Equivariant Principal Bundles Over Spheres and Cohomogeneity One Manifolds

EQUIVARIANT PRINCIPAL BUNDLES OVER SPHERES AND COHOMOGENEITY ONE MANIFOLDS

IAN HAMBLETON and JEAN-CLAUDE HAUSMANN

1. Introduction

Let P and G be Lie groups. A principal …P; G†-bundle is a locally trivial, principal G-bundle p: E ! X such that E and X are left P-spaces. The projection map p is P-equivariant and g…e ´ g† ˆ …ge† ´ g, where g 2 P and g 2 G, acts on e 2 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…P; G† for principal …P; G†-bundles [5]; so the classi®cation of equivariant bundles in particular cases can be approached by studying the P-equivariant homotopy type of B…P; G†. If the structural group G of the bundle is abelian, then the main result of [18] states that equivariant bundles over a P-space X are classi®ed by the ordinary homotopy classes of maps ‰X ´

EP; BGŠ. In practice, this program leads to an obstruction theory rather than a classi®cation. See, however, the results of Lashof in the special cases where P acts semi-freely [14] or transitively [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; P†, together with equivariant gauge theory [1, 8, 9, 10]. By an isotropy representation at a P-®xed point x

2 X we mean the homomorphism a

: P ! G de®ned by the formula

g ´ e

ˆ e

´ a…g†

where e

2 p

…x

†. The homomorphism a is independent of the choice of e

up to conjugation in G. The relationship between the local invariants and the homotopy classi®cation (in the form of a Localization Theorem?) deserves further study.

In this paper, we use the second approach for P ˆ SO…n† acting in the standard way on X ˆ S

. In this concrete situation, we obtain a complete classi®cation by relatively elementary geometric methods. It turns out that the local isotropy representations at the north and south poles of S

explicitly determine the classi®cation of …SO…n†; G† principal bundles over S

(for short …n; G†-bundles).

One surprising consequence is that the set E…n; G† of …n; G†-bundles is ®nite for n > 3. In contrast, the set of (non-equivariant) principal G-bundles over S

is often in®nite. A detailed statement of these results is given in the next section and their proofs, essentially self-contained, are explained in §§ 3 to 6. Several examples are given in § 7. In § 8 we show how these results ®t into the more general setting of equivariant …P; G†-bundles over certain P-manifolds studied by

Received 24 April 2001; revised 19 December 2001.

2000Mathematics Subject Classi®cation55R91.

The authors thank the Swiss National Fund for Scienti®c Research, the Universite de GeneÁve, and the Max Planck Institut fuÈr Mathematik in Bonn for hospitality and support.

DOI: 10.1112/S0024611502013722

K. JaÈnich [13] and E. Straume [25]. In particular, we obtain a classi®cation of …P; G†-bundles over manifolds with cohomogeneity 1.

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

2. Statement of results

Let S

be the n-dimensional sphere of radius 1 in R

. We consider the action on S

of the group SO…n†, by orthogonal transformations ®xing the poles …0; . . . ; 0; 61†.

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 speci®ed, all maps between manifolds are smooth of class C

.

By a G-principal bundle h over S

, we mean, as usual, a smooth map p: E ! S

from a manifold E ˆ E…h† and a free right action E ´ G ! E so that p…z ´ g† ˆ p…z† with the standard local triviality condition. The isomorphism classes of G-bundles over S

are in bijection with p

…G†= p

…G†, the quotient of the homotopy group p

…G† (based at the identity element e of G) by the action of p

…G† induced by the conjugation of G on itself. The bijection associates to a bundle h the class C…h† :ˆ ‰¶…id

†Š 2 p

…G† = p

…G†, where

¶: p

…S

† ! p

…G† is the boundary operator in the homotopy exact sequence of h [23, Theorem 18.5].

An SO…n†-equivariant principal G-bundle y over S

(or an …n; G†-bundle for short) is a G-principal bundle y

over S

together with a left action SO…n† ´ E…y† ! E…y† commutingwith the free right action of G and such that the projection to S

is SO…n†-equivariant (we write E…y† for E…y

†). Two …n; G†- bundles y

and y

are isomorphic if there exists a diffeomorphism h: E…y

† ! E…y

† which is both SO…n† and G-equivariant and which commutes with the projections to S

, inducingthe identity on S

. We will compute the set E…n; G† of isomorphism classes of …n; G†-bundles.

Let y be an …n; G†-bundle. Choose points a; b 2 E…y† such that p…a† ˆ …0; . . . ; 1†

and p…b† ˆ …0; . . . ; 1†. Let a and b be the maps from SO…n† to G determined by the formulae A ´ a ˆ a ´ a…A† and A ´ b ˆ b ´ b…A†. We shall prove in Lemma 3.2 that a and b are smooth homomorphisms and that their classes in R…n; G† depend only on

‰yŠ 2 E…n; G†. We call a and b the isotropy representations (associated to a and b).

This de®nes a map J: E…n; G† ! R…n; G† ´ R…n; G† by J…y† :ˆ …‰aŠ; ‰bŠ†.

When n ˆ 2 and G is connected, J…y† is a complete invariant which, in particular, determines the (non-equivariant) isomorphism class of y

. More precisely, let w: R…2; G† ´ R…2; G† ! p

…G† be the map determined by w…a; b†…z† :ˆ ‰a…z†b…z†

Š. To de®ne w we use the identi®cation SO…2† ˆ S

and note that w is well de®ned if G is connected.

Theorem A. Suppose that G is a connected Lie group.Then, (i) the map J: E…2; G† ! R…2; G† ´ R…2; G† is a bijection;

(ii) if J…y† ˆ …‰aŠ; ‰bŠ†, then w…a; b† ˆ C…y

†.

We shall now generalize Theorem A by allowing n > 2 and G to be any Lie group.

In general, J is then neither injective nor surjective and C…y

† is not determined by

J…y†. Consider SO…n 1† as the subgroup of SO…n† ®xingthe last coordinate. The

restriction ‰mŠ 7! ‰mj

Š gives a map Res: R…n; G† ! R…n 1; G†. Denote by

R…n; G† ´

R…n; G † the set of …‰aŠ; ‰bŠ† 2 R…n; G† ´ R…n; G† such that Res‰aŠ ˆ Res‰bŠ. If J: H ! G is a group homomorphism, we denote by Z

Ì G the centralizer of J…H†.

Theorem B. Let G be any Lie group. Then the following hold.

(i) The image of J is R…n; G† ´

R…n; G†.

(ii) Let a; b: SO…n† ! G be two smooth homomorphisms such that aj

ˆ bj

ˆ: g. Then J

…‰aŠ; ‰bŠ† is in bijection with the set of double cosets p

…Z

†n p

…Z

† =p

…Z

†.

