{θ}) ↦ (ε, r_{θ}^{p}) and any non-trivial ρ_{+} is conjugate to ρ̃_{+} : (r_{θ}, ε) ↦ (r_{θ}^{q}, ε), where p, q ∈ N_{odd}, the set of positive odd integers. Both ρ̃_{±} restrict on H to the identification ρ of H with the diagonal matrices of SO(3). One has Z_{ρ} = H, Zρ̃_{−} = {diag (1, 1, 1), diag (1, −1, −1)} and Zρ̃_{+} = {diag (1, 1, 1), diag (−1, −1, 1)}. Therefore, the set of double cosets π_{0}(Zρ̃_{−})\π_{0}(Z_{ρ})/π_{0}(Zρ̃_{+}) is trivial and E([S^{4} → I], SO(3)) is in bijection, via the map J of theorem 8.12, with {0, 0} ∪ N_{odd} × N_{odd}.

In [7, § 3], (SO(3), SO(3))-bundles P_{p,q} over S^{4} are constructed for p, q ∈ Z with p ≡ 3(mod 4) (those come from (S^{3} × S^{3})-bundles). These bundles satisfy J(P_{p,q}) = (|p|, |q|), so, up to isomorphism, the sign of p and q does not matter.

Example 8.19. This is the complex analogue of Example 8.18. One considers the action of SU(3) on the traceless (3 × 3)-Hermitian matrixes by conjugation and restrict it to the unit sphere. One thus gets a special SU(3)-manifold over I diffeomorphic to S^{7}. The isotropy groups are H = S(U(1) × U(1) × U(1)), U_{−} = S(U(1) × U(2)) and U_{+} = S(U(2) × U(1)). As in Example 1, any homomorphism ρ_{±}: U_{±} → G is trivial unless its restriction to H is injective.

For G = SU(3), one checks that any non trivial homomorphism of U_{±} to G is conjugate to the standard inclusion. As Z(H) = H in SU(3), the map J is injective and E(S^{7}, SU(3)) consists of two elements.

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Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 4K1, Canada

E-mail address: ian@math.mcmaster.ca

Section de Mathématiques Université de Genève, B.P. 240 CH-1211 Genève 24, Switzerland

E-mail address: hausmann@math.unige.ch