Ann. Global Anal. Geom. Vol. 6, No. 2 (1988), 141-163

On extensions of principal bundles* KIRILL MACKENZIE

A principal bundle P(B, G x N) whose group is a direct product is itself a direct product PI(B, G) x B P2(B, N) of principal bundles (see 2.7). In this paper we study the more general situation of principal bundles Q(B, H) whose group is given as an extension N >,- H -, G. In this situation we have an "extension of principal bundles" N >-* Q(B, H)

-

P(B, G)

(*)

where P = (Q x G)/H _ Q/N is (what we call) the produced bundle. P(B, G) may be thought of as a factor of Q(B, H). The obvious candidate for the second factor is the "transverse bundle" Q(P, N, n) but it is clear that some sort of compatibility condition between two bundles P(B, G) and Q(P, N) is needed before they define an extension (*). Since G in (*) does not generally act on Q (or on H) it may be hard to see what this compatibility condition might be. This paper is founded on the observation ([10]) that the group G in (*), though it does not act on Q(P, N) does act in a natural fashion on the groupoid associated to Q(P, N). Further, the bundle P(B, G), and a Lie groupoid Y on base P and with group N, equipped with an action of G which satisfies certain minimal requirements, define an extension (*). In [10] we called such groupoids Y "PBG-groupoids"and showed that there is a bijective correspondence between (equivalence classes of) extensions (*) and (equivalence classes of) PBG-groupoidson P(B, G). Thus Y may be regarded as the missing "second factor" in the extension. One may also regard (*) as a lifting of P(B, G) to the group H. In this point of view, the PBG-groupoidconcept allows one to study liftings in which P(B, G) and N are prescribed, but in which H is allowed to emerge at a later stage, and can be any extension of G by N. Lie groupoids are essentially equivalent to principal bundles. The word "essentially" is important here: The two concepts are sufficiently close to being truely equivalent to ensure that the general connection theory of principal bundles translates completely to Lie groupoids (a detailed account of this translation is given in [9]), but are sufficiently far from true equivalence that their automorphism groups do not correspond. Now the distinctive additional structure of a PBG-groupoid is a group action, and this action is of a type which does not transfer to the corresponding principal bundle. As a result, * Research supported by the Science and Engineering Reseach Council under grant no. GR/D/30938 10 AnnalsBd. 6, Heft 2 (1988)

142

K.Mackenzie

PBG-groupoids carry an equivariant version of connection theory which does not correspond to the familiar concept of equivariance for bundles. Much of this paper is devoted to arguing that this equivariant connection theory is the appropriate language in which to study the geometry of (*). In § 1 we run through the correspondence between extensions of principal bundles and PBG-groupoids. This material comes largely from [10], to which we refer the reader for several of the proofs. § 2 gives examples and some special cases. Most notable here is a classification of extensions (*) in which N is discrete and abelian and P is connected; equivalence classes of such extensions are classified by G-equivariant morphisms H (P, Z) -* N. Here again we see the role of equivariance conditions. § 3 gives the equivariant connection theory referred to above and also gives some elementary results on the relationship between the characteristic classes of Q(B, H) and P(B, G) in (*). These results are at present restricted to the case where N, H and G are all compact and connected and this case does not allow the role of equivariance to be fully displayed; we expect the method to be much more generally applicable.

§ 1. Extensions of principal bundles and their PBG-groupoids In this section we describe how an extension of principal bundles may be replaced by a single Lie groupoid together with a Lie group action. Since Lie groupoids are essentially equivalent to principal bundles, what we get is a description of the whole principal bundle extension in terms of a suitably equivariant version of standard bundle theory; this is the key to all that follows. Most of the material of this section comes from [10]. We begin with a brief recollection of the concept of Lie groupoid (see [9] for further details and references). Throughout the paper we work with manifolds which are C ®, real, and have countably many components. Definition 1.1. Let B be a manifold. A Lie groupoid on base B is a manifold 2 together with two surjective submersions a, fi: - B, called the source and targetmaps, a smooth map e: B - 2, x -- x, called the object inclusion map, and a smooth partialmultiplication, from 2 * Q = {(aq, ) e £2 x Q 1al = f/3} to 2, denoted by juxtaposition, all such that (i) (q/) = a(,) and fP(t1) = f(q1) for all (, r) e Q*Q2;

(/1) = (rq) for all 5, ,1, r e 2 such that a(4) = ]3(i/) and a(q) = (,); ca(x)= (x) = x for all x e B; Xx = 5 and y9 = for all 5 e Q2 with ca(,) = x, fl(,) = y; for each 5 e Q there is a unique inverse -' with a(-') = fi(,), f(-') = a() and -'l = ff, 5.~ = ~a; and act), called the anchor of £2, is a surjective (vi) the map (, at): 2 -- B x B, -* ( a, submersion. // (ii) (iii) (iv) (v)

It follows, as it does for Lie groups, that £2 - £2, 5 -- ', is smooth. It is now easy to see that 2x = (fi, a)- (x, x) is a Lie group, called the vertex group at x e B, and each

On extensions of principal bundles

143

2x = c-(x) is a principal bundle on base B with respect to the restriction 2x x QX - x as action and the restriction fix: Ox - B of / to Ox as projection. This is the vertexprincipal bundle at x. Given a principal bundle P(B, G, p) there is an associated Lie groupoid (P x P)/G on B, which is the quotient manifold of P x P under the action (u2, u,) g = (u2g, ug). The orbits are denoted by u2, u1>. The groupoid structure is P((U2 , Ul>) = p(U2 ) ,

=

, u>,

0a(U2 , ul>) = p(ul)

any u with p(u) = x,

and
ul> = u3 , ulg> , where

u

= u2g .

Any vertex bundle of (P x P)/G is isomorphic to P(B, G), but not usually in any natural way. Conversely, every Lie groupoid Q is isomorphic to (2x x Qx)/Qx under <(/, 6> [- i -', but there is not usually any natural choice for x. See [9, II 1.19] for further details and references. Note that in [9] the anchor (, ) is denoted by [fl, oc]. Definition 1.2. Let and Q' be Lie groupoids on B and B', respectively. Then a morphism from 2 to Q' is a pair of smooth maps p: -- Q', (po: B - B', such that ' o q0 = 0o oc, 13' o o = %oo 3 and (p(q) = Tp(,q) (p(5) for all (, ) e * 2. We then say that p is a morphism over (p,; if B = B' and p, = idB, then p is over B, or is base-preserving. // If p: - Q' is a morphism of Lie groupoids then the restriction p: x -* 2' is a principal bundle morphism over : B - B' with respect to px: f2l -* Q2. Conversely, if fOCo, p): P(B, G) -- P'(B', G') is a morphism of principal bundles, then u2, u, - (u2), f(u)> is a morphism of the associated Lie groupoids. To get a true equivalence one needs to.deal with morphisms which fix specified base-points (see, for example, [9, 11 1.19]); in general any two "conjugate" principal bundle morphisms correspond to a single morphism of the associated Lie groupoids. For example, all the automorphisms P(B, G) - P(B, G), u [-* uh, x ~- x, g [-* h- 'gh, for fixed h EG, correspond to the identity automorphism of (P x P)/G. It is for this reason that the Lie groupoid language gives intrinsic formulations of concepts which for principal bundles are defined "up to conjugacy" - the holonomy subbundles of a connection are a typical example (compare [9, II 7.14]). Attached to every Lie groupoid Q is a Lie group bundle on B which we denote I1 and call the inner group bundle, or gauge group bundle. This is the union of all the vertex groups of 2, that is Io:

U 1x , xeB

and may be regarded as the kernel of (/, ): - B x B, in a natural sense. If {ai: Ui - Qb} is a section-atlas for some vertex bundle Qb(B, 2', /?b), then the inner automorphisims 0i = : U X ' IQalui defined by (x, .) -Hai(x) aui(x) -, are Lie group bundle charts for I12. If 2 = (P x P)/G, then IQ2 is naturally isomorphic to (P x G)//G, the Lie group bundle associated to P(B, G) 10-

144

K. Mackenzie

through the inner automorphism action of G on itself. Elements of(P x G)//G are denoted by u, g>, u P, g e G, with uh, h-'gh> = u, g> for h e G. The injection is PxG PxP G--<, g> <-u, u>. G G Definition 1.3. (i) An extension of principalbundles is a sequence N

A>Q(B, H, p') -- P(B, G, p)

(I)

in which 7r denotes both a surjective submersion Q - P and a surjective Lie group morphism H -. G (necessarily a submersion), such that nr(idB, 7r) is a morphism of principal bundles; where denotes an injective Lie group morphism N -* H, and where

N A. H

G

is an extension of Lie groups. (ii) An extension of Lie groupoids is a sequence (2) M a-* - 2 in which 7r is a base-preserving morphism of Lie groupoids and a surjective submersion, M is a Lie group bundle on the same base, and is an injective immersion whose values lie in I and which, considered as M -* I, is a morphism of Lie group bundles; finally we require that (2) be exact in the sense that, for E q, (4) is an identity of £2 iff C lies in the image of i. // One could equally well describe (1) as a lifting of P(B, G) to the group H. Now, given (1), one can construct an extension of Lie groupoids QxN QxQ , PxP -- >-* --- (3) H H G where (Q x Q)/H and (P x P)/G are the associated Lie groupoids and (Q x N)//H is the Lie group bundle associated to Q(B, H) through the action of H on N by (the restrictions of) inner automorphisms. The maps are i: -*

and

i: FH <(v 2), n(v)> .

Converselly, assume given (2). Then for any chosen b E B, Mb

Ihb(B,

b) b

£2b(B,

2b)

(4)

is clearly an extension of principal bundles. The constructions (1) t- (3), (2) - (4) are easily seen to be mutually inverse, providing that base-points are chosen consistently. For the concepts of equivalence appropriate to these notions of extension, see [1 1, § 4]. Now that the preparatory remarks have been made, our purpose is to show that principal bundle extensions are themselves equivalent to Lie groupoids together with an additional structure. Consider then the extension (1). Observe that Q(P, N, ) is itself a principal bundle; call it the transverse bundle. Here N acts on Q via , but in practice we write merely vn for v(n).

On extensions of principal bundles

145

Denote by Y the associated groupoid (Q x Q)/N on P. Call Y the transverse groupoid of (1). Its inner group bundle is given as follows. Proposition 1.4. IY <(, n>N - ((v),

(Q x N)//N is naturally isomorphic to p*((Q x N)//IIH) under

)).

Proof. Here the superscripts N and H indicate the group with respect to which the orbit is taken. We verify the surjectivity; the rest is similar. Take (u, )H = H; thus N is mapped to (u, H). // The groupoid Y can now be presented in the following form, which we shall regard as canonical: P [QXN -

Y

P x P.

(5)

The injection is (u, v, n>H) ½ vnh-', vh-'>, where h E H has 7r(v) = ur(h). The submersion is the anchor - ((v 2), 7r(v1 )) of Y. Now comes the crucial additional structure. We define a right action of G on Y by g = v2 h, vh> where 7t(h) = g. Clearly this is well-defined. Proposition 1.5. This action interacts with the algebraic structure of Y in the following ways: (i) (ii)

(vg) = A(v)g,

(vg) = (v)g;

- - i q:

(iii) (v'v) g = (v'g) (vg); (iv) (vg) - ' = v-,g; where v, v' e Y, g E G, u

P and i(v') = #(v).

Proof. Here a, # denote the source and target projections of Y. For (iii), let v = (v2, I, >; then v' can be written in the form (v3, 2 >.Now (v'v) g = (v3, vl> g = (U3h, vlh> where 7r(h) = g, and (v'g) (vg) = v3h, v2h> = . The other parts are similar. // Proposition 1.6. The action of G on Y is free andprincipal with respect to the map q: Y-

H

(Uv2,vi>N

(

Proof. Suppose that )N g = )N . Then N = (v 2 , 1>)N where ir(h) = g. It follows that h E N and so g = 7n(h) = 1. The other assertion is similar. // Thus Y((Q x Q)/H, G, q) is itself a principal bundle. For later purposes we need to describe the action of G on IY in terms applicable to (5). Notice that IY is G-stable by 1.5 (i). The proof of the following proposition is given in [10].

146

K. Mackenzie

Proposition 1.7. (i) Regarding IY as (Q x N)//N, the action of G becomes
n>Ng = N where 7r(h) = g.

ii) Regarding IY as p*((Q x N)//H), as in 1.4, the action of G becomes (u, H) g = (ug, ") . // Thus the action of G on p*((Q x N)//H) induced from Y is the natural action of G which exists on any inverse-image bundle across p: P -o B. The sequence (5) is now G-equivariant, with the actions on the first and last terms being given solely by the action of G on P. 1.6 suggests that it is possible to recover the original extensions (1) and (2) from their transverse groupoids. We therefore abstract the properties of transverse groupoids into the following concept. Until the remarks at the end of the section, let P(B, G, p) be a fixed principal bundle. Definition 1.8. A PBG-groupoid on P(B, G) is a Lie groupoid Y on base P together with a right action of G on the manifold Y such that the conditions (i)-(iv) of 1.5 hold. // Equally well, one could say that g e G is to act by a Lie groupoid automorphism Y Y over Rg: P - P, u + ug. The terminology is intended to suggest that Y is both a G-groupoid and a principal G-bundle. The key theorem is the following; for the proof see [10]. Theorem 1.9. Let Y be a PBG-groupoidon P(B, G). Then the action of G on Y isfree and the quotient manifold Y/G exists. // Further, if the action of G on P is proper, or properly discontinuous, then the same is true of the action of G on Y. From 1.9 it follows that Y( Y/G, G) is itself a principal bundle. Denote elements of Y/G by and the map v H by q. Then one can define a groupoid structure on Y/G with base B by ],'() = p(f(v)),

c'() = p(()),

x = <>i)

where p(u) = x,

and =
Y G

PxP G

On extensions of principal bundles

147

The corresponding extension of principal bundles is IY ,._, Y Y P7 G lb b[ ,t -'bJ l]-p(BG -I-Gb,~GI, ~bL ,,,,(7)