2.1. Remark. The compatibility statement in Part (i) of Theorem B was also observed by K. Grove and W. Ziller [7, Proposition 1.6]. In § 8, Theorem B is extended to a more general setting, to include equivariant principal bundles over

`special' P-manifolds in the sense of JaÈnich [13]. In particular, this provides a classi®cation 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 ˆ 2. To determine C…y

† as in Part (ii) of Theorem A, we must choose particular representatives of ‰aŠ and ‰bŠ (in general, J…y† does not determine y

: see examples 7.2 and 7.5). An isotropic lifting for y is a smooth curve e

c: ‰ 1; 1Š ! E…y† lifting the meridian arc c…t† ˆ …0; . . . ; cos…

pt†; sin…

pt †† and such that B ´ e c…t † ˆ e c…t†a…B† for all B 2 SO…n 1†. Isotropic lifting always exists (see Lemma 3.5). Choosing a :ˆ e c… 1† and b :ˆ e c…1† leads to isotropy representations a; b: SO…n† ! G such that aj

ˆ bj

. The map w…a; b†: SO…n† ! G constructed as in Theorem A then satis®es w…a; b†…AB† ˆ w…a; b†…A† when B 2 SO…n 1†. It thus induces a map

w…a; b†: S

> SO…n† = SO…n 1† ! G:

Note that w is well de®ned since a and b are actual homomorphisms and not conjugacy classes.

Proposition C. Let y be an …n; G†-bundle. Let a; b: SO…n† ! G be the isotropy representation associated to the end points of an isotropic lifting. Then,

‰ w…a; b†Š ˆ C…y

† in p

…G†= p

…G†.

We shall prove two consequences of Theorem B and Proposition C which emphasize the contrast between the cases n ˆ 2 and n > 3.

Proposition D. Let h be a principal G-bundle over S

with G a non-trivial Lie group. Then there exist in®nitely many y 2 E…2; G† such that y

> h.

Proposition E. For G a compact Lie group, the set E…n; G † is ®nite when n > 3.

These results are proved in § 5, while the earlier sections are devoted to

preliminary material. In § 6, we determine which …n; G†-bundles come from an

SO…n ‡ 1†-equivariant bundle. Examples are given in § 7.

3. Preliminary constructions

3.1. J is well de®ned. This follows from the following lemma.

3.2. Lemma. Let y be an …n; G†-bundle. Let a; b 2 E…y† such that p…a† ˆ …0; . . . ; 1† and p…b† ˆ …0; . . . ; 1†. Let b and a be the maps from SO…n† to G determined by the formulae A ´ a ˆ a ´ a…A† and A ´ b ˆ b ´ b…A†. Then a and b are smooth homomorphisms and their class in R…n; G† depends only on ‰yŠ 2 E…n; G†.

Proof. Let A; B 2 SO…n†. One has

a ´ a…BA† ˆ …BA† ´ a ˆ B ´ …A ´ a† ˆ B ´ …a ´ a…A††

ˆ …B ´ a† ´ a…A† ˆ a ´ …a…B†a…A††:

Therefore, a and, similarly, b are homomorphisms. They are smooth because the action of SO…n† is smooth. If a

is another choice instead of a, there exists g 2 G such that a

ˆ a ´ g and one has

a ´ …ga

…A†† ˆ a

´ a

…A† ˆ A ´ a

ˆ A ´ a ´ g ˆ a ´ …a…A†g†;

…3:3†

whence a

…A† ˆ g

a…A†g. This proves that the class of …a; b† in R…n; G† ´ R…n; G†

does not depend on the choice of a and b. Now, if h: E…y† ! < E…y

† is an …SO…n†; G†-equivariant diffeomorphism over the identity of S

, then, by choosing a

:ˆ h…a† and b

:ˆ h…b†, one has …a

; b

† ˆ …a; b†. The proof of Lemma 3.2 is then complete. . . . . A 3.4. Isotropic liftings. Let I :ˆ ‰ 1; 1Š and c: I ! S

be the parametrisation of the meridian arc c…t † ˆ …0; . . . ; cos…

pt †; sin…

pt††. Let e c: I ! E ˆ E…y† be a (smooth) lifting of c. As c…t† is ®xed by SO…n 1†, one has B ´ e c…t † ˆ e c…t† ´ a

…B†, for B 2 SO…n 1†. As in the proof of Lemma 3.2, one checks that this gives a smooth path a

…t 2 I † of homomorphisms from SO…n 1† to G, which depends on the lifting e c. Call e c isotropic if a

is constant: a

…B† ˆ a…B† for all B 2 SO…n 1†.

3.5. Lemma. 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 af®ne space. If a curve u…t † in E…y† is horizontal for an SO…n†-invariant connection, then u…t † ´ g and A ´ u…t † are horizontal. Lemma 3.5 then follows from the following.

3.6. Lemma. Let y be an …n; G†-bundle endowed with an SO…n†-invariant connection. Then, any lifting e c of c which is horizontal is isotropic.

Proof. If e c is a horizontal lifting, then so are B ´ e c and e c ´ a…B† for B 2 SO…n 1†.

As B ´ e c… 1† ˆ e c… 1† ´ a…B†, one has B ´ e c…t † ˆ e c…t† ´ a…B† for all t 2 I. . . . . .A 3.7. The …n; G†-bundles y

. If X is a topological space, the unreduced suspension SX is

SX :ˆ I ´ X= f… 1; x† , … 1; x

† and …1; x† , …1; x

†; " x; x

2 X g:

We denote by C X the image of ‰ 1; 1† ´ X in S X and by C

X that of … 1; 1Š ´ X.

Let …a; b† be a pair of smooth homomorphisms from SO…n† to G. De®ne the space E b

by

E b

:ˆ I ´ SO…n† ´ G= f… 1; A

; g† , … 1; A; a…A

A

†g† and …1; A

; g† , …1; A; b…A

A

†g†; "A 2 SO…n†g:

The space E b

admits an obvious free right action of G and a map p: E b

! SSO…n†.

This makes a principal G-bundle over SSO…n†; indeed, trivializations on C

SO…n†

are given by b

J : ‰t ; A; gŠ 7! …‰t; AŠ; a…A†g† if 1 < t < 1;

b

J

: ‰t ; A; gŠ 7! …‰t; AŠ; b…A†g† if 1 < t < 1:

…3:8†

The change of trivializations is b

J ± J b

…‰t ; AŠ; g† ˆ …‰t ; AŠ; a…A†b…A†

g†:

…3:9†

Now, suppose that aj

ˆ b j

. Form the space E

as the quotient E

:ˆ E b

= f‰t ; AB; gŠ , ‰t ; A; a…B†gŠ; "B 2 SO…n 1†g:

Let «: SO…n† ! S

be the map which associates to a matrix its last column;

it is also the projection «: SO…n† ! SO…n†= SO…n 1† > S

. There are a map p: E

! S S

, given by p…‰t; A; gŠ† ˆ ‰t; «…A†Š, and a free G-action, given by

‰t; A; gŠ ´ g

:ˆ ‰t; A; gg

Š. As above, we check that this de®nes a G-principal bundle over SS

; the trivializations J b

descend to trivializations Ï J

over C

S

.

The map ‰t ; AŠ 7! A ´ c…t† descends to a homeomorphism f : SS

! < S

. By replacing p by f ± p, we obtain a (topological) principal G-bundle

y

: E

! p S

:

Let S

be the punctured spheres S

:ˆ f …C

S

†. The trivializations given by the compositions

J

: p

…S

† J Ï

! C

S

´ G f ´ id ! S

´ G …3:10†

are homeomorphisms from p

…S

† onto manifolds. The change of trivialization is a diffeomorphism, being obtained by conjugating that of (3.9) by f. Therefore, J

produce a smooth manifold structure on E

. The map p and the G action are smooth. One checks that the map

SO…n† ´ E b

! E b

given by C ´ ‰t; A; gŠ :ˆ ‰t; CA; gŠ

descends to a smooth SO…n†-action on E

which makes y

an …n; G†-bundle.

