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)
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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-
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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].
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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