(7)

)

where b e B is some reference-point. Choose u e P with (uo) = b. Then Yo,

,

G

v



b

is a diffeomorphism. The preimage of (Y/G)b under this map is U yuog

gcG

U O 0

and with the group structure *' the map

= (Ag')

A where A YUo9,

' e

yUO-

becomes an isomorphism of Lie groups. Thus (7) can be presented as (F

Yo -* Yo [B, U Yu0vg1 gG

-uO J

P(B, G)

(7)'

The correspondence between PBG-groupoids and extensions of Lie groupoids, or of principal bundles, can be made suitably functorial. However we will not need the details of the various notions of equivalence here. We merely note that a morphism of PBG-groupoids Y - Y' over a common principal bundle P(B, G) is a morphism of Lie groupoids over P which is G-equivariant. Remarks. (1) It is worthwhile stressing that the PBG-groupoid concept is a genuinely groupoid concept and admits no simple formulation in terms of principal bundles. The problem is, that although Lie groupoids and principal bundles are sufficiently close to being equivalent for connection theory (to take one example) to transfer readily to Lie groupoids, one cannot transfer a general group action on a Lie groupoid to its corresponding principal bundles. To transfer a single automorphism is easy enough, because one need only pick at most two reference-points in the bundle, but to transfer an action of the PBG-groupoid type would require reference-points in all fibres, and this cannot be done smoothly unless the groupoid is trivializable. (2) The concept of PBG-groupoid should also be distinguished from the action on a Lie groupoid corresponding to an equivariant bundle. Let Q(X, N, 7r) be a principal bundle on which a Lie group G acts on the left in the standard sense (for example, [4, 6.28]): G acts on Q and X so that is equivariant and the N and G actions on Q commute. Then G acts on the associated groupoid Y = (Q x Q)/N by g), and this is an action by Lie groupoid automorphisms over G x X X, satisfying the four conditions of 1.5 (in their left-hand form).

148

K. Mackenzie

However in this case there is a map 0: G x X -- Y, (g, x) - (gv, v>, where t(v) = x, which has the following three properties: (i) 0(g,x) E Yx, for g G, x X; (ii) 0(g2, g1x) 0(gl, x) = 0(g2g1, x), for l, g2 e G, x X; (iii) 0(g, Pfv) v 0(g, av)' = gv, for g e G, v Y. Conversely any Lie groupoid Y on a base Xwith a group action which satisfies (i)-(iv) of 1.5, and for which there exists such a map 0, arises from an action * of G on any vertex bundle Yx(X, Yx), namely

g*v = (g, Bv) v, and this action makes Yx(X, Yx) an equivariant principal bundle.' The conditions (i)-(iii) are far from artificial - (i) and (ii) simply state that 0 is a groupoid morphism from the action groupoid G x X (see [9, 1 1.6]) to Y, and (iii) is a kind of generalized homotopy condition (see [6, p. 46]) between the given action and the trivial one. //

§ 2. Examples and special cases Example 2.1. Consider the extension of principal bundles Z2 by SU(2) (S2, U(1))

A S0(3) (S2, S0(2)).

Here Ad maps the unit quaternion u SU(2) to the map q -- uqu-', restricted to 3 H. On the group level, Ad is the double covering which maps el E U(I) to O RI cos 2 sin 20

-sin 20 cos 20J

The transverse bundle SU(2)(SO(3),Z 2) has associated groupoid Y = (SU(2) x SU(2))/Z 2 ; this may be regarded equally well as the quotient of SU(2) x SU(2) over the normal subgroup { (l, 1)}, and Y is therefore diffeomorphic to SO(4). Not that the action of S0(2) on Y is u2, u1> z = , where either (consistent) choice of root may be made. Thus SO(4) is, at one and the same time, a Lie group, a principal bundle over the Stiefel manifold V3(4, 2), and a Lie groupoid over S0(3). Some further comments on this superabundance of structures are made below. // Example 2.2. Consider the bundle SO(4)S 2 x S2, SO0(2) x S0(2)) where we take SO(4) to be the usual matrix group, and SO(2) x SO(2) to be the subgroup R(0)

0) |

R

On extensions of principal bundles

149

where /cos0 -sin \ cos OJ'

R() = (I) ( s in 0

Denote the element of SO(2) x SO(2) above by (R(O), R(p)). Take the bundle P(S2 x S2, SO(2)) obtained from this by the morphism 7r1 : SO(2) x SO(2) SO(2), (R(O), R((p)) F-R(0); in [9] we call this the produced bundle. It is the quotient of SO(4) x SO(2) by the action (A, R(O)) (R((p), R(O)) = (A(R((p), R(O/)), R(O - qp)) of SO(2) x SO(2). To identify P, let V = V(4, 2) be SO(4)/SO(2) where the subgroup is {(I, R(f)) I e R} SO(4). Denote the orbit of A e SO(4) under this action by A and map P to V by

* A(R(O), 1).

It is easy to check that this is a diffeomorphism and that the produced bundle is the universal bundle V(S 2 x S 2, SO(2)). This gives an extension with the property that the group extension S0(2) - S0(2) x S0(2) -* S0(2). is the direct product, but the transverse bundle SO(4) (V, SO(2)) is not trivializable. More generally there is an extension SO(k) >-* SO(n) (G

k,

SO(k) x SO(n - k)) --* V k(G,

and likewise with the complex and quaternionic cases.

k,

SO(n - k))

//

Example 2.3. [10] Let Q be any a-connected Lie groupoid on base B. Then the monodromy groupoid MO (see [9], II § 6] for details and references) is an extension of 2, F

-

MO --*

where F is a certain bundle of discrete groups, whose fibre-type is the fundamental group of a typical -fibre of Q. In principal bundle terms, if P(B, G) is given with P connected, then there is a principal bundle P(B, G) on the same base, where P is the universal cover of P, and G is a certain group locally isomorphic to G, and this is an extension ,P

P(B,

) - P(B, G) .