3.11. Proof of Part (i) of Theorem B. Let y be an …n; G†-bundle. By Lemma 3.5 there exists an isotropic lifting e c: I ! E…y† of c. Choosing a :ˆ e c… 1† and b :ˆ e c…1† produces a representative …a; b† of J…y† with aj

ˆ bj

. Therefore, the image of J is contained in R…n; G† ´

R…n; G†.

Conversely, a class P 2 R…n; G† ´

R…n; G† is represented by a pair …a; b†

with aj

ˆ bj

. Let 1 be the identity matrix in SO…n† and e be the

unit element of G. Computing J…y

† with the points a :ˆ ‰ 1; 1; eŠ and b :ˆ ‰1; 1; eŠ

in E

shows that J …y

† ˆ P. . . . . A

4. The map e J

Let g: SO…n 1† ! G be a smooth homomorphism. De®ne a set R

…n; G† as follows: an element of R

…n; G† is represented by a pair …a; b† of smooth homomorphisms from SO…n† to G such that aj

ˆ bj

ˆ g. Two pairs …a

; b

† and …a

; b

† represent the same element of R

…n; G† if there is a smooth path fg

j t 2 ‰ 1; 1Šg in the centralizer Z

of g…SO…n 1†† such that a

…A† ˆ g

a

…A†g

and b

…A† ˆ g

b

…A†g

. There is an obvious map j: R

…n; G† ! R…n; G† ´

R…n; G†.

Part (i) of Theorem B, already proven in (3.11), permits us to de®ne a map J: E…n; G† ! R…n 1; G† by J…y† :ˆ Res‰aŠ ˆ Res‰bŠ. We shall now compute the preimage J

…‰gŠ†.

4.1. Proposition. Let g: SO…n 1† ! G be a smooth homomorphism. Then there exists a bijection e J

: J

…‰gŠ† ! < R

…n; G† such that j ± e J

ˆ J.

The proof divides into several steps.

4.2. De®nition of e J

. Let y be an …n; G†-bundle with J…y† ˆ ‰gŠ. Choose, using Lemma 3.5, an isotropic lifting e c

: I ! E…y† of c. As J…y† ˆ ‰gŠ, the constant path a

: SO…n 1† ! G associated to e c

is conjugated to g: there exists g 2 G such that a

…B† ˆ g g…B†g

. Let e c :ˆ e c

´ g. As in equation (3.3), one checks that e c is g-isotropic, that is, a

ˆ g. Choosing a :ˆ e c… 1† and b :ˆ e c…1†

then produces a pair …a; b† of smooth homomorphisms from SO…n† to G which represents a class e J

…y† in R

…n; G†.

To see that e J

is well de®ned, let e c

be another g-isotropic lifting of c. The smooth path t 7! g

2 G de®ned by e c

…t † ˆ e c…t † ´ g

satis®es

g…B† ˆ a

…B† ˆ g

a

…B†g

ˆ g

g…B†g

for all B 2 SO…n 1†. Therefore, g

2 Z

. One has a

…A† ˆ g

a

…A†g

and b

…A† ˆ g

b

…A†g

, for all A 2 SO…n†, which proves that e J

…y† does not depend on the choice of a g-isotropic lifting.

Now, if h: E…y† ! < E…y

† is an …SO…n†; G†-equivariant diffeomorphism over the identity of S

and e c: I ! E…y† is a g-isotropic lifting for y, then c

:ˆ h ± e c is a g-isotropic lifting for y

giving …a

; b

† ˆ …a; b†. This proves that e J

is well de®ned.

4.3. Surjectivity of e J

. Let …a; b† represent a class P in R

…n; G†. One checks that e J

…y

† ˆ P, using the fact that the path t 7! ‰t ; 1; eŠ is a g-isotropic lifting for y

.

4.4. Injectivity of e J

. Let g: SO…n 1† ! G be a smooth homomorphism,

and let y be an …n; G†-bundle with J…y† ˆ ‰gŠ. There exists a 2 E…y† with

p…a† ˆ …0; . . . ; 1† and B ´ a ˆ a ´ g…B† for all B 2 SO…n 1†. Choose an SO…n†-

invariant connection on y and let e c be a horizontal lifting of c with e c… 1† ˆ a. By

Lemma 3.6, e c is g-isotropic. If b: SO…n† ! G is de®ned by A ´ e c…1† ˆ e c…1† ´ b…A†,

then …a; b† represents e J

…y†.

Consider the map l: b E b

! E…y† given by b l…‰t ; A; gŠ† :ˆ A ´ e c…t† ´ g:

The map b l descends to a continuous map l: E

! E…y† which is both SO…n†

and G-equivariant and which covers the identity of S

. Therefore y and y

are isomorphic as topological …n; G†-bundles. What remains to prove is that l is a diffeomorphism, which is clear except possibly in a neighbourhood of the ®bers E

above the north and south poles.

The connection on y provides a smooth trivialization of y restricted to the punctured sphere S

(see § 3.7) in the following way. Consider the map s : p

…S

† ! E assigning to z the end point in E of the horizontal path through z above the meridian arc through p…z†. De®ne the G-equivariant map j : p

…S

† ! G by s …z† ˆ a ´ j …z†.

The required trivialization t : p

…S

† ! S

´ G is t …z† :ˆ … p…z†; j …z††.

Take the trivialization J for y

de®ned in formulae (3.10) of § 3.7. As e c is horizontal, one has

t ± l ± J

…x; g† ˆ …x; g†:

This, and the same for E

, prove that l is a diffeomorphism. We have thus established that if e J

…y† is represented by …a; b† then the …n; G†-bundle y is isomorphic to y

, which proves the injectivity of e J

.

The proof of Proposition 4.1 is now complete. . . . .A

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 a; b: SO…n† ! G be two smooth homomorphisms such that aj

ˆ bj

ˆ g. The pair …a; b† de®nes a class ‰a; bŠ 2 R

…n; G†.

The group Z

´ Z

acts on R

…n; G† by

…g; h† ´ ‰a; bŠ :ˆ ‰gag

; hbh

Š:

The set J

…‰aŠ; ‰bŠ† Ì J

…‰gŠ† is in bijection with an orbit of the above action via the bijection e J

: J

…‰gŠ† ! < R

…n; G† of Proposition 4.1. Let `,' be the equivalence relation on Z

´ Z

de®ned by …g

; h

† , …g

; h

† if and only if …g

; h

†‰a; bŠ ˆ …g

; h

†‰a; bŠ. Let

f: Z

´ Z

! p

…Z

†n p

…Z

†= p

…Z

†

be the map de®ned by f…g; h† :ˆ ‰g

hŠ. Part (ii) of Theorem B then follows from the following lemma.

5.2. Lemma. The equivalence …g

; h

† , …g

; h

† holds if and onlyif f…g

; h

† ˆ f…g

; h

†.

