топология / Tom Dieck T. Algebraic topology (EMS, 2008)

Chapter 14

Bundles

Bundles (also called fibre bundles) are one of the main objects and tools in topology and geometry. They are locally trivial maps with some additional structure. A basic example in geometry is the tangent bundle of a smooth manifold and its associated principal bundle. They codify the global information that is contained in the transitions functions (coordinate changes).

The classification of bundles is reduced to a homotopy problem. This is achieved via universal bundles and classifying spaces. We construct for each topological group G the universal G-principal bundle EG ! BG over the so-called classifying space BG. The isomorphism classes of numerable bundles over X are then in bijection with the homotopy set ŒX; BG .

The classification of vector bundles is equivalent to the classification of their associated principal bundles. A similar equivalence holds between n-fold covering spaces and principal bundles for the symmetric group Sn. This leads to a different setting for the classification of coverings.

From the set of (complex) vector bundles over a space X and their linear algebra one constructs the Grothendieck ring K.X/. The famous Bott periodicity theorem in one of its formulations is used to make the functor K.X/ part of a cohomology theory, the so-called topological K-theory. Unfortunately lack of space prevents us from developing this very important aspect.

Classifying spaces and universal bundles have other uses, and the reader may search in the literature for information.

The cohomology ring H .BG/ of the classifying space BG of a discrete group G is also called the cohomology of the group. There is a purely algebraic theory which deals with such objects.

If X is a G-space, one can form the associated bundle EG G X ! BG. This bundle over BG can be interpreted as an invariant of the transformation group X. The cohomology of EG G X is a module over the cohomology ring H .BG/ (Borel-cohomology). The module structure contains some information about the transformation group X, e.g., about its fixed point set (see [7], [43]).

14.1 Principal Bundles

Let G be a topological group. In the general theory we use multiplicative notation and denote the unit element of G by e. Let r W E G ! E, .x; g/ 7!xg be a

continuous right action of G on E, and p W E ! B a continuous map. The pair

.p; r/ is called a (right) G -principal bundle if the following axioms hold:

(1)For x 2 E and g 2 G we have p.xg/ D p.x/.

(2)For each b 2 B there exists an open neighbourhood U of b in B and a G- homeomorphism ' W p 1.U / ! U G which is a trivialization of p over U with typical fibre G. Here G acts on U G by ..u; h/; g/ 7!.u; hg/.

If we talk about a G-principal bundle p W E ! B, we understand a given action of G on E. From the axioms we see that G acts freely on E. The map p factors through the orbit map q W E ! E=G and induces a continuous bijection h W E=G ! B. Since q and p are open maps, hence quotient maps, h is a homeomorphism. Thus G-principal bundles can be identified with suitable free right G-spaces. In contrast to an arbitrary locally trivial map with typical fibre G, the local trivializations in a principal bundle have to be compatible with the group action. In a similar manner

we define left principal bundles.

A G-principal bundle with a discrete group G is called a G -principal covering. The continuity of the action r is in this case equivalent to the continuity of all right translations rg W E ! E, x 7!xg. This is due to the fact that E G is homeomorphic to the topological sum qg2G E fgg, if G is discrete.

Let E G ! E be a free action and set C.E/ D f.x; xg/ j x 2 E; g 2 Gg. We call t D tE W C.E/ ! G, .x; xg/ 7!g the translation map of the action.

(14.1.1) Lemma. Let p W E ! E=G be locally trivial. Then the translation map is continuous.

Proof. Let W D p 1.U / E be a G-stable open set which admits a trivialization W U G ! W . The pre-image of .W W /\ C.E/ under is f.u; g; u; h/ j

u 2 U; g; h 2 Gg, and t ı . / is the continuous map .u; g; u; h/ 7!g 1h.

A free G-action on E is called weakly proper if the translation map is continuous. It is called proper if, in addition, C.E/ is closed in E E.

(14.1.2) Proposition. A free action of G on E is weakly proper if and only if

0 W E G ! C.E/, .x; g/ 7!.x; xg/ is a homeomorphism.

Proof. The map W C.E/ ! E G, .x; y/ 7!.x; tE .x; y// is a set-theoretical inverse of 0. It is continuous if and only if tE is continuous.

Let E carry a free right G-action and F a left G-action. We have a commutative diagram

Pp

with orbit maps P and p and q D pr1 =G.

330 Chapter 14. Bundles

(14.1.3) Proposition. A free right G action on E is weakly proper if and only if for each left G-space F the diagram is a topological pullback.

Proof. We compare the diagram with the canonical pullback

b

X E

ap

with X D f..x; f /; y/ 2 .E G F / E j p.x/ D p.y/g and a D pr1, b D

homeomorphism E G G ! E, .x; g/ 7!xg transforms q into p, X into C.E/

is continuous. One verifies that Q induces a map W X ! E F . The equalities

(14.1.4) Proposition. Let G act freely and weakly properly on E. The sections of q W E G F ! E=G correspond bijectively to the maps f W E ! F with the property f .xg/ D g 1f .x/; here we assign to f the section sf W x 7!.x; f .x//.

Proof. It should be clear that sf is a continuous section. Conversely, let s W B ! E G F be a continuous section. We use the pullback diagram displayed before (14.1.3). It yields an induced section of pr1 which is determined by the conditions

pr1 ı D id and P ı D s ı p. Let f D pr2 ı W E ! F . Then pr1 .xg/ D xg D pr1. .x/g/, since is a section and pr1 a G-map. (The right action on

E F is .e; f; g/ 7!.eg; g 1f /.) The equalities P .xg/ D sp.xg/ D sp.x/ D

P .x/ D P . .x/g/ hold, since is an induced section and P the orbit map. Since

(14.1.5) Proposition. Let the free G-action on E be weakly proper. Then the orbit map p W E ! E=G D B is isomorphic to pr W B G ! B, if and only if p has a section.

Proof. Let s be a section of p. Then B G ! E, .b; g/ 7!s.b/g and E ! B G, x 7!.p.x/; t.spx; x// are inverse G-homeomorphisms, compatible with the projections to B. Conversely, pr has a section and hence also the isomorphic map p.

(14.1.6) Proposition. Let X and Y be free G-spaces and ˆ W X ! Y a G-map. If ' D ˆ=G is a homeomorphism and Y weakly proper, then ˆ is a homeomorphism.

Proof. X is weakly proper, since the translation map of X is obtained from the translation map of Y by composition with ˆ ˆ. We have to find an inverse ‰ W Y ! X. By (14.1.4), it corresponds to a section of Y W .Y X/=G ! Y =G. We have the section s W x 7!.x; ˆ.x// of X W .X Y /=G ! X=G. Let be the inverse of '. With the interchange map W .X Y /=G ! .Y X/=G we formD ı s ı . One verifies that is a section of Y .

Let a commutative diagram below with principal G-bundles p and q be given, and let F be a G-map. Then F or .F; f / is called a bundle map.

YF X

qp

f

CB

If f is a homeomorphism, then F is a homeomorphism (see (14.1.6)). If f is the identity, then F is called a bundle isomorphism.

Given a principal bundle p W X ! B and a map f W C ! B, we have a pullback diagram as above with Y D f.c; x/ j f .c/ D p.x/g C X. The maps q and F are the restrictions of the projections onto the factors. The G-action on Y is

.c; x/g D .c; xg/. If p is trivial over V , then q is trivial over f 1.V /. Therefore q is a principal bundle, called the bundle induced from p by f . Also F is a bundle

map. From the universal property of a pullback we see, that the bundle map diagram above is a pullback.

(14.1.7) Proposition. Let U be a right G-space. The following are equivalent:

(1)There exists a G-map f W U ! G.

(2)There exists a subset A U such that m W A G ! U , .a; g/ 7!ag is a homeomorphism.

(3)The orbit map p W U ! U=G is G-homeomorphic over U=G to the projection pr W U=G G ! U=G.

(4)U is a free G-space, p W U ! U=G has a section, and tU is continuous.

Proof. .1/ ) .2/. Let A D f 1.e/ and v W U ! A,x 7!x f .x/ 1.x/. Then

.v; f / W U ! A G is an inverse of m.

332 Chapter 14. Bundles

.2/ ) .3/. The G-homeomorphism m induces a homeomorphism m=G of the orbit spaces. We have the homeomorphism " W A ! .A G/=G, a 7!.a; e/. With these data m ı ." id/ ı m=G D pr.

.3/ ) .4/. If ' W U=G G ! U is a G-homeomorphism over U=G, then the G-action is free and x 7!'.x; e/ is a section of p. The translation map of U=G G is continuous and hence, via ', also tU .

.4/ ) .1/. Let s W U=G ! G be a section. Then U ! G, u 7!tU .sp.u/; u/

f W U ! G into the G-space G with right translation action. A right G-space is called locally trivial if it has an open covering by trivial G-subspaces.

(14.1.8) Proposition. The total space E of a G-principal bundle is locally trivial.

i.e., the orbit map, is called a Hopf fibration. There is a similar Z=2-principal bundle Sn 1 ! RP n 1 onto the real projective space and an S3-principal bundle S4n 1 ! HP n 1 onto the quaternionic projective space. Þ

(14.1.10) Proposition. Let f W X ! Y be a G-map and pY W Y ! Y =G a G- principal bundle. Then the diagram

is a pullback.

Proof. Let U Y be a G-set with a G-map h W U ! G. Then h ı f W f 1.U / ! U ! G is a G-map. Hence pX is a G-principal bundle. The diagram is therefore a bundle map and hence a pullback.

We say, a map f W X ! Y has local sections if each y 2 Y has an open neighbourhood V and a section s W V ! X of f over V ; the latter means f s.v/ D v for all v 2 V .

(14.1.11) Proposition. Let G be a topological group and H a subgroup. The quotient map q W G ! G=H is an H -principal bundle if and only if q has a section over some neighbourhood of the unit coset.

Proof. A locally trivial map has local sections. Conversely, let s W U ! G be a section of q over U . The map tG is continuous. For each k 2 G we have the H -

Actions with the properties of the next proposition were called earlier properly discontinuous.

(14.1.12) Proposition. Let E G ! E, .x; g/ 7!xg be a free right action of the discrete group G. The following assertions are equivalent:

(1)The orbit map p W E ! E=G is a G-principal covering.

(2)Each x 2 E has a neighbourhood U , such that U \ Ug D ; for each g 6De.

(3)The set t 1.e/ is open in C .

(4)The map t is continuous.

Proof. (1) ) (4) holds by (14.1.1).

(4)) (3). The set feg G is open, since G is discrete.

(3)) (2). We have t.x; x/ D e. Since t 1.e/ is open, there exists an open neighbourhood U of x in E such that .U U / \ C t 1.e/. Let U \ Ug ¤ ;,

say v D ug for u; v 2 U . Then .u; v/ D .u; ug/ 2 .U U / \ C , hence t.u; ug/ D g D e.

(2) ) (1). Let U be open. Then U G ! UG, .u; g/ 7!ug is a G-homeo- morphism, hence UG an open trivial G-subspace.

(14.1.13) Example. Let G be a closed discrete subgroup of the topological group E. Then the action G E ! E, .g; x/ 7!gx is free and has property (4) of the previous proposition. Examples are Z R or Z C. Þ

(14.1.14) Example. The map g W R ! S1, t 7!exp.2 it/ has kernel Z. Therefore there exists a bijective map f W R=Z ! S1 such that fp D g. Since g is an open map, g is a quotient map and therefore f a homeomorphism. By the previous example, g is therefore a covering. Similarly exp W C ! C is seen to be a covering. Þ

(14.1.15) Example. Let G be a Lie group and H a closed subgroup. Then the quotient map p W G ! G=H is an H -principal bundle. In the chapter on differentiable manifolds we show that G=H carries the structure of a smooth manifold such that p is a submersion. A submersion has always smooth local sections. Þ

We construct locally trivial bundles from principal bundles. Let p W E ! B be a right G-principal bundle and F a left G-space. The projection E F ! E induces, via passage to orbit spaces, q W E G F ! B. The map q is locally trivial

334 Chapter 14. Bundles

with typical fibre F . A bundle chart ' W p 1.U / ! U G of p yields a bundle chart

q 1.U / D p 1.U / G F ' G id .U G/ G F Š U F

for q. We call q the associated fibre bundle with typical fibre F , and G is said to be the structure group of this fibre bundle. The structure group contains additional

information: The local trivializations have the property that the transition functions are given by homeomorphisms of the fibre which arise from an action of an element of the group G.

Let p W Y ! B be a right G-principal bundle. It may happen that there exists

a right H -principal bundle q W X ! Y for a subgroup H G and a G-homeo- morphism W X H G ! Y over B. In that case .q; / is called a reduction of the structure of p. One can consider more generally a similar problem for

homomorphisms ˛ W H ! G.

(14.1.16) Example. Let E ! B be a G-principal bundle and H G a subgroup. Then E G G=H ! E=H , .x; gH / 7!xgH is a homeomorphism. Therefore E=H ! E=G, xH 7!xG is isomorphic to the associated bundle E G G=H ! E=G. From a subgroup K H G we obtain in this manner G=K ! G=H as a bundle with structure group H and fibre H=K, if G ! G=H has local sections.

If X is a G-space and H C G, then G=H acts on X=H by .xH; gH / 7!xgH . The quotient map X=H ! X=G induces a homeomorphism .X=H /=.G=H / Š

One can “topologise” various algebraic notions in analogy to the passage from groups to topological groups. An important notion in this respect is that of a small

category.

A (small) topological category C consists of an object space Ob.C / and a morphism space Mor.C / such that the structure data

Mor2.C / D f.ˇ; ˛/ 2 Mor.C / Mor.C / j s.ˇ/ D r.˛/g

carries the subspace topology of Mor.C / Mor.C /. For these data the usual axioms of a category hold. In a groupoid each morphism has an inverse. In a topological

336 Chapter 14. Bundles

R-vector space on each fibre Xb D p 1.b/. A bundle chart or a trivialization over

the open basic set U B for these data is a homeomorphism ' W p 1.U / ! U Rn over U which is fibrewise linear. A set of bundle charts is a bundle atlas, if their

basic domains cover B. The data p W X ! B together with the vector space structures on the fibres are an n-dimensional real vector bundle over the space B if a

bundle atlas exists. Thus a vector bundle is in particular a locally trivial map. In a

similar manner one defines complex vector bundles or quaternionic vector bundles. A vector bundle W E. / ! B is called numerable, if there exists a numerable

covering U of B such that is trivial over the members U 2 U. (This notion is also used for other types of locally trivial bundles.) A bundle has finite type, if it has a finite bundle atlas.

Let .U; '/ and .V; / be bundle charts for p. Then the transition map is

' 1 W .U \ V / Rn ! .U \ V / Rn; .x; v/ 7!.x; gx .v//

with gx 2 GLn.R/. The assignment g W U \ V ! GLn.R/, x 7!gx is continuous. A bundle atlas is said to be orienting, if the gx have positive determinant. If an orienting atlas exists, then the bundle is orientable. An orientation of a vector

bundle p W E ! B consists of a vector space orientation of each fibre p 1.b/ with the property: For each x 2 B there exists a bundle chart .U; '/ about x such that

' transports for each b 2 U the given orientation on p 1.b/ into the standard orientation of Rn. A chart with this property is called positive with respect to the

given orientation. The positive charts form an orienting atlas, and for each orienting atlas there exists a unique orientation such that its charts are positive with respect to the orientation. A complex vector space has a canonical orientation. If one uses this orientation in each fibre, then the bundle, considered as a real bundle, is an oriented bundle.

Let W E. / ! B and W E. / ! C be real vector bundles. A bundle morphism ! over ' W B ! C is a commutative diagram

E. / ˆ E. /

'

BC

with a map ˆ which is fibrewise linear. If ˆ is bijective on fibres, then we call the bundle morphism a bundle map. Thus we have categories of vector bundles with

bundle morphisms or bundle maps as morphisms. The trivial n-dimensional bundle is the product bundle pr W B Rn ! B. More generally, we call a bundle trivial if it is isomorphic to the product bundle.

(14.2.1) Proposition. A bundle map over the identity is a bundle isomorphism.

Proof. We have to show that the inverse of ˆ is continuous. Via bundle charts this can be reduced to a bundle map between trivial bundles

(14.2.2) Proposition. If the previous diagram is a pullback in TOP and a vector bundle, then there exists a unique structure of a vector bundle on such that the diagram is a bundle map.

Proof. Since ˆ is bijective on fibres, we define the vector space structures in such a way that ˆ becomes fibrewise linear. It remains to show the existence of bundle charts.

If is the product bundle, then can be taken as product bundle. If has a bundle chart over V , then has a bundle chart over ' 1.V /, by transitivity of pullbacks.

We call in (14.2.2) the bundle induced from along ' and write occasionallyD ' in this situation. The previous considerations show that a bundle map is a pullback. (Compare the analogous situation for principal bundles.)

A bundle morphism over id.B/ has for each b 2 B a rank, the rank of the linear map between the fibres over b. It is then clear what we mean by a bundle morphism of constant rank.

A subset E0 E of an n-dimensional real bundle p W E ! B is a k-dimensional subbundle of p, if there exists an atlas of bundle charts (called adapted charts)

' W p 1.U / ! U Rn such that '.E0 \ p 1.U // D U .Rk 0/. The restriction p W E0 ! B is then a vector bundle.

A vector bundle is to be considered as a continuous family of vector spaces. We can apply constructions and notions from linear algebra to these vector spaces. We begin with kernels, cokernels, and images.

(14.2.3) Proposition. Let ˛ W 1 ! 2 be a bundle morphism over B of constant rank and ˛b the induced linear map on the fibres over b. Then the following hold:

(1) Ker ˛ D Sb2B Ker.˛b/ E. 1/ is a subbundle of 1.

S

/E. 2/ is a subbundle of 2.

(3)Suppose Coker.˛/ D Sb2B E. 2/b= Im.˛b/ carries the quotient topology from E. 2/. Then, with the canonical projection onto B, Coker.˛/ is a vector bundle.

Proof. The problem in all three cases is the existence of bundle charts. This is a local problem. Therefore it suffices to consider morphisms

˛ W B Rm ! B Rn; .b; v/ 7!.b; ˛b.v//