The transverse bundle is P(P, 7irP) and the PBG-groupoidis the fundamental groupoid

n(P) of P with the natural action of G. This example includes 2.1.

//

Example 2.4. Let G be a Lie group, H a closed subgroup, and N a closed normal subgroup of H. Then there is an extension of principal bundles N - G(G/H, H) -* G/N(G/H, H/N)

150

K. Mackenzie

in which not only both bundles in the extension, but also the transverse bundle G(G/N, N) are equivariant. The transverse groupoid is accordingly isomorphic to the action groupoid G x (GIN) ([9, II 1.12]). In these terms the action of H/N on Y = G x (GIN) becomes (g, g'N) hN = (g, g'hN). This example includes 2.2. Suppose further that N = Z is a central subgroup of G. Then the quotient space Y = (G x G)/Z is the quotient group of G x G over Az = {(z, z) Iz E Z}, which is normal in G x G. Thus Y also has a group structure, and it is easy to check that the two structures commute; that is, Y is a -groupoid in the sense of [2]. Thirdly, Y is a principal bundle over 0 = (G x G)/H and it is clear that the action of H/Z on Y is right-multiplication from the subgroup { E Y Ih e H}, which is isomorphic to H/Z. So Y(O, H/Z) is also a homogeneous bundle; in this sense all three structures mutually commute. This example includes 2.1. Note also the case where G is compact and connected, and T is a maximal torus with normalizer N and Weyl group W = NIT. There is an extension T >-- G(G/N, N) - GIT(G/N, W) and the Weyl group acts on the groupoid (G x G)/N. // Example 2.5. Any principal bundle P(B, G) defines an extension Go >- P(B, G) -* P/G(B, ItG) where Go is the identity component of G. Following Pradines, we call P/Go(B, 7rOG) the squeezed bundle corresponding to P(B, G). If P is connected then the squeezed bundle is the covering of B corresponding to the map t 1B -* 7rG from the homotopy sequence. If P is simply-connected then this map is an isomorphism and we have GO, - P(B, G) B(B,

7rnlB

)

with transverse bundle P(B, Go). Here P - B can also be regarded as the natural lift of P -*B. // We now describe some special cases in which simplifications occur. Firstly consider an extension in which the quotient bundle is trivial: N >-* Q(B, H) * B x G(B, G). Restrict the transverse bundle to B x {I }; this gives a bundle QB(B, N). Proposition 2.6. Q(B, H) is isomorphic to the produced bundle (QB x H)/N(B, H). Proof. The isomorphism is (v, h)> * vh.

//

Thus the transverse bundle quotients down to B and gives a reduction of Q(B, H) to N. This result is of course a form of [8, I 5.6]. Given bundles P,(B, Gt, p,) and P2(B, G2, P2) on the same base, there is a direct product bundle P1 xB P2(B, G1 x G2) whose total space is the pull back t(u,, u2) E P 2 Ip,(ul) = p2 (U2 )} and whose group is the direct product group. In [1] this is called the spliced bundle. It corresponds to the direct product of Lie groupoids on the same base [9, III 1.24].

On extensions of principal bundles

151

Proposition 2.7. Let Q(B, G x N) be a principalbundle with group a directproduct. Then Q(B, G x N) is isomorphic to the direct product bundle Q/N(B, G) x B Q/G(B, N). Proof. Denoting the elements of Q/N by v ½- (v>N, G). //

v>N, etc., the map Q , Q/N x

B

Q/G is

For example, in 2.2 one has S0(4) - V x B V as principal bundles. Next we consider extensions by a discrete abelian group. Recall ([7]) that an extension of Lie groups D >- H -* G in which D is discrete abelian and G is connected is automatically central and that the group of equivalence classes of all such extensions is isomorphic to Hom (rG, D). We now give in 2.8 a surprisingly similar result for bundle extensions. We consider extensions D >, Q(B, H) - P(B, G)

(1)

in which D is discrete abelian and P is connected. Since G need not be connected, there is a not necessarily trivial action of G on D, which quotients to an action of 7roG on D. Since 7rG is itself a quotient of 7r,B, Qcan also be considered to be an action of 7r 1 B on D. The corresponding extension of Lie groupoids is PxD G

QxQ H

,

PxP G

notice that determines the action of (P x P)/G on (P x D)//G ([9, II 4.8]). We fix e and write Opext (P(B, G), D, o) for the set of equivalence classes of extensions which induce Q; it has a natural abelian group structure. Consider the PBG-groupoid corresponding to (1), namely QxQ D Let CY denote the c-identity-component subgroupoid of Y; it is stable under G. Fix u e P and consider PxD -*Y--*PxP,

Y- =

CYL 17(P)U -P Since 1n(P). is a universal cover of P, and CY I some other cover, there is a unique morphism of covers ¢p: I1(P) - C YIl which makes the diagram commute and maps identities iuto identities. Since p must respect the deck transformations, it induces a groupoid morphism p: 1n(P) -- C Y. Fix g E G and consider ½ p(g) g-', H(P) -- C Y; by uniqueness, this must be equal to p. So p is a morphism of PBG-groupoids H(P) CY Y. The restriction 'p+ of p to (P x )//7r -+ P x D, where n7= r1P, maps conjugation by n(P) to conjugation by Y and must therefore be of the form (u, 1>) - (c(D),f()), where c: P - P is the covering and f: r - D is a morphism (compare [9, II 4.10]). Since D is abelian, f induces f: 7zab - D, where tab = 7r/[r, 7r]. There is an action of G on x7b induced from the sequence r,b >- G/[7r, r] -- G obtained by quotienting -*

152

K. Mackenzie

n >- G -* G over [, It], and an action of G on P x D by (u, d) g = (ud, Q(g-') (d)). Now the G-equivariance of qp implies thatf: Ita, - D is G-equivariant. Theorem 2.8. The niap

Opext (P(B, G), D, e) -+ Hom (ab

D)G

which maps (1) to f above, is an isomorphism. Proof. Notice thatf is uniquely obtained from p and that qp is the unique PBG-groupoid morphism (P) -- Y over P. Take anyf Hom (ab D)G and letf: 7n - D be the associated morphism. Note that the G-equivariance off implies that (glg-) = e(g) (fll)) for any g E G lying above g e G, and any 1 7r. Let Q = (P x D)/I be the produced principal bundle using f: 7 - D. Thus elements of Q are of the form (ii, d>, E P, d e D, with = < ,f() d> for 1e 7r. Let Y = (Q x Q)/D be the associated groupoid. Elements of Y are of the form <, <,

d>>

with <, > =

2, d2 >, >.

Define an action of G on Y by <<2, d2>, <,, d,>> g = <<2, g, e(g-') (d2 )>, <,g, Q(g-') (d,)>> where g E G lies above g. It is a routine exercise to check that this is well-defined and makes Y a PBG-groupoid on P(B, G). It therefore induces an extension (1) and it is easy to check that the (1) so found induces the given. Thus the map Opext -- Hom of the statement is surjective. The proof that it is a morphism is straightforward. Lastly, if (1) is given with = 1, the constant morphism, then c(5), for 5 e 17(P), depends only on the endpoints of . Therefore, (P induces a morphism A: P x P -+ , which is G-equivariant because p is, and this quotients to a groupoid morphism (P x P)/G YI/G (Q x Q)/H which gives a splitting of the extension. The connectivity properties of Q can be expressed in terms off: namely, 7rQ - D/im(f) and 1, Q kerf The point of the PBG-groupoid apparatus in 2.8 is that we do not have to construct the group H seperately. Working directly with the bundle P(B, G) leads to problems with the construction of H because G need not be connected. The same comment applies to the following result, which was proved in the run-up to 2.8. Proposition 2.9. Let M >- *

Q2

On extensions of principal bundles

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be an exact sequence of Lie groupoids with Q a-connectedandM a bundle of discrete groups. Then there is a unique morphism M - 4i front the monodromy groupoid M2 of 2 to qP such that

Q

commutes. commutes. //

Next we consider extensions N - Q(B, H) -

P(B, G)

(2)

in which each of N, H, G is compact and connected. By [13], the group extension is equivalent to N ,-

G xN D

G

where D is defined in terms of a morphismf: lG - ZN to be {(, f(l)-')E x NIl l7r G}. (Here ZN is the centre of N.) Let I = imf _ ZN and consider the obvious map H G x (N/I). This is a covering with kernel I. Taking the produced bundle Q = (Q x G x (N/II)/H(B, G x (N/I)) and applying 2.7 to it gives the following result, which is presumably folklore. Proposition 2.10. Let N >-- Q(B, H) -- P(B, G) be an extension of principal bundles with N, H andG all compact and connected. Then Q(B, H) is locally isomorphic to a direct product P(B, G) x B Q/G(B, N/I). // If follows that the Lie algebroid of Q(B, H) (see §3 and [9]) is the direct product of the Lie algebroids of P(B, G) and Q/G(B, N/I), and hence all the infinitesimal properties of Q(B, H), most notably its connection theory, can be reduced to that of P(B, G) and Q/G(B, N/II). See also 3.8. Lastly we classify those principal bundle extensions which are semi-direct on the group level. Definition 2.11. An extension of principal bundles N "- Q(B, H) - P(B, G)

(3)

is group-semi-direct if N -* H

G

is semi-direct as a group extension. A second group-semi-direct extension N - Q'(B, H) -P(B,

G)

(3)'

with the same group extension is strongly equivalent to (3) if there exists a diffeomorphism p: Q - Q' such that p(idB, id,): Q(B, H) - Q'(B, H) is an isomorphism of principal bundles and 7r' o = (and p o = ). //

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Theorem 2.12.Let P(B, G,p) be aprincipalbundle, A an abelianLie group and e a representation of G on A. Denote by G Ix A the semi-direct product corresponding to Q. Then the set of strong equivalence classes of group-semi-direct extension A >- Q(B, G

(4)

A) -- P(B, G)

is in natural bijective correspondence with H'(B, (P x A) I//G) and, with the Baer sum operation on the set of strong equivalence classes, this is an isomorphism. Proof. Much of the proof is routine, and we give only the basic formulas for each step. Given (4), let = {Ui} be a simple open cover of B. Then there are local sections ai : Ui - P and i: Ui -, Q and one can redefine Ti if necessary so that r oti = a,. Now define vi:p(U) - Q by vi(ai(x) g) = i(x) (g, 1) where throughout we are taking G ix A with the operation (g, a,) (g2, a2) = (gg 2, e(g2) (al) a2) It follows that v(ug) = v(u) (g, 1) for u ep-'(U), g G, and so for the transition functions nij: p-(Uij)- A defined by vinij = Vj, we have nij(ug) = (g-1) (nj(u)) . From this it follows that n.. can be considered as a local section of(P x A)//G and defines an element of 2'(, (P x A)//G). Now take a second group-semi-direct extension A

-*

(4)'

Q'(B, G v A) .- P(B, G),

strongly equivalent to (4) under go: Q -* Q'. Taking q = {U} as before, we choose first {ai: Ui -P} and then {ti: U - Q} and {T': Ui -* Q'}, with nr o T = ai. Writing oTpi = T[k, for ki: Ui -* G Ix A, it follows that ki takes values in A. Now with vi and v' constructed as before we have

( o v) (i(x) g) = vI(au(x) g) g(g

)

(ki(X))

and the map ai(x) g ½+Q(g-') (ki(x)), p-(Ui) - A, is clearly G-equivariant and defines an element of CO(, (P x A)//G) whose coboundary is the difference between {ntj} and {n'j}. The proof that different choices of {ai} lead to cohomologous cocycles is similar. Now suppose we are given n E H(B, (P x A)//G). Let 1 be a simple open cover of B and choose representative nj: p- (Uij) /A. From these one constructs a principal bundle Q(P, A) whose elements are equivalence classes i, u, a>, where u p-l(Ui), a A, and = (I, u, nji(u) a> for u ep-'(Uij). Define an action of G Ix A on

Q by (g, a') = i, ug, (g-')(a)a' . It is straightforward to check that this is well-defined, is a free action, and is principal with respect to the projection Q -- B, i, u, a> -, p(u). Thus we obtain a group-semi-

On extensions of principal bundles

155

direct extension (4), and one can check that any choice of local sections ai: Ui - P and i: Ui - Q, x - , defines the original cocycle {nij}. Lastly, suppose {nlj} is a second representative of n, cohomologous to {nj} under n'. = r- nij.r where the r: p-'(Ui)- A are G-equivariant. Then Q - Q', ) - i, u, ri(u)-1 a>' is a strong equivalence. The processes of taking the inductive limit, and of checking that the Baer sum corresponds to the operation in HI(B, (P x A)//G) are routine. // Notice that the zero element of the group of extensions is A >-. P x A(B, G

K A)

-* P(B, G)

where G Ix A acts on P x A by (u, a) (g, a') = (ug, Q(g-') (a) a'). This extension is characterized by the existence of a principal bundle morphism P(B, G) -o P x A(B, G Ix A) which is right-inverse to the projection. Indeed the groupoid corresponding to P x A(B, G !x A) is the semi-direct product groupoid. Calculations of the same general type as in 2.12 are given in [3]. The method can be extended to non-abelian group-semi-direct extensions by using the approach of [11].