Proof. Suppose that …g

; h

† , …g

; h

†. This means that there exist s ; s

2 Z

, with ‰s Š ˆ ‰s

Š in p

…Z

†, such that the equality

…g

ag

; h

bh

† ˆ …s g

ag

s

; s

h

bh

s

†

holds in Z

´ Z

. This implies that

g

ˆ s g

A and h

ˆ s

h

B

with A 2 Z

and B 2 Z

(the centralizers of the images of a and b). Therefore g

h

ˆ A

g

s

s

h

B, which implies that f…g

; h

† ˆ f…g

; h

†.

To prove the converse, observe that (i) …g; h† , …Cg; Ch† for C 2 Z

,

(ii) …g; h† , …gA; hB† for A 2 Z

and B 2 Z

,

(iii) …g; h† , …g; uh† for u in the identity component of Z

.

Suppose that f…g

; h

† ˆ f…g

; h

†. This means that there are A 2 Z

, B 2 Z

and u in the identity component of Z

such that g

h

ˆ A

g

uh

B (u can be put in the middle since the identity component of Z

is a normal subgroup of Z

).

One then has

…g

; h

† , …e; g

h

† ˆ …e; A

g

uh

B† , …g

A; uh

B† , …g

; h

†: A Proof of Proposition C. L et y, a and b be as in the statement of Proposition C. Let g :ˆ aj

ˆ b j

. Then, y 2 J

…‰gŠ† and, by § 4.2, one has e J

…y† ˆ ‰a; bŠ

in R

…n; G†. By § 4.4, y ˆ y

. Therefore, C…y

† ˆ C…y

† ˆ ‰ w…a; b†Š, where the last equality comes from equation (3.9) of § 3.7 and the fact that C…y

† can be represented by its characteristic map [23, Theorem 18.4].. . . .A Proof of Theorem A. Since SO…1† is trivial, Z

ˆ G which is supposed to be connected. Therefore, Part (i) is a particular case of Part (i) of Theorem B. Let c map I to S

parametrizing the meridian arc, as in § 3.4. Let a; b: SO…2† ! G be two homomorphisms representing J…y†. One can ®nd a and b so that a and b are the isotropy representations associated to a and b. As G is connected, the submanifold P

:ˆ p

…c…I†† of E…y† is connected and there is a smooth lifting e c of c such that e c… 1† ˆ a and e c…1† ˆ b. As SO…1† is trivial, e c is isotropic. Part (ii) of Theorem A then follows from Proposition C. . . . .A Proof of Proposition D. Recall that any element of p

…G; e† can be represented by a homomorphism (a geodesic in a maximal compact subgroup K of G, with a K -bi-invariant Riemannian metric, being a 1-parameter subgroup [11, Chapter IV, § 6]). Therefore, if h is a G-bundle over S

, there exists a homomorphism a: S

! G such that C…h† ˆ ‰aŠ. For q 2 N, let a

: S

! G be given by a

…z† :ˆ a…z†

. If a is not trivial, the classes ‰a

Š 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 a

is q times those representing a.

Suppose ®rst that h is not trivial. Hence, a is not trivial and ‰a

; a

Š are all different classes in R

…2; G† with ‰ w…a

; a

†Š ˆ C…h†. The result then follows from Propositions C and 4.1.

When h is trivial, one takes any non-trivial homomorphism a: SO…2† ! G. The

classes ‰a

; a

Š in R

…2; G† represent in®nitely many distinct SO…2†-equivariant

G-bundles y

with trivial y

. . . . . A

Proof of Proposition E. If n > 3, the group SO…n† is semi-simple and the set

R…n; G† is ®nite. The latter follows from the following known results:

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

(ii) the Lie algebra of G contains only ®nitely many semi-simple Lie subalgebras, up to inner automorphism [21, Proposition 12.1];

(iii) there are only ®nitely many homomorphisms between two semi-simple Lie algebras, modulo inner automorphisms.

If G is compact, then the group Z

is compact and p

…Z

† is ®nite. Proposition E then follows from Theorem B. . . . . A 5.3. Remark. To remove the hypothesis `G compact' from Proposition E, it is enough to consider the case G connected. Indeed, R

…n; G† is a quotient of R

…n; G

†, where G

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 p

…Z…H†† is ®nite. We do not know whether this is 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

. Let d: SO…n† ! < SO…n† be the conjugation by the diagonal …n ´ n†-matrix diag…1; . . . ; 1; 1† (or, equivalently, diag… 1; . . . ; 1; 1†). If a: SO…n† ! G is a smooth homomorphism, observe that Res‰aŠ ˆ Res‰a ± dŠ in R…n 1; G†.

6.1. Theorem. Let y be an …n; G†-bundle. If y comes from an SO…n ‡ 1†- equivariant G-bundle then J…y† is of the form …‰aŠ; ‰a ± dŠ†.

For any ‰aŠ 2 R…n; G† there is a unique y 2 E…n; G† which comes from an SO…n ‡ 1†-equivariant G-bundle and is such that J…y† ˆ …‰aŠ; ‰a ± dŠ†.

Proof. For v 2 ‰0; pŠ , let R

2 SO…n ‡ 1† be the rotation of angle v in the plane of the last two coordinates. Let R :ˆ R

, the diagonal matrix with entries …1; . . . ; 1; 1; 1†.

Let y be an SO…n ‡ 1†-equivariant bundle. Choose a; b 2 E…y†, with p…a† ˆ …0; . . . ; 1† and let b :ˆ R ´ a. For A 2 SO…n† , one has R

AR ˆ d…A† and

bb…A† ˆ A ´ b ˆ A ´ …R ´ a† ˆ R ´ …R

AR† ´ a …6:2†

ˆ R ´ aa…d…A†† ˆ ba…d…A††;

whence b ˆ a ± d, which proves Part (i).

Let a: SO…n† ! G be a smooth homomorphism and set g :ˆ aj

. Suppose that y is an SO…n ‡ 1†-equivariant G-bundle with a 2 E…y† such that p…a† ˆ …0; . . . ; 1† and A ´ a ˆ aa…A† for A 2 SO…n†. Then y 2 J

…‰gŠ†. Let c: I ! E…y† be the curve c…t † :ˆ R

´ a, where v…t† :ˆ

p…t 1†. Using the relation R

BR

ˆ B for B 2 SO…n 1†, one checks, as in equation (6.2), that c is a g-isotropic lifting. By § 4.2 and equation (6.2), one has e J

…y† ˆ …‰aŠ; ‰a ± dŠ† in R

…n; G †. By Proposition 4.1, this proves that y is determined by a, which proves the uniqueness statement of Part (ii).

It remains to construct, for a smooth homomorphism a: SO…n† ! G, an

SO…n ‡ 1†-equivariant G-bundle with J…y† ˆ …‰aŠ; ‰a ± dŠ†. Consider the map

p: SO…n ‡ 1† ! S

sending a matrix to its last column vector. This makes an SO…n ‡ 1†-equivariant SO…n†-bundle (the principal SO…n†-bundle associated to the tangent bundle of S

). Let y be the G-bundle obtained by the Borel construction, using the homomorphism a ± d: SO…n† ! G,

E…y† :ˆ SO…n ‡ 1† ´ G = f…BA; g† ˆ …B; a…d…A††g†g:

…6:3†

This is an SO…n ‡ 1†-equivariant G-bundle. Choosing a :ˆ …R; e† and b :ˆ …I; e†, one sees that

A ´ a ˆ …AR; e† ˆ …Rd…A†; e† ˆ …R; a…A†e† ˆ aa…A†

and

A ´ b ˆ …A; e† ˆ …I ; a…d…A††e† ˆ ba…d…A††:

Therefore, y is an SO…n ‡ 1†-equivariant G-bundle with J…y† ˆ …‰aŠ; ‰a ± dŠ†. . . A 6.4. Remark. Theorem 6.1 and its proof show that any SO…n ‡ 1†-equivariant G-bundle is derived from the tangent bundle to S

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 D X the diagonal in X ´ X.

7.1. n ˆ 2 and G ˆ U…m†. A homomorphism a: S

! U…m† has, up to conjugacy, a unique diagonal form a…z† ˆ diag…z

; . . . ; z

†, with p

> . . . > p

. The same holds for b. By Theorem A, E…2; U …m†† is then in bijection with the set of pairs … p; q† of m-tuples of non-increasing integers. In p

…U…m†† ˆ Z, one has ‰aŠ ˆ P

iˆ1

p

and ‰bŠ ˆ P

iˆ1

q

, so, by Proposition C, C…y

† ˆ X

iˆ1

… p

q

†:

If one wishes instead to characterize y

by its ®rst Chern number c…y

† 2 H

…S

† ˆ Z, then c…y

† ˆ C…y

† [19, p. 445].

For instance, if t is the unit tangent bundle over S

with the natural action, then a…z† ˆ z

, b…z† ˆ z, and so C…t† ˆ 2 and c…t† ˆ 2 ˆ x…S

†.

Note that if y comes from an SO…3†-bundle, then, by Theorem 6.1, one has q

ˆ p

(as for t above). In particular c…y

† must be even (see also [10, (6.3)]).

7.2. n ˆ 2 and G ˆ O…2†. For q 2 Z , let a

: SO…2† ! O…2† be the homo- morphism A ! A

. The set R…2; O…2†† is in bijection with N given by a

7! j qj.

The same recipe produces a bijection p

…O…2†; e† =p

…O…2†† > N. L et y

:ˆ y

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 y

, with the only relation y

ˆ y

. One has J…y

† ˆ …j pj; j qj† and C…y

† ˆ j p qj. Therefore, J

…r; s† contains one element if rs ˆ 0 and two otherwise.

7.3. n ˆ 2 and G ˆ SO…m† with m > 3. A maximal torus of SO…m† is formed

by matrices containing 2-blocks concentrated around the diagonal, and so is

isomorphic to SO…2†

where k ˆ ‰

mŠ. 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-increasing integers. The bundle y

is determined by its second Stiefel±Whitney number w …y

† 2 Z

which is then given by

w…y

† ˆ

X

iˆ1

… p

q

† …mod 2†:

Again, y comes from an SO…3†-equivariant bundle if and only if q

ˆ p

and then y

is trivial.

7.4. n ˆ 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,Chapter 6,Corollary 2.8],for any embedding w: G a U…m†,the representations w ± a and w ± b are conjugate in U…m†. For G a compact classical group other than SO…2 m†,this implies that a and b are conjugate in G [20,Proposition 8,p. 56]. Therefore,the image of J is the diagonal D R…n; G†. We do not know whether this is true for G ˆ SO…2 m† (see [20,Remark,p. 57] for a possible source of counter-examples).

7.5. n ˆ 2k ‡ 1 > 3 and G ˆ 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 i: SO…n 1† Ì SO…n† denotes the inclusion,then Z

contains two elements, represented by the identity matrix and the diagonal matrix D :ˆ diag… 1; . . . ; 1; 1†.

Let d be the inner automorphism of SO…n† given by the conjugation by D. By Theorem B,the set E…n; SO…n†† for n ˆ 2k ‡ 1 > 3 then contains three elements as follows.

(i) The trivial bundle S

´ SO…n† with the action A ´ …z; B† ˆ …A ´ z; B†. The isotropy representations are both trivial.

(ii) The trivial bundle S

´ SO…n† with the action A ´ …z; B† ˆ …A ´ z; AB†. It is characterized by J …y† ˆ ‰i Š and e J

…y† ˆ …‰idŠ; ‰idŠ†. This does not come from an SO…n ‡ 1†-equivariant bundle.

(iii) The principal SO…n†-bundle TS

associated with the tangent bundle of S

. It is characterized by J …TS

† ˆ ‰ iŠ and e J

…TS

† ˆ …‰idŠ; ‰dŠ†. This comes from an SO…n ‡ 1†-equivariant bundle.

Observe that …‰idŠ; ‰dŠ† ˆ …‰dŠ; ‰idŠ† in R

…n; SO…n††. By Proposition C,this implies that C…TS

† ˆ C…TS

†. This proves again the classical fact that C…TS

† 2 p

…SO…2k ‡ 1†† is of order 2 [4,Corollary IV.1.11].

7.6. n ˆ 2k > 6 and G ˆ SO…n†. The set R…n; SO…n†† contains three elements, represented by the trivial homomorphism,the identity id of SO…n†,and the conjugation d by the diagonal matrix with entries … 1; . . . ; 1; 1†. The non-trivial homomorphisms restrict to the inclusion i: SO…n 1† Ì SO…n†. The group Z

is trivial. Therefore, J is injective and the set E…n; SO…n†† for n ˆ 2 k then contains

®ve elements,as follows.

(i) The trivial bundles S

´ SO…n† with the actions A ´ …z; B† ˆ …A ´ z; B†,

A ´ …z; B† ˆ …A ´ z; AB† and A ´ …z; B† ˆ …A ´ z; d…A†B†. Their images under J give the

diagonal D R…n; SO…n††.

(ii) The principal SO…n†-bundle TS

associated with the tangent bundle of S

. One has J…TS

† ˆ …‰idŠ; ‰dŠ†.

(iii) The …n; G†-bundle TS

with J… TS

† ˆ …‰dŠ; ‰idŠ†. Its underlying SO…n†-principal bundle is stably trivial with Euler number 2.

The trivial bundle with action A ´ …z; B† ˆ …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 ˆ 4 and G ˆ SO…3†. The groups SO…4† and SO…3† are built up out of the unit quaternions S

by SO…4† > …S

´ S

† = f…1; 1†; … 1; 1†g and SO…3†>

S

= f61g. Recall that these isomorphisms are constructed as follows: the orthogonal transformation A

2 SO…4† associated to … p; q† 2 S

´ S

is A

…x† :ˆ pxq, where x 2 H is a quaternion and H is made isomorphic to R

by choosing …i ; j; k; 1† as a basis. The correspondence p ! A

then induces the inclusion i: SO…3† Ì SO…4†. The automorphism d of SO…4† de®ned in § 6 is just the conjugation by D :ˆ diag… 1; . . . ; 1; 1†. One checks easily that d…A

† ˆ A

,since Dx ˆ x is the quaternion conjugation.

The non-equivariant isomorphism class of an SO…3†-principal bundle h is characterized by C…h† 2 p

…SO…3†† ˆ p

…S

† ˆ Z. It is also determined by its ®rst Pontrjagin number p…h† 2 4Z ,with the relation p…h† ˆ 4C…h†.

The set R…4; SO…3†† contains three elements represented by the trivial homo- morphism and those induced by the projections S

´ S

! S

given by j

… p; q† :ˆ p and j

… p; q† :ˆ q. The last two restrict over SO…3† to the identity id of SO…3†. The group Z

being trivial,the map J is injective. This shows that the set E…4; SO…3††

contains ®ve elements.