§ 3. Transverse connections and Chern-Weil Theory We begin with a brief review of the approach to connection theory used in [9]. For a principal bundle P(B, G), the Atiyah sequence is Px G

j TP p >-+-* -- * TB G

where TP/G is the quotient of the tangent bundle over the group action Xg = T(Rg) (), (P x g)/G is the adjoint bundle, associated to P(B, G) via Ad: G Aut(g), p. is the quotient of T(p) and j is the fundamental vector field map u, X> - (. Sections of TP/G can be identified with vector fields on P which are invariant under G and so F(TP/G)acquires a Lie bracket [ , ]. A connection in P(B, G) is now a map y: TB -+ TP/G right-inverse to p* and the corresponding map co: TP/G - (P x g)/G with j o = id and y op* + o co = id is precisely the quotient of the corresponding connection form in A'(P, g). The same quotienting process applied to the curvature form 2 e A2 (P, g) gives a map TP/G i TP/G - (P x g)/G which pulls down to TB i TB -- (P x g)/G and is then given by X Y y[X, Y] - [yX, y Y]; we denote this by Ry. Every connection y in P(B, G) induces a connection V in (P x g)/G by j(V (V)) = [yX, jV], called the adjoint connection. For a Lie groupoid 2, the Atiyah sequence of any corresponding principal bundle can be constructed by a process directly generalizing that by which the Lie algebra of

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a Lie group is found and is correspondingly called the Lie algebroidof Q; in this context we use the notation LO >- A2 -* TB. See [9, A §§ 2,3 II § 3] for a detailed account and references. We will also need to consider general Lie algebroids, not necessarily attached to any principal bundle or Lie groupoid. Definition 3.1. A Lie algebroid on base B is a vector bundle A on B together with a map a: A TB, called the anchor, and an alternating bracket [ , ] on FA which satisfies the Jacobi identity and is such that (i) a[X, Y] = [aX, x Y], (ii) [X,fY] = fiX, IY + a(X) () Y for all X, Y e FA and f: B - R. A is transitive if a is a surjective submersion and is totally intransitive if a = 0. A morphism of Lie algebroids over a common base B is a vector bundle map p: A - A' such that a' o (p = a and which preserves the brackets. // References to the history of this concept are given in 9, II §2]. We normally denote a transitive Lie algebroid by L

>

A

TB ,

wherej need not be the inclusion. The kernel L is the adjoint bundle of A; it is a Lie algebra bundle ([9, IV 1.4]). One can easily develop a version of connection theory for abstract transitive Lie algebroids ([9, III §5, IV §11). Now suppose given an extension of principal bundles N

Q(B, H, q)

-

P(B, G, p) .

(I)

One has the corresponding extension of Atiyah sequences K

TQ . Q H

TP T G

no

(2)

where K = (Q x 91)/H and n, is the quotient of T(n). There is also the Lie algebroid of Y, LY >-, AY -, TP,

AY

TQ

(3)

and it is easy to see that G acts freely on this. sequence by Lie algebroid automorphisms and that the quotient is precisely (2), ([10, §31). In particular, G acts freely on TQ/N

On extensions of principal bundles

157

with quotient TQ/H. Conversely, given (2) one can construct an inverse-image sequence

']*

[ H

-] - p-[

TP

(4)

and this is canonically isomorphic to (3) ([10, 4.2]). With all this preparation done, the following key result is immediate. Proposition 3.2. Let X: TP/G - TQ/H be a right-inverse to .* in (2). Then the induced map X = P*(X): TP - TQ/N is a G-equivariant connection in Q(P, N). Conversely, any G-equivariant connection y: TP -* TQ/N = A Y is X where X is the induced map TP/G - TQ/H. We call Xa transverseconnection in (1). We stress again that this concept of equivariance applies to the Atiyah sequence of Q(P, N), and to the groupoid (Q x Q)/N, but not to Q(P, N) itself; this concept of equivariant connection is fundamentally different from that appropriate to homogeneous bundles G(G/H, H) (see, for example, [4, 6.281). It is for this reason that the use of the groupoid Y is essential. All the familiar concepts of connection theory can be formulated for Lie groupoids ([9, III §§5, 7]) and are, for G-equivariant connections in Y, suitably equivariant; for example, the holonomy subgroupoid of a G-equivariant connection in Y quotients down to a subgroupoid of = (Q x Q)/H and one thus obtains a generalized Ambrose-Singer theorem applicable to transverse connections (see also [12]). Although G does not in general act on Q, it is quite possible to define a G-equivariant connection in terms of the bundles alone: a connection form co e A'(Q, 9l) is G-equivariantif (Rh)* co = (Adh -1) co for all h e H. However this definition becomes extremely awkward when one tries to deduce the equivariance properties of associated concepts, such as path-lifting and holonomy, and it also, of course, makes no sense for extensions of Lie algebroids which do not arise form extensions of principal bundles. We begin with a few small results to illustrate the techniques. Proposition 3.3. Let

N

Q(B, H)

P(B, G)

be an extension of principal bundles. If the extension admits a flat transverse connection and P is simply-connected, then it is semi-direct. Proof.A flat transverse connection Xinduces a flat equivariant connection Xin Q(P, N). Since P is simply-connected, the bundle is trivial; in terms of Y = (Q x Q)/N this means that there exists a morphism 0: P x P -, Y. Since Xis equivariant, 0 is also and therefore quotients to a morphism of groupoids (P x P)/G Y/G (Q x Q)/H which is rightinverse to 7n. // In connection With 3.3 the following result is interesting. 11 Annals Bd. 6, Heft 2 (1988)

158

K. Mackenzie

Proposition 3A. Let N >- Q(B, H) -P(B,

G)

be an extension of principalbundles with G compact. Then if Q(P, N) admits aflat connection, it admits a flat equivariant connection. Proof. Let y: TP - A Y be any connection in Q(P, N) and define X: TP -- A Y by X(X) = y(Xg) g- dg where I dg is the bi-invariant normalized integral for G. For fixed X T(P)u, the map g [- y(Xg) g-' takes values in the vector space A Y ,. Now X is certainly G-equivariant connection and it is routine to check that the curvatures are related by RX(X, Y) = Ry(Xg, Yg) g-' dg.

//

Similarly one can prove that if there is a map a: P -+ Q with 7n a = id, and G is compact, then the induced groupoid morphism 0: P x P -- Y, (u2, u1) a(u2) a(u1)-1, can be averaged over G to give a groupoid splitting. What is most interesting about these results is that it is only G that need be compact, not N nor H. 3.3 is a consequence of III 6.5 of [9], and in fact goes back to Pradines [12], but the present proof is more direct than that of [9]. Given any extension N

-*

Q(B, H) -* P(B, G)

with P connected there is a natural pull-back extension N >-* Q * P(B, H * G)-

P(B, G),

where Q * = {(v, u) E Q x P I n(v) = c(a)} and H* = {(h, ) H x I (h) = c(g)} acts on Q * P by (v, u) (h, g) = (vh, ui). (Here c denotes both the covering P - P and the associated morphism r -+ G.) So if N - Q(B, H) -* P(B, G)admits a flat transverse connection, then the pullback to P(B, G) is semi-direct. The next result is an infinitesimal version of 2.6. Proposition 3.5. Let K > A'-* A be an extension of transitive Lie algebroids. Suppose that y': TB -* A' is a connection in A' such that rcoy': TB -* A is flat. Then L' >- A' -- TB reduces to a Lie algebroid with adjoint bundle K. Proof. Since 7r o y' is flat, the curvature Ry, of y' takes values in K. Because K is an ideal of A', the adjoint connection V' = V"' of y' in L' restricts to K. Now applying IV 3.20 of [9], the hypotheses of which are easily seen to be satisfied, we obtain a Lie algebroid structure on TB K with respect to which TB e K -- A', X D V I- y'(X) + V is an injective morphism of Lie algebroids. //

On extensions of principal bundles

159

Example 3.6. ([8, IX §4]). Let M be a Kahler manifold of dimension n and U(M) (M, U(n)) the bundle of unitary frames. There is an extension of principal bundles SU(n) >- U(M) (M, U(n))

de,

-p

U(M)/SU(n) (M, U(1))

where we are using det to denote U(M) - U(M)/SU(n), as well as the morphism U(n) -* U(1). Let 2 be the curvature form of the canonical Riemannian connection and Qthe Ricci form. Then e, regarded as a ()-valued 2-form on M, is precisely - 2 times the curvature of the image of the canonical Riemannian connection under det. If the Ricci form vanishes and M is simply-connected, then U(M)/SU(n) (M, U(1)) is trivializable and we can apply 2.6. It follows that the holonomy group of M is contained in SU(n), which is a slight refinement of 4.6 of [8]. // We come now to the main business of this section, which is the "transverse" ChernWeil theory associated to an extension of principal bundles. Until 3.7 we consider an arbitrary extension of principal bundles N

(1)

Q(B, H) .L P(B, G).

Our purpose is to show that there is a G-equivariant form of the Chem-Weil morphism for Q(P, N, 7r). Consider the Lie algebroid of the associated PBG-groupoid, LY >- AY --* TP,

AY -

N

,

LY N

N

(2)

Let Sk( Y) denote the vector bundle Symmk (L Y, P x R) whose sections are the symmetric k-linear maps (L Y)k __ P x R. There is a natural action of Y on Sk(Y) induced by the adjoint action of Y on L Y. Namely, for any u, v e P, p e Sk( Y)u and v e Y, V(f) = (P o(Ad v-l)k

(compare [9, III 1.25]). Fix a reference-point u e P and identify Y with N. Then, by [9, II 4.15], the space (FSk(Y))" of Y-invariant sections is naturally isomorphic with Sk(91)N, the space of symmetric k-linear maps f: 91 k - R such that f(Ad n-' V, ... , Ad n - 'Vk) = f(V, ... , Vk) for V 1, ... , Vk 9 and n e N. We denote Sk(91)N by Ik(N) and identify it with (FSk(Y)Y without comment. Give I(N) =

Ik(N) k>O

the algebra structure of [8, XII § 1]. Now G acts on each Ik(N) by (f g) (V ,, ... , V) = f(Ad h - V, ... , Ad h - 'Vk) where h e Hhas ir(h) = g. Denote by Ik(N)G the set of elements invariant under G. Clearly /(N)G =

) Ik(N)G k>O

1I'

160

K. Mackenzie

is a subalgebra of I(N). In terms of the isomorphism Ik(N)

(FSk( Y))y ,

P(N)G corresponds to those maps F: (L )k P x R for which F(Vlg,..., Vkg) = F(V ,,..., Vk) g for all V1 ,...., VkEL Y and geG. Here Gacts on P x R by (u, t) g = (ug, t) and in terms of the isomorphism L Y p*K, where K = (Q x 9)/H, the action of G of L Y becomes (u, V) g = (ug, V) (compare 1.7(ii)). Such maps F: (L y)k -_ p x R quotient down to K - B x R and this give in fact an isomorphism IP(N)G

- (F Symmk (K, B x R)) ®

where = (Q x Q)/H acts on Symmk(K, B x R) by 5(qp) = go o (Ad -)k. Not let X: TP - A Y be any connection in Y, and takef e P(N). Using the curvature R,: TP $ TP - L Y and regarding f as being in (F Symmk (L Y, P x R))Y we define f(R) e A 2k(P) by the usual ([8]) formula 1 f(R,) (X, ..., X 2k) = (2k)!

ef(Rx(Xa(l), Xa( 2)) ...

Rx(Xa(2 k-1 ) Xa( 2 k)))

One obtains in this way the standard Chern-Weil morphism w: I(N) - H*(P), where H* denotes de Rham cohomology with real coefficients, for the bundle Q(P, N). Now suppose that X is an equivariant connection and that f belongs to P(N)G. It is immediate thatf(Rx) is an equivariant form, that is, that f(Rx) (X , g, ... , X2kg) = f(R) (X,, ... , X2,) for all X e TP and g E G. We denote the space of G-equivariant n-forms on P by A"(P)G; it is well-known that these are closed under exterior differentiation and define the equivariant cohomology H*(P)G ([4, Chapter IV]). (We refer to this cohomology as equivariant, rather than invariant, for uniformity with the general case considered in [9, IV §21.) One needs to prove that for a fixed f P(N)G and differing choices of equivariant connection X, the classes f(Rx,) and fRx 2) are cohomologous in the equivariant cohomology. The set of equivariant connections is cetainly convex and one need only check that each stage of the standard proof that the Chern-Weil morphism is welldefined, is valid equivariantly. This is routine. We thereby have an equivariant ChernWeil morphism G W: I(N) -_ H(p)G

Notice that H(P)G is naturally isomorphic to the Lie algebroid cohomology .5(TP/G, B x R), with coefficients in the trivial representation [9, IV 2.7]. For simplicity we denote B(TP/G, B x R) by H(TP/G). One can now regard wVas a map (F Symmk (K, B x R)) - H(TP/G)

On extensions of principal bundles

161

and it is possible to define w entirely in terms of the extension of Lie algebroids K

> A

1i

Qx9 H

-

AQ

11

11

TQ H

TP G

Such a construction, but for an extension of abstract Lie algebroids K >- A' - A, not necessarily associated with any extension of principal bundles, was given by Teleman [14]. In his context, A' acts on Symmk(K, B x R) by k

X'(f) (V,,

*--,

k) = a'(X') (

k))

f(...

, ad (X'(Vi)), .. ,

Vk)),

i=l

where X' e FA', f e r Symmk(K, B x R), V,,..., VkE FK, and where a' is the anchor of A'. In the case of an extension K >-* Ad -* An arising from an extension of principal bundles, this is the derivative of the action of 0 on Symmk (K, B x R) (compare [9, III 4.8]). coincides with that invariant It follows- that the space of sections invariant under under A' = A4 if and only if N is connected (compare [9, IV 1.19]). We now wish to describe the relationship between the Chern-Weil morphisms w: I(G) - H(B) for P(B, G) and w': I(H) - H(B) for Q(B, H), and the transverse Chern-Weil morphism w: I(N)G - H(P)G. Since t is a principal bundle morphism there is a commutative diagram I(G)

-

H(B)

I(H) H(B)

where the top map sends fe Ik(G) to fo I(H) I(N)G.

ik E

I(H). There is also a restriction map Q:

Proposition 3.7. The diagram I(H)

--

I(N)G

H(B) .a* H(P)G commutes. Proof. Because Q(P, N) is a reduction of the pullback bundle p*Q(P, H), there are two diagrams I(H)I(H) I(H) wI Iw" and H(B) P* H(P) H(P)-

1w"

I(N) li H(P)

where w" and · are the Chem-Weil morphism of p*Q(P, H) and Q(P, N) respectively and p* is the map induced on cohomology by p: P -, B. Now one need only put these

162

K. Mackenzie

together and observe that I(H)-- , I(N) takes values in I(N)G, that p* = a* takes values in H(P)G, and that w is the restriction of v. // Now suppose that each of N, H, G is compact and connected. Then 9 has an Ad Hstable complement in b and so Q above is surjective. It follows now from 3.7 that a* restricts to a surjection of characteristic algebras a*: Ch(Q(B, H)) -* Ch(Q(P, N))G where Ch(Q(B, H)) is the image of w' and Ch(Q(P, N))G is the image of w. It is wellknown that forfe I(G), the formsf(R), for y a connection in P(B, G), are the projections to B of exact forms on P, and so it follows that Ch+(P(B, G)) = im w+ is contained in the kernel of a*. It would be interesting to know if Ch+(P(B, G)) is the whole kernel. In favour of this is the fact that, since G is compact and connected, the triviality of Ch+(P(B, G)) is equivalent to the injectivity of a*: H(B) -+ H(P) ([5, § 9.18]). We now summarize all this: Q(B, H) -* P(B, G) be an extension of principal bundles with Theorem 3.8. Let N N, H, G all compact and connected. Then a*: Ch(Q(B, H)) -- Ch(Q(P, N))G is a surjective algebra morphism and Ch+(P(B,G)) is contained in its kernel. // Hence if all the characteristic classes of P(B, G) vanish, then a* gives an isomorphism between Ch(Q(B, H)) and Ch(Q(P, N))G ([5], 9.18). Example 3.9. Let Ad: U(2) -* SO(3) be the adjoint map, with kernel U (1) the centre of U (2), and consider an extension of principal bundles U(1)

Q(B, U(2)) -* P(B, S0(3)).

Denote the generators of I(U (2)) by f, and f 2 ; thus f,(X) = -tr iX, f2(X) = det iX. 2 where 11IIis the standard' Denote the generator of I(SO (3)) by g,; thus g,(X) = X11XX ID (3). Lastly, denote the generator of I(U (1)s0(3)) = I(U (1)) by e; norm on R3 thus e(it) = t. Now it is easy to see that, up to normalization factors, w'(f) E H2 (B) is mapped to w(e) e H2 (P)S0(3) = H 2(p) under a*, and w'(f2) = w(g,) lies in ker a*. // 3.8 is of course an easy result, reflecting the two facts that extensions of compact connected groups are locally semi-direct, and that H(P)G - H(P) for G compact. On the other hand, one can only be certain of capturing all characteristic classes by the ChernWeil construction if the groups concerned are compact, so the hypotheses are reasonable. It may well be that the most fruitful application of the methods developed here will be to extensions of noncompact groups, treated by classifying space techniques. We hope to deal with this on another occassion.

On extensions of principal bundles

163

References [1] D. D. BLEECKER, Gauge theory and variationalprinciples. Addison-Wesley, Reading, Mass. 1981. [2] R. BROWN and C. B. SPENCER, "-groupoids, crossed modules and the fundamental groupoid of a topological group". Proc. Konin, Nederl. Akad. van Wetens. A, 79, 296-302, 1976. [3] W. GREUB and H. R. PETRY, "On the lifting of structure groups". Differential GeometricalMethods in MathematicalPhysics 11, 1977. Ed. K. Bleuler, H. R. Petry and A. Reetz. Lecture Notes in Mathematics # 676, Springer-Verlag, Berlin, 1978. [4] W. GREUB, S. HALPERIN and R. VANSTONE, Connections, Curvatureand Cohomology, Vol. II. Academic Press, New York, 1973. [5] W. GREUB, S. HALPERIN and R. VANSTONE, Connections, Curvature and Cohomology, Vol. III. Academic Press, New York, 1976. [6] P. J. HIGGINS, Notes on Categoriesand Groupoids. van Nostrand-Reinhold, London, 1971. [7] G. HOCHSCHILD, "Group extensions of Lie groups, II". Ann. of Math. (2) 54, 537-551, 1951. [8] S. KOBAYASHI and K. NOMIZU, Foundations of differential geometry, Vols. I, II. Interscience, New York, 1963 and 1969. [9] K. MACKENZIE, Lie groupoids and Lie algebroidsin differential geometry. LMS Lecture Notes Series, no. 124. Cambridge Univ. Press. 1987. [10] K. MACKENZIE, "Integrability obstructions for extensions of Lie algebroids". Cahiers de Topol. et Gom. Diff Categoriques, (to appear). [11] K. MACKENZIE, Classification of principal bundles and Lie groupoids with prescribed gauge group bundle. Preprint. [12] J. PRADINES, "Theorie de Lie pour les groupoides diffbrentiables. Calcul differentiel dans la categorie des groupoides infinit6simaux". C.R. Acad. Sci. Paris. t. 264, 245-248, Serie A, 1967. [13] A. SHAPIRO, "Group extensions of compact Lie groups' . Ann. of Math. (2) 50, 581-586, 1949. [14] N. TELEMAN, "A characteristic ring of a Lie algebra extension". Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8) 52, 498-506 and 708-711, 1972. KIRILL MACKENZIE

Department of Mathematical Sciences University of Durham Science Laboratories South Road Durham DHI 3LE England. Communicated by T. J. Willmore

(Received October 15, 1987)