(i) The trivial bundles S

´ SO…3† with the actions A ´ …z; B† ˆ …A ´ z; B† and A ´ …z; B† ˆ …A ´ z; j

…A†B† for i ˆ 1; 2. Their images by J give the diagonal D R…4; SO…3††.

(ii) The principal SO…3†-bundle H: RP

! S

coming from the quaternionic Hopf bundle S

! S

; the SO…4†-action comes from the SU…2† ´ SU…2†-action on S

given by … p; q† ´ …z

; z

† ˆ … pz

; qz

†. We have J …H† ˆ …‰j

Š; ‰j

Š† and p…H

† ˆ 4.

(iii) The …n; G†-bundle H with J… H† ˆ …‰j

Š; ‰j

Š† and p…H

† ˆ 4.

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

7.8. n ˆ 4 and G ˆ SO…4†. With the notation of § 7.7,the set R…4; SO…4††

contains ®ve elements represented by:

(i) the trivial homomorphism;

(ii) those induced by j

… p; q† :ˆ … p; p† and j

… p; q† :ˆ …q; q†;

(iii) the identity id of SO…4†;

(iv) the homomorphism d… p; q† :ˆ …q; p†.

The non-trivial homomorphisms all restrict to i over SO…3†. The group Z

is trivial and then J is injective. The non-equivariant isomorphism class of an SO…4†-principal bundle h is characterized by C…h† 2 p

…SO…4†† ˆ p

…S

´ S

† ˆ Z Z. More usually,one takes the pair of integers … p…h†; e…h†† formed by the ®rst Pontrjagin number (in 2 Z) and the Euler number of h. The linear map which sends C…h† to … p…h†; e…h†† has matrix

. This can be checked using the examples below.

The set E…4; SO…3†† contains the following seventeen elements.

(i) The trivial bundles S

´ SO…3† with the ®ve actions using the above representations. Their images by J are just the diagonal elements D R…4; SO…4††.

(ii) The principal SO …4†-bundle H c whose total space is E… H† c :ˆ

R P

´

SO…4†. One has J … H† ˆ …‰j c

Š; ‰j

Š† and C… H c

† ˆ …1; 1†; therefore its characteristic classes are … p…H

†; e…H

†† ˆ …4; 0†. One also has its `inverse'

H, with c J … H† ˆ …‰j c

Š; ‰j

Š† and … p…H

†; e…H

†† ˆ … 4; 0†.

(iii) The principal SO…4†-bundle TS

associated with the tangent bundle of S

. One has J …TS

† ˆ …‰idŠ; ‰dŠ†. Therefore, C…TS

† ˆ …1; 1† and … p…T

†; e…T

†† ˆ …0; 2†. Again, one can consider its inverse.

(iv) The …n; G†-bundles y

…i ˆ 1; 2† with J…y

† ˆ …‰idŠ; ‰j

Š† and their inverses y

. They satisfy C…y

† ˆ …0; 1† and C…y

† ˆ …1; 0†, or, equivalently,

… p…y

†; e…y

†† ˆ … 2; 1† and … p…y

†; e…y

†† ˆ …2; 1†:

(v) The …n; G†-bundle y

…i ˆ 1; 2† with J…y

† ˆ …‰dŠ; ‰j

Š† and their inverses.

They satisfy C…y

† ˆ … 1; 0† and C …y

† ˆ …0; 1†, or, equivalently, … p…y

†; e…y

†† ˆ … 2; 1† and … p…y

†; e…y

†† ˆ …2; 1†:

Only the trivial bundle with action A ´ …z; B† ˆ …A ´ z; B† and the bundles in (ii) and (iii) come from SO…5†-equivariant bundles.

7.9. G ˆ 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,

p

…U …m†† < p

…U …m ‡ k†† < 0 if n is odd, Z if n is even.

Problem. Which integers occur as C…y

† for an …n; U…m††-bundle y?

8. A more general setting

The orthogonal action of SO…n† on S

is an example of the special P-manifolds de®ned by JaÈnich [13, 1.2]. Other examples include the `cohomogeneity 1' actions studied by E. Straume [25] (see [7] for a recent application and other references).

In this section we give the classi®cation of equivariant …P; G†-bundles over special P-manifolds. We will assume in this section that P and G are both compact Lie groups.

Let X be a smooth, connected, closed n-dimensional manifold with a smooth P-action. Choose a P-invariant Riemannian metric on X, and then each tangent space T

X contains a P

-invariant subspace V

perpendicular to the orbit P ´ x.

Then X is called a special P-manifold if for each x 2 X the representation of P

on the normal space V

is the direct sum of a trivial representation and a transitive representation.

It follows that the orbit space M ˆ X = P admits a natural structure as a

topological manifold with boundary [13, 1.3], with dimension equal to

n dim…P= 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; p: X ! M† is called a special P-manifold over M .

Special P-manifolds over M were classi®ed by JaÈnich [13, 3.2], and independently by W.-C. Hsiang and W.-Y. Hsiang [12] (see also [2, V.5, VI.6]).

Let ¶M

ˆ fB

g

denote the set of boundary components of M. An admissible isotropy group system …H; U

† over M consists of a closed subgroup H of P and a set U

ˆ fU

g

of closed subgroups in P containing H, with the property that for each a 2 A there exists a transitive representation in which H appears as the isotropy group of a non-zero vector. Let G ˆ N …H †= H and Q

ˆ N…U

† Ç N …H †= H for each a 2 A. The idea of the classi®cation is to re-construct X from the unique principal G-bundle P over M such that Pj

ˆ fx 2 X j P

ˆ H g, and a reduction of the structural group of Pj

to Q

over each of the boundary components B

.

Let B

´ ‰0; 1Š be a collar neighbourhood of some component B

in M, and let Y

ˆ p

…B

´ ‰0; 1Š† denote its pre-image in X. The key fact is the following identi®cation of Y

as a P-space.

8.1. Theorem (Tube Theorem [2, V.4.2]). Let Y ˆ Y

, B ˆ B

and Q ˆ Q

. There exist a right Q-principal bundle Q ˆ Q

over B, and a P-equivariant diffeomorphism

M

´

Q ! < Y

commuting with the projection to ‰0; 1Š, where M

denotes the mapping cylinder of the canonical projection w: P = H ! P = U.

Let X

ˆ X S

a

p

…B

´ ‰0;

†† and M

ˆ X

=P. The Tube Theorem shows that X is P-diffeomorphic to the union

X ˆ X

È [

a

Y

ˆ P = H ´

P È [

a

M

´

Q

with the identi®cation on the overlaps B

´ ‰

; 1Š induced by a reduction of structural groups Pj

> G ´

Q

.

Two special P-manifolds …X

; p

† and …X

; p

† over M are called isomorphic when there exists a P-equivariant diffeomorphism f : X

! X

such that the induced diffeomorphism f : M ! M is the identity. By the smooth isotopy lifting theorem of G. Schwarz [22, Corollary 2.4] this is equivalent to JaÈnich's original de®nition where f was assumed to be only isotopic to the identity, by an isotopy

®xing ¶M pointwise. Let S‰H ; U

Š denote the set of isomorphism classes of special P-manifolds over M , with isotropy group system ®ne equivalent to …H ; U

† (cf. [13, § 2]).

8.2. Theorem [13, 3.2]. Let …H; U

† be an admissible isotropy group system over M, where M is a smooth, compact connected manifold with boundary. Then

S‰H ; U

Š > ‰M; ¶M

; BG; BQ

Š:

In order to classify equivariant …P; G†-bundles …E; p† over a special P-manifold

X, called special …P; G†-bundles for short, we will generalize the results of

Lashof [16] to describe the bundles over P-spaces X

and Y

with one orbit type,

and then follow JaÈnich's method to glue the pieces together.

First some general de®nitions: if r: H ! G is a (smooth) homomorphism, ‰ rŠ

denotes the set of homomorphisms r

: H ! G such that r

…h† ˆ gr…h†g

for some g 2 G and all h 2 H. We will say that the ®bre over x belongs to ‰r Š if for each z 2 p

…x† there exists r

2 ‰ rŠ such that hz ˆ z ´ r

…h† for all h 2 H. Then let

X

ˆ fx 2 X

j the fibre over x belongs to ‰ rŠg.

Let X

ˆ X

Ç X

and notice that X

Ì P ˆ fx 2 X j P

ˆ H g. By [16, Lemma 1.2], the space

E

ˆ fz 2 E j hz ˆ z ´ r…h†; "h 2 H g

is a Z

-bundle over X

, where Z

is the centralizer of r in G. The group G b ˆ P ´ G has a left action on the total space E given by the formula …g; g† ´ z ˆ gz ´ g

for any …g; g† 2 G b and any z 2 E. Let us set

H h ri :ˆ f…h; r…h†† j h 2 H g Ì P ´ G and

Gh ri :ˆ N…H h ri† =H h ri:

Then E

is just the ®xed set of H hri in E under this action.

Two special …P; G†-bundles …E

; p

† and …E

; p

† over M are called equivalent when there exists a P-equivariant G-bundle isomorphism f: E

! E

, covering a P-equivariant diffeomorphism f : X

! X

, so that the induced diffeomorphism f : M ! M is the identity.

We now give the classi®cation of the part of …E; p† lying over M

. After the remarks above, we see that it follows directly from the Slice Theorem [2, II.5.8].

8.3. Theorem [16, 1.9]. Let r: H ! G be a smooth homomorphism.The equivalence classes of …P; G†-equivariant bundles …E

; p

† over M

with all ®bres belonging to ‰ rŠ are in bijection with the homotopy classes of maps ‰M

; BGhr iŠ.

We next de®ne an admissible …P; G†-isotropy group system over M to be a set …H ; U

; r; r

†, where H and U

are as above, r: H ! G is a homomorphism, and r

ˆ fr

g

is a set of homomorphisms r

: U

! G such that r

j

ˆ r.

We de®ne

Q

h r

i ˆ N …U

h r

i† Ç N…H h ri† = H h ri and Q

h r

i ˆ fQ

h r

ig

:

A special …P; G†-bundle …E; p† realizes an admissible …P; G†-isotropy group system …H; U

; r; r

† over M if

(i) there exist points fy

2 Y

g such that P

ˆ U

,

(ii) for each y

, the normal space V

to the orbit P ´ y

has a point z

2 V

with P

ˆ H,

(iii) the images c

…t† of rays in V

joining z

to y

have isotropic liftings e c

…t† to E such that e c

…t † Ì E

for 0 < t < 1 and e c

…1† Ì E

.

It is not dif®cult to check (following [13, § 2]) that every special …P; G†-bundle …E; p†

over M realizes some admissible …P ; G†-isotropy group system …H; U

; r; r

†.

This isotropy group system is unique up to a natural notion of equivalence,

extending the `®ne-orbit structure' of JaÈnich.

We say that two isotropy group systems …H; U

; r; r

† and …H

; U

; r

; r

† are

®ne equivalent if the following conditions hold:

(i) there exists an element g 2 P ´ G such that H

hr

i ˆ gH h rig

, and (ii) there exist n

2 NH h ri such that U

hr

i ˆ … gn

†U

h r

i…gn

†

.

Let S‰H; U

; r; r

Š denote the set of equivalence classes of special …P; G†-bundles over M realizing the given …P; G†-isotropy group system, up to ®ne equivalence.

8.4. Theorem. Let …H; U

; r; r

† be an admissible …P; G†-isotropy group system over M, where M is a smooth, compact connected manifold with boundary. Then

S‰H; U

; r; r

Š > ‰M; ¶M

; BGhri; BQ

hr

iŠ:

Proof. Suppose that we are given an admissible …P; G†-isotropy group system over M . Let …E; p† be a special …P; G†-bundle over M realizing the given …P; G†-isotropy group system. By restricting the bundle to M

, we get a map q

: M

! BGhri classifying the principal Gh ri-bundle Ph ri (which completely determines …E

; p

†) by Theorem 8.3. We can apply the Tube Theorem 8.1 to the G b ˆ P ´ G action on p

…Y

† Ì E, since z 2 E

means that G b

ˆ H hri and similarly for z 2 E

. This identi®es the restriction of our bundle to the part over B

´ ‰0; 1Š as M

´

Q hr

i, where

J

: G b = H hri ! G b = U

h r

i

is the G-equivariant projection, and b Qh r

i is a principal right Q

h r

i-bundle over B

. The classifying map q

for Phri therefore extends to a map

q: …M; ¶M

† ! …BG hri; BQ

h r

i†

where the notation means that each boundary component B

is mapped into the a-component of BQ

hr

i. The restriction of q to B

classi®es Q hr

i. This shows that …E; p† is determined up to equivalence by q.

Conversely, if we are given a map q as above we can reconstruct a special …P; G†-equivariant bundle over M realizing the isotropy group system, up to ®ne equivalence. It can be checked that this bundle is unique up to equivalence. . . A 8.5. Remark. Note that a special P ´ G-manifold over M is a special …P; 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 P ´ G-action. This observation shows that Theorem 8.4 follows from JaÈnich's results.

8.6. Corollary. Let P and G be compact Lie groups. The set ofspecial …P; G†-equivariant bundles over M is ®nite, provided that dim M < 1 and the isotropy groups are semi-simple.

Proof. This is proved using [21], as for the ®niteness of E…n; G†. . . . . A

To conclude this section, we discuss the connection between these results and

Theorem B. Let X be a special P-manifold over M , and let E…X; F† denote the

set of bundle isomorphism classes of principal …P; G†-bundles over X with

isotropy group system ®ne equivalent to F :ˆ …H; U

; r; r

†. Here a bundle isomorphism is a P-equivariant G-bundle isomorphism f: E

! E

covering the identity on X.

8.7. Lemma. There is a commutative diagram

E…X; ? ? ? y F† v ! S‰H; U

; r; r

Š l ! S‰H; U

Š

? ?

? y <

? ?

? y <

‰M; ¶M

; BZ

; BZ

Š u ! ‰M; ¶M

; BGh ri; BQ

h r

iŠ ! ‰M; ¶M

; BG; BQ

Š where the horizontal composites have images represented by the element ‰X Š.

Proof. The exact sequences

1 ! Z

! Ghr i ! G and 1 ! Z

! Q

hr i ! Q

of groups induce a map

u: ‰M; ¶M

; BZ

; BZ

Š ! ‰M; ¶M

; BGh ri; BQ

h r

iŠ:

By Theorem 8.4 and Theorem 8.2, we also have a surjective map v: E…X; F† ! ‰M; ¶M

; BZ

; BZ

Š

induced by u and our construction of equivariant bundles. . . . . A 8.8. Theorem. Let X be a special P-manifold over M, and let F ˆ …H; U

; r; r

† be an admissible isotropy group system.Then

v: E…X; F† > ‰M; ¶M

; BZ

; BZ

Š:

Proof. The map v is given in the diagram above, and we have already observed that it is surjective. Suppose that y

; y

2 E…X; F† with v…y

† ˆ v…y

†.

Then we have a continuous map

…M ´ I; ¶…M ´ I †† ! …BZ

; BZ

†

realizing the homotopy between the classifying maps for y

and y

. By the surjectivity of v for …P; G†-bundles over M ´ I , we get a bundle …E; p† over X ´ I which restricts to y

and y

at the ends X ´ ¶I. Since E > E

´ I, we get y

> y

. . . . . A Let Aut…X† be the group of P-equivariant isotopy classes of P-equivariant diffeomorphisms of X over the identity of M. The group Aut…X† acts on E…X; F†

by pulling back: f ´ y :ˆ … f

†

y. This action is well de®ned by the equivariant Covering Homotopy Theorem of Palais [2, II.7.3]. If

l: S‰H ; U

; r; r

Š ! S‰H; U

Š denotes the natural forgetful map, then there is an induced map

w: E…X; F† ! l

…X†

given by applying our stronger equivalence relation on bundles (which allows f to

cover a self-diffeomorphism of X).

8.9. Proposition. The map w induces a bijection between l

…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 de®ned by regarding a …P; G†-bundle over X as an element of l

…X†, and this is well de®ned since the equivalent relation in S‰H; U

; r; r

Š 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 l

…X†, then there exists a P-equivariant diffeomorphism h: X ! X

covering the identity on M. Then E :ˆ h

…E

† is an equivalent element in l

…X†, and is a bundle over X , so our correspondence is surjective. . . . . A These results and Theorem 8.4 can sometimes be used for explicit classi®cation of equivariant …P; G†-bundles over special P-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 P-manifolds over I :ˆ ‰ 1; 1Š , and extend Theorem B to this setting. Examples include actions with cohomogeneity 1 on spheres, classi®ed in [25, Theorem C, Table II]. The two components f61g of ¶I are denoted f6g, and the notation F ˆ …H; U

; r; r

† will be used for the admissible …P; G†-isotropy group systems, as well as G, Q

, etc. The classi®cation of special P-manifolds over I takes the following form.

8.10. Theorem. Let …H; U

† be an admissible isotropy group system over I.

Then S‰H; U

Š is in bijection with the double cosets p

…Q †n p

…G†= p

…Q

†.

Proof. This follows directly from Theorem 8.2, since

‰I; ¶I ; BG; BQ

Š > p

…BQ † n p

…BG†= p

…BQ

† > p

…Q †n p

…G†= p

…Q

†: A 8.11. Remark. The bijection of Theorem 8.10 can be seen in a more constructive way. Given a special P-manifold X over I , we can choose an H-meridian c: I ! X, that is, a smooth section of X ! I so that P

ˆ H for t in the interior of I;this can be obtained from a smooth section of the (trivial) principal G-bundle P ! I.

Let U

:ˆ P

. 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 P-manifold X can be reconstructed as a quotient of I ´ P:

X ˆ …I ´ P = H† =f…61; g† , …61; gu

†; " u

2 U

g:

Therefore, X is determined by the subgroups U

(compare [7, Proposition 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 P-manifold X

over I (proved as in § 4.3). In this way, S‰H; U

Š is a quotient of p

…N …H †† ´ p

…N …H††. The diagonal group acts trivially, and by carefully examining the relevant equivalence relation one sees that S‰H; U

Š is in bijection with p

…Q †n p

…G†= p

…Q

†.

Let X be a special P-manifold over I . We will now compute the set E…X; G † of

isomorphism classes of P-equivariant principal G-bundles over X. Fix an

H-meridian isotropy group system …H; U

† for X induced by a smooth H-meridian c: I ! X. If y ˆ …E; p† is a …P; G†-bundle over X , then we can choose an isotropic lift e c: I ! E, and obtain an admissible isotropy group system F ˆ …H; U

; r; r

† for the bundle.

Since the composition of an isotropic lift with a bundle isomorphism is again isotropic, the conjugacy classes ‰ r

Š 2 R…U

; G† depend only on ‰yŠ 2 E…X; G†.

This de®nes a map

J: E…X; G† ! R…U ; G† ´ R…U

; G†:

We write R…U ; G† ´

R…U

; G† for the set of pairs

…‰ r Š; ‰ r

Š† 2 R…U ; G† ´ R…U

; G†

such that Res ‰ r Š ˆ Res‰ r

Š in R…H; G†.

Theorem B generalizes to special manifolds over I as follows.

8.12. Theorem. Let X be a special P-manifold over I realizing the isotropy group system …H; U

†. The set E…X; G† of isomorphism classes of …P; G†-bundles over X is determined by the following properties.

(i) The image of J is R…U ; G† ´

R…U

; G†.

(ii) Let r

: U

! G be two smooth homomorphisms such that r j

ˆ r

j

ˆ: r.

Then there is a bijection between J

…‰ r Š; ‰ r

Š† and the set of double cosets p

…Z

† n p

…Z

† = p

…Z

†.

Proof. Since r

comes from an admissible system, the image of J is contained in R…U ; G† ´

R…U

; G†. On the other hand, let r

: U

! G be two smooth homomorphisms such that r j

ˆ r

j

ˆ: r. The special …P ´ G†- manifold constructed as in Remark 8.11, with isotropy group system …H hri; U

h r

i† (associated to an Hh ri-meridian) is a …P; G†-bundle y over X (the special P-manifold with H-meridian isotropy group system …H ; U

†). One has J…y† ˆ …‰ r Š; ‰ r

Š†, which proves Part (i).

If …E; p† 2 J

…‰ r Š; ‰ r

Š† then its isotropy group system is ®ne equivalent to F ˆ …H; U

; r; r

†. In the notation introduced earlier, we have

E…X; F† ˆ J

…‰ r Š; ‰ r

Š†

and the result now follows from Theorem 8.8.. . . . A We also have a more explicit version of Proposition 8.9. Note that J… f ´ y† ˆ J…y†, for f 2 Aut…X†; so we must investigate the action of Aut…X† on a pre-image E…X; F† > J

…‰ r Š; ‰ r

Š†. The group Aut…X† has a homotopy description:

choose a base point …·† 2 Q n G=Q

which corresponds to the class of X in p

…Q †n p

…G† = p

…Q

†.

8.13. Proposition (Bredon [2, V.7.3, VI.6.4]). There is a group anti-isomorphism Aut…X† > ‰I; ¶I; G; Q

Š

:

Proof. The pointed maps …I; ¶I † ! …G; Q

† send ¶I into the component of Q n G= Q

containing the base point …·†. Let f 2 Aut…X †. Using our H-meridian c, one de®nes a smooth path d: I ! P = H by the formula

f …c…t †† ˆ d…t† ´ c…t †: