Chapter VIII Characteristic Homomorphism for Σ-bundles

Chapter VIII

Characteristic Homomorphism for 1.-bundles I n this chapter r = R or C and vector spaces and vector bundles are defined over r. We continue the linear and multilinear conventions announced at the beginning of Chapter VII.

SI. Z-bundles 8.1, Definition.

Let F be a vector space and write F'."

=

0" F* @ 0".

A linear isomorphism a : F 5 H induces the isomorphism

0" (a*)-' 0 0" a : F'"

N

5H'.'

;

it will be denoted by a P , q , or simply by a. If ,$ = ( M , n-, B , F ) is a vector bundle, we shall write

(cf. sec. 7.8). T h e fibre of this bundle at x E B is the space F,P*Q.A bundle map cp: 4 -+ that restricts to isomorphisms in the fibres induces the bundle maps c p p t q : @ q -+ v p l q defined by

cpy

= (cp,)"'",

x E B.

We shall frequently denote cppvq simply by v. Thus, if CT is a cross-section , cp#a is the pulled-back cross-section in [ P , q . in q p ~ q then I n particular, if {( U, , is a coordinate representation for ,$,then {( U,, +,P,'J)} is a coordinate representation for ,$p*'J, 372

373

1 . Z-bundles

Definition:

A Z-bundle is a pair ( f , Z,) where:

(i) ( = ( M , T , B, F ) is a smooth vector bundle. (ii) Zf = ( u l , ..., u,) is a finite ordered set of cross-sections ui E Sec f p i i * i , subject to the following condition: vi

(iii) There is a finite ordered system ZF= (vl , ..., nm) of tensors E F p i l * i and there is a coordinate representation {( U, , $,)} of f such that xE

+,(x, ui) = ui(x),

U, , i

=

1, ..., m.

A coordinate representation for 5 that satisfies condition (iii) will be called a Z-coordinate representation for the Z-bundle (5, Zf). Thus {( U, , qa)}is a Z-coordinate representation if and only if for each i and a, qfui (regarded as anP>*i-valuedfunction in U,) is the constant function

u, --+

vi

.

Remark: A Z-structure in a vector bundle can be regarded as a reduction of the structure group from GL(F)to an algebraic subgroup (cf. article 7).

Now suppose ( f , Z6) and (7, Zq) are Z-bundles, where Zf = ( u l , ..., u,) and Z,, = ( T ,~ ..., T ,) . Then a Z-homomorphism between these Z-bundles is a bundle map, v: f + 7, restricting to isomorphisms in the fibres and satisfying i = 1, ..., m. @(T~)= ui, 8.2. 0-deformable cross-sections. Let f = ( M , rr, B, F ) be a vector bundle, and let Z, = (ul,..., u,,) be some ordered set of cross-sections: ui E Sec ( p i ? * ( , i = 1, ..., m . Definition: T h e set Zf is called 0-deformable if, for each pair x, y E B, there is a linear isomorphism, %.,:

Fx

rr

F, ,

+

such that mx,,(ui(x))= u,(y),

i

=

1, ..., m.

Theorem I: Let ( = ( M , T , B, F ) be a vector bundle, and let ZE= (ul , ..., uVn)(ui E Sec ( p i % ) . Then ( f , 2,) is a Z-bundle if and only if ZEis 0-deformable.

3 74

VIII. Characteristic Homomorphism for Zbundles

Proof: If (4, Zc) is a Z-bundle, then Zc is obviously 0-deformable. Conversely, assume ZEis 0-deformable. We must show that (t,ZE) admits a 2’-coordinate representation. Since ZEis 0-deformable, we may Q ~ linear isomorphisms, choose a fixed set of tensors vi E F P ~ Iand N

a,: F A F,

,

x E B,

satisfying a,(vi) = ui(x), i = 1, ..., m. T o construct a 2’-coordinate representation, we lose no generality in assuming M = B x F. Thus the ui are smooth maps ui : B -+ F P d * q r . Now fix a E B and use a, to arrange that .,(a)

=

oi,

i

=

1,

..., m.

Then consider the vector space, H

and let u: B

+H

= F”1-q‘ @

... @F”m4n,

be the smooth map given by U(X)

= (u,(x), ..., u,(x)),

x E B.

Set .(a) = v and let K be the isotropy subgroup of GL(F)at v (with respect to the obvious representation of GL(F)in H ) . Then (cf. sec. 3.5) the smooth map A, : GL(F)--t H given by A,(cp) = ~ ( u )induces an embedding Av: GL(F)/K+ H. Moreover, the image of A , is the orbit of GL(F)through v. Now, since ZEis 0-deformable, each vector o(x) lies on the orbit of GL(F)through v ; i.e., Im u C Im A;, . Thus we can apply Theorem I, sec. 3.7, to obtain a smooth map, B + GL(F)/K,such that the diagram,

7:

commutes.

1. Z-bundles

375

Let rK:GL(F)+ GL(F)/K be the projection. Since .rr, is a bundle projection, (cf. sec. 2.13) there is a neighbourhood U, of a in B and a smooth map, w,: U , 4 GL(F),such that 7TK

0

w a = 7.

This implies that (w,(x))u = CJ(X), x E U , . Hence a Z-coordinate representation {( U , , q ~ , ) } , for ~~

8.3, Examples: 1. vector bundles.

ZF=

0.

5 is given by

In this case the Z-bundles are just

2. Riemannian bundles: A Riemannian bundle ( E , g) is a Z-bundle. (This follows from Theorem I, sec. 8.2, which in this case coincides with Proposition V, sec. 2.17, volume I). We may choose ZF= (( , )), where ( , ) is an inner product in F ; then a 2-coordinate representation is a Riemannian coordinate representation.

3. Oriented Riemannian bundles: Let 5 be an oriented Riemannian vector bundle with Riemannian metricg, . Let A , be the unique positive normed determinant function in E (cf. sec. 2.19, volume I). Set Zt = (g, , A t ) . Then (5, Zt) is a Z-bundle (apply Theorem I, which in this case coincides with Proposition VIII, sec. 2.19, volume I). 4. Complex bundles as real 2-bundles: Let be the underlying real bundle of a complex vector bundle 5 (cf. sec. 2.22, volume I). Let it E Sec LcR(= Sec (k') be defined by

i,(x)(z) = iz,

Z€F,

Set ZER = (it).Then ( t a , Zta)is a Z-bundle. Its Z-coordinate representations coincide with the coordinate representations of the complex bundle 4.

5. Whitney sums: Let The projections,

4,

p<: 4 0 7

-

be vector bundles over the same base. f 7

pn:

4 OT

-

T?

may be regarded as strong bundle maps of the bundle may write P,

, p,, E S

e c & , = Sec(4 0T ) ~ , ' .

4 0r]. Thus we

376

VIII. Characteristic Homomorphism for Cbundles

It follows from Theorem I (or sec. 2.8, volume I) that, with

ZEon = ( p c , pn), the pair (8 07,2,0,,) is a 2-bundle.

6 . Cross-sections in L, : Let = ( M , n, B, F ) be a vector bundle and let u E Sec L , . Then the set ZC= (0) is 0-deformable if and only if for each x, y E B, there is a linear isomorphism a : F, 5 F, such that 01

0

u(x)

0

01-1

= u(y).

This is equivalent to the following two conditions: For all x, y

(1)

E

B,

o(x) and u(y) have the same characteristic polynomial p ( t ) .

(2) If p ( t ) = pl(t)71 factors, then rank[p,(u(x))]j

=

p n L ( t y m is the decomposition of p into prime j = I,

rank[p,(u(y))]i,

..., Y , ,

i = 1, ..., m.

Moreover, these two conditions are equivalent to the conditions: (1 ’) u(x) and u( y ) have the same minimal polynomial p(t). is the decomposition of p into prime (2’) If p(t) = pl(t)fl pWL(t)L factors, then rank[p,(u(x))]j

=

j = 1, ..., 1, , i

rank[p,(u(y))]j,

=

1,

..., m.

Thus we can apply Theorem I , sec. 8.2, to obtain Proposition I:

are equivalent:

Let u E SecL, and set Z, = (u). Then the following

(i) u satisfies (1) and (2). (ii) u satisfies (1’) and (2’). (iii) ( f , 2,) is a Z-bundle. Corollary: Assume that B is connected. Suppose T E Sec L, and assumef(t) is a polynomial with no repeated roots such that ~ ( T ( x ) )= 0,

X E

B.

Then ( f , ( T ) ) is a Z-bundle. Proof: Write f ( t ) =fi(t) .*.fm(t),where the fi are irreducible and relatively prime, and set T $ ( x ) = ~ $ ( T ( x()i)= 1, ..., m).Then rank(T,(x>j) = rank T ~ ( x ) ,

i = 1, ..., m, j

=

1,2, ...,

1 . Z-bundles

377

and m

1 rank

T ~ ( x )=

i=l

Now fix a

E

(m - 1) dim F , ,

XE

B.

B. Then there is a neighbourhood U of a such that

rank ~ ~ ( > x )rank T ~ ( u ) ,

xE

U, i

=

1, ..., m.

Since the sum of these ranks is constant, we obtain rank

T~(x= )

rank T ~ ( u ) ,

xE

U, i

=

1,

..., m.

Thus these ranks are locally constant, and since B is connected, they are constant. I t follows easily that T satisfies (1') and (2').

Q.E.D. Examples: 7 . Algebra bundles: A vector bundle 5 is called an algebra bundle if each fibre F, , and the typical fibre F are algebras (not necessarily associative), and if [ admits a coordinate representation {( U, , +,)} such that each map

is an isomorphism of algebras. I n particular, if 71 is any vector bundle then A T and L, are algebra bundles. Algebra bundles may be regarded as Z-bundles as follows. Regard the multiplication in F, , p,:F,

OF, +Fx

9

as an element of F:*'. Then the hypothesis above implies that x t-+ pr is a cross-section in f Z > and l that (5,Z,) is a Z-bundle where ZE= (p). Let 5 be a vector bundle and let p E Sec 527'. T h u s p makes each fibre F', into an algebra. Theorem I, sec. 8.2, implies that (5,p ) is an algebra bundle if and only if the algebras F, ( X E B ) are all isomorphic. 8 . Lie algebra bundles: An algebra bundle is called a Lie algebra bundle if the fibres F and F, are Lie algebras. As an example, consider the bundle L , with Lie product given by [01,/3]

=010/3-/300,

O L , / ~ E L ~X , ,E B .

378

VIII. Characteristic Homomorphism for 2-bundles

8.4. The associated Lie algebra bundle. Let 5 = ( M , r, B , F ) , Z6= (ol, ..., 0,) be a Z-bundle. If x E B, let G(zhbe the closed subgroup of GL(F,) consisting of those F which satisfy

i

~J(u~(x)) = ui(x),

= 1,

..., m.

Its Lie algebra, I?(,), is the subalgebra of LF, consisting of the linear transformations $J which satisfy O(+)(ui(x))= 0,

i

=

1, ..., m,

where 8(#) is the linear transformation of FiVp defined as in sec. 7.8 (cf. sec. 1.9). Next, fix a set of tensors oiE F P i * q i such that the set ZF = (v1 ,..., v,) corresponds to Ztunder a Z-coordinate representation. T h e subgroup G of GL(F) consisting of the automorphisms that fix each zli is called the structure group of the 2-bundle ([, 20. Its Lie algebra is denoted by E. Now we shall construct a Lie algebra bundle (cf. Example 8, sec. 8.3),

whose fibre at x is the Lie algebra E(,) . I t will be called the Lie algebra bundle associated with the Z-bundle (8, Z6). I n fact, let {(U, ,ya)) be a Z-coordinate representation for (4, 2€). It determines the coordinate representation (( U , , 4,)) for L, given by

Since ~ , , ~ ( z l = ~ ) oi(x) (for each i, a), it follows that Lie algebra isomorphism @a,x:

E

Y

+

@m,,

restricts to a

E(x).

Thus the E(,) are the fibres of a subbundle, f E , of L , with coordinate representation {( U, , 4,)). tEis the desired Lie algebra bundle. A Z-homomorphism 9:(5, .ZE) ( r ) , 2?) induces the bundle map 9: L6 3 L, which corresponds to + l under the isomorphisms v* @ r). Thus @ restricts to a bundle map, LEg [* @ f , L, --f

and each (qE),is a Lie algebra isomorphism.

1 . 2-bundles

379

In particular, suppose (t,Z,) is a complex Z-bundle (I'= C). Then L , is a complex vector bundle. Since E ( z bconsists of the elements i,h E LFzsatisfying i = I , ..., m, O($)(a,(x)) = 0, it follows that E(%)is a complex subalgebra of LFz. Thus tEis a complex subbundle of L , . Finally, if cp: .$ -+ 77 is a Z-homomorphism, then each map (g)& is complex linear. Hence cpE is a homomorphism of complex vector bundles.

s2.

8.5. Definition.

%connections

Let f be a vector bundle and let

Zt= (q, ...,urn),

ui E Sec ~ " + ' i ,

i

=

1,

be an ordered set of cross-sections. A Z-connection in connection, V, satisfying Vui=O,

i=l,

..., m,

(4, Z;)

is a linear

..., m.

Theorem 11: Let f be a vector bundle over a connected base B and , ) be an ordered set of cross-sections. let Zlp = (al , ..., a Then ( f , ZJ is a Z-bundle if and only if it admits a 2-connection.

Proof: Assume (8, Zc)is a Z-bundle. Choose a 2-coordinate representation {( U, , cpJ} for .$ and let {p,} be a partition of unity subordinate to the open cover (U,]. Further, let V, be the linear connection in the restriction of .$ to U, corresponding to 6 under cp, . Then (since cpzai is constant) V,ui

Thus setting V

=

x,p ,

*

= 0,

for each i, a.

V, (Example 3, sec. 7.11) we find

Vai=O,

i = 1,..., m.

Conversely, let V be a Z-connection in 4. Since B is connected, Proposition VIII of sec. 7.17, shows that Zpis 0-deformable. Now Theorem I, sec. 8.2, implies that ( f , ,YE) is a Z-bundle.

Q.E.D.

If v: 5-v is a homomorphism of Z-bundles and if V, is a Z-connection in 7, then the pull-back of V, is a Z-connection in E. 8.6. Examples: 1, nection.

Zt=

0 ; any linear connection is a Z-con-

2. Riemannian connections: Suppose Zt= (( , )), where ( , ) is a Riemannian metric. Then the C-connections are precisely the Riemannian connections (cf. sec. 7.23). 380

381

2. Z-connections

If in addition 5 is orientable, let A , be the positive normed determinant function. Then (cf. sec. 7.24) a Riemannian connection V satisfies V(d,) = 0. It follows that the Riemannian connections are the Z-connections in the 2-bundle (6, Z,) with Z, = ({ , ), A , ) . 3. Complex bundles: The linear connections in a complex bundle 6 coincide with the Z-connections in the corresponding Z-bundle ( t RZER) , of Example 4, sec. 8.3. 4. Algebra bundles: Let ZE= ( p ) , where p E Sec t2,' makes 6 into an algebra bundle (cf. Example 7, sec. 8.3). Denote the induced multiplication in A ( B ; 5) by Then a linear connection V in 6 is a Z-connection if and only if 0.

V(u

* T)

= Vu

*

r

+ u - Vr,

u, r E Sec 6.

5. Trivial Z-bundles: Consider the trivial Z-bundle ( B x F, ZF), where ZF = (vl , ..., v,) is a set of tensors in F, regarded as constant cross-sections. Every linear connection in B x F is of the form

Let E be the Lie algebra of the subgroup G C GL(F)which fixes the voi. Then E is a subalgebra of LF, and V is a Z-connection if and only if Y E A 1 ( B ;E ) . 8.7. The bundle E L . Proposition 11: Let V be a Z-connection in a Z-bundle (5, Z,), where Z, = (ul,..., u,). Then

(1) T h e connection, 6, induced in L , restricts to a linear connection in the associated Lie algebra bundle t E . (2) T h e curvature, R, of V takes values in tE; R E A2(B;tE). Proof: (1) Let

T

E Sec

tE;we must show that % E A'(B; tE).

This is equivalent to (cf. sec. 7.8)

(e,(97))(ui) = 0,

i

= 1,

..., m.

But (cf. Example 4,sec. 7.12) B*(Qr) = V

0

e*(T)

- e*(T)

0

v

382

VIII. Characteristic Homomorphism for Z-bundles

and (0*(7))(cri) = 0 = Vui ,

i = 1,2, ..., m.

(2) Recall from Example 4, sec. 7.16, that B,(R) is the curvature of the connection induced in (Pi3qt. It follows that O,(R)(uJ

and so R takes values in (,

=

.

V20i = 0 ,

i

=

1,

...,m, Q.E.D.

s3. Invariant subbundles 2-bundle with f = ( M , 77 , B , F ) and (Pt,Qi. ZF = (ul , ..., vnt) is a set of tensors (uiE F P i , q i ) corresponding to the ui under a 2-coordinate representation. V denotes a Z-connection in (5,Zc) with curvature R. G is the structure group of (8, Z f )(cf. sec. 8.4); its Lie algebra is denoted by E. If the underlying coefficient field F is C, then E is a complex Lie algebra. I n this case E* is the complex dual of E. T h e adjoint representation of G in E is given by I n this article

(5,Z f )denotes a

Zc = (ul , ...,u n l ):

ui is a cross-section in

(Ad?J)($) =Q)0407J-1,

VEG,

*EE;

the contragredient representation, AdQ,of G in E* is defined by Adb(T)

=

(Ad p-')*.

Note that the contragredient representation of Ad was defined in sec. 1.9 to be a representation in the space Hom(E; R). However, if E is complex, a complex structure is induced in Hom(E; R). I n this case, there is a natural identification of the complex spaces E* and Hom(E; R), and under this identification the two definitions of Ad Qcoincide. We continue to use the notation established in sec. 8.4, and note that, if = C, then the associated Lie algebra bundle, f E , is a complex vector bundle. We remind the reader that in this case all linear and multilinear operations with respect to 4, and E are taken over C.

r

eE,

8.8. Invariant subbundles, T h e representation Ad6 of G extends to representations of G in OPE* and W E * . I n particular, we have the invariant subspaces (OPE*), and ( W E * ) , . Similarly, for x E B , representations of G(z) in OPE&, and W E & , are defined; their invariant subspaces are denoted by (OPE&,),and ( W E $ , ) , . Now let {( U , , q,)} be a C-coordinate representation for (5, Zf). Recall (sec. 8.4) that a coordinate representation {( U , , $,)} for L , is given by $b,Z(U)

= %,x

O

0O 6 . x

,

0

EL,

9

x6

and that it restricts to a coordinate representation for restricts to an isomorphism of Lie groups G 5 G(z). 383

u,

>

f E .

Further,

$u,z

VIII. Characteristic Homomorphism for Z-bundles

384

Next, let {( U, , $,)} also denote the induced coordinate representations and V P f g . Then, for x E U , , Faex restricts to isomorphisms for *,,,:

(ODE*), -=+(@”E&), c=

,

(V”E*), 5 (V’E,*,,),

x,,x:

*

It follows that the spaces (OPE&,),and (VPEG,), are the fibres of subbundles, (@”t$), C @”t,* and ( V ” t z ) ,C V’t;, with coordinate representations { ( U , , $,)} and {( U, , X,)}, respectively. Definition: (OPE;),and ( VPtZ), are called the p t h invariant tensor bundle, and the p t h invariant symmetric bundle associated with (6, Zc). A cross-section of (Opt:), (respectively, (Vp[g),) is called invariant. Proposition 111: There are unique strong bundle isomorphisms,

4: B x ( W E * ) , 5 (@”[:),

X: B x (V”E*), 5 (V’f;), ,

,

with the following property: If {( U , ,y,)} is any Z-coordinate representaand X,,x are the maps defined above, then tion for 5, and $,

=

A,,

and

X,

= Xu,,,

XE

U,.

and X,,x are independent of Proof: I t is sufficient to show that the choice of U, and of the choice of coordinate representation. Since the union of two Z-coordinate representations is again a Z-coordinate representation, it is sufficient to show that

#,, Write y;:

0

= *B,X

and

= T.

Then r

$;,\

0

xu.2

E

= XB.,

Ad r : E

II

_-t

I t follows that the induced isomorphisms satisfy 0

xE

u, n u,

*

G and

$B,x =

$is\

,

$B,x =

E.

$a,z, $B,x:

OPE* 5 OPE&,

0” Ad(.r)h.

Restricting this equation to (OPE*),yields

*i,;a. *BOX Similarly, X,,x = XB,x.

= 1,

XE

u, n up. Q.E.D.

385

3. Invariant subbundles

Identify (ODE*), and ( V P E * ) , with the constant cross-sections of the trivial bundles. As in secs. 7.8 and 7.10, write m

,

p=o

and S e c ( 0 tf), =

(VE*), =

p=0

m

0sec(@”t;),

p=o

2 (V”E*), m

(0 E*), = 1(@”I?*),

Sec(Vt;),

,

m

=

0Sec(Vpt;), . p=0

These spaces are all graded associative algebras. Moreover, induce canonical homomorphisms $*:

(0 E*), + Sec 0ti

#

and

x

X,: (VE*), -+ Sec V f z .

and

Remark: T h e constructions of this article depend only on the isomorphism class of the pair ( F , ZF).I n fact, suppose ( F l , ZF1)is a second pair, in the same isomorphism class, with corresponding Lie algebra El C LF1 . Then there is an isomorphism a: F 3 F , carrying ZF to ZF,. I t induces isomorphisms

0E* - 0E: = -+

and

VE* -% VE:.

and

( VE*),

These restrict to isomorphisms,

(0 E*), 5 (0 E:),

c?=

(VE:), ,

which are independent of the choice of u. Moreover, the diagrams,

(0 E*)l

(0 E31

(VE*),

,

YE:),

commute; this shows that, up to canonical isomorphism, independent of the choice of F and ZF.

,

#* and

X , are

8.9. 8-connections. Recall that the connection, 9, in L , induced by V restricts to a linear connection in f E (cf. Proposition 11, (I), sec. 8.7). Hence it determines a linear connection, V, in the bundles and v p g .

VIII. Characteristic Homomorphism for Z-bundles

386

Proposition IV:

#: B x

T h e inclusions,

(ODE*), -+ @”(:

and

X: B x (V’E*),

--f

V”(;,

are connection preserving with respect to the standard connection, 6 , and V. Proof: It is sufficient to consider the case that

and

( = ( B x F, T ,B , F )

u i ( x ) = ( x , vi),

xE

B, i

=

1,

..., m.

+

I n this case V = 6 Y, where Y E A1(B;E) (cf. Example 5, sec. 8.6). Moreover the total space of f E is B x E and the induced connection in f E is given by

% = 87

+ [Y, T I ,

T

E

- Y ( B ;E ) .

is given by V = 6 Thus the induced connection in is the LOPE*-valuedI-form defined by

+

Yp,

where

Y p

Y f ( x h)(x, ; @ .. @ x”)

-c z1 9

=

@

... @ ad*(Y(x; h))xi @ ... @ x p ,

i=l

xE

B , h E T,(B), xi E E*.

I n view of Proposition IX, sec. 1.8, it follows that v

Y’(x; h)(v) = 0,

E

(@”I?*),

.

This proves the proposition for I); the proof for X is identical. Q.E.D. Corollary: T h e inclusions,

#*: (0 E*), -+

Sec

0

and

X,: ( VE*)I-+ Sec V(,*,

of sec. 8.8 are isomorphisms into the graded subalgebras of invariant, V-parallel cross-sections. If B is connected they are surjective. + (7,Z,,) be a homomorph8.10. Homomorphisms. Let q ~ :(5, Zt) ism of Z-bundles (cf. sec. 8.1). It induces a bundle map, Y E : f E -+ 7 E , whose fibre maps (vE)$ are isomorphisms. Thus the linear isomorphisms,

and

V”((q~~)z)-’,

3. Invariant subbundles

define bundle maps bundle maps,

QPeg

-

-+ Q

P q g and

(O”t*,),(OD&

and

VP&$

387

+VP7;.

(V”t,*),

-

These restrict to

(VP&

9

and all these bundle maps will be denoted by y E . Moreover, all the vEinduce the same map vB:B -+ P (B, the base of T ) . The same argument as that given in Proposition 111, sec. 8.8, shows that the diagrams,

commute. Thus the diagrams,

also commute.

s4. Characteristic homomorphism

I n this article we continue the notation conventions of article 3. 8.11. The homomorphisms Pe and y C . Define a linear map,

(recall that R

E A2(B;f E )

is the curvature of the Z-connection V).

Lemma I: /3( is an algebra homomorphism. I t factors over the canonical projection (rS)*:Sec 0s; -+ Sec Veg to yield a commutative diagram,

of algebra homomorphisms. Proof: Let A iE Sec

(i = 1,2). Then (cf. Lemma IV, sec. 7.8)

T h u s Bt is a homomorphism. 388

4. Characteristic homomorphism

389

Since R is a 2-form, Im BE is a graded subalgebra of the commutative algebra C, A2p(B;F ) ; in particular,

PP(4AP s ( 4

=P

P(4

A

4 E See 06:

PE(fll),

*

Hence Bs factors as desired. Extend

/3(

Q.E.D.

and y f to homomorphisms,

by setting and

Ps(@ A A ) = Q, A Pc(A)

(1E

ys(@ A

9)= Q,

A

ys(E),

Sec @,*, 9E Sec V e f ,

@ E A(B).

T h e analogue of Lemma I holds and

Lemma 11: T h e maps

pE and y s satisfy

Proof: T h e second relation follows trivially from the first. T o prove the first, fix A ~ S e @cp , $ g . According to the Bianchi identity (cf. Proposition VI, sec. 7.15), V R = 0. Thus it follows from Example 4, sec. 7.12, that PP(V(l)=
Q.E.D.

-

66 #* : (@E*), 0

and ysI = yr

0

Q R)

s(4

Finally, set =

.**

x* : (VE*),

r)

A(B;q.

390

VIII. Characteristic Homomorphism for Zbundles

Proposition V: The linear maps morphisms. They make the diagram,

/3:

and 7: are algebra homo-

commute, and satisfy s o p :

=

0,

0.

so,:=

Proof: The first part of the proposition follows from Lemma I, together with the commutative diagram

4 Sec @S$ (BE*), a

-.1

(VE*),

1 7

(ns)*

Sec

L

vtf .

The last relation is a consequence of Lemma 11, and the corollary to Proposition IV, sec. 8.9.

Q.E.D.

Note that 7; is not the restriction of

/3:

to SP(E), .

4. Characteristic homomorphism

391

8.13. Characteristic homomorphism. Proposition V of sec. 8.1 1 shows that the differential forms in Imy; are closed. Thus we can compose y; with the projection Z ( B ; I')--t H ( B ; I') ( Z ( B ; I') = ker 6) to obtain a homomorphism

h,: (vE*),-,H ( B ; r).

It is called the characteristic homomorphism of the 2-bundle (t,Z,). Its image is called the characteristic subalgebra. T h e restriction of h, to ( V p E * ) I will be denoted by h;, h:: (VPE*): -+H 2 p ( B ;

Clearly, h; = 0 if 2p

r).

> dim B.

Theorem 111: T h e homomorphism h, is independent of the choice of connection, and hence an invariant of the Z-bundle ( E , Z& Theorem IV: Let 9): (6, ZJ -+ (9,2,J be a homomorphism of Z-bundles inducing vB: B -+ P (P,the base of 9).Then the diagram,

commutes. T h e proofs of both theorems depend on Lemma 111: Let 9): (-+ be a connection preserving 2-homomorphism with respect to Z-connections V and 6. Then the diagram,

9

commutes, where y,' and y; are defined as in sec. 8.11 via 6 and

v.

392

VIII. Characteristic Homomorphism for C-bundles

Proof: I n view of sec. 7.15, the curvatures of V and R

v satisfy

=

(where ‘ p E : tE + q E is the map induced by ‘p between the Lie algebra bundles). Thus we can apply Lemma I, sec. 7.5, and the commutative diagram at the end of sec. 8.10 to obtain

Q.E.D. Proof of Theorem 111: Consider the Z-bundle,

5‘ x R with Zcxw= (6,,

=

(11.1 x R , r x

L,

B x R,F),

..., ern) given by x E B,

Ci(x, t ) = ( ~ ~ ( xt)),,

Z-homomorphisms p: p(z, t ) = z

4xR and

-+

t

t and j , :

E

R, i = 1, ..., m.

(4

[

x R are defined by

z E M , t E R.

j,(z) = ( z , t ) ,

T h e induced maps p B : B x R -+ B and i f : B + B x R are given by ps(x, t ) = x

and

x E B,

it(x) = (x, t ) ,

t

E R.

Now let 0, and V, be any two 2-connections in 8. Let 9, and v, be the pull-backs (via p) of these connections to 5 x R. Define a Z-connection, V, in 4 x R by setting

v = tv, + (1 - t)V0

*

Because p 0 j , = L and p 0 j , = L, it follows that j o (respectively, j,) is connection preserving with respect to V, and V (respectively, with respect to 0,and V). Now let h: , h i , and h E X Idenote the characteristic homomorphisms defined via V, , V, , and V. Since it = if, Lemma 111 gives Q.E.D.

4. Characteristic homomorphism

393

Proof of Theorem IV: Choose any Z-connection, V, in 7 and give 4 thepull-back of V via q~ According to Theorem 111, h, and h, can be defined via these connections; hence Lemma I11 implies that h,

= g;"B

0

h,, .

Q.E.D. 8.14. Smooth functions. I n this section [ denotes a real vector = R). Exactly as in sec. 6.22 we extend h, to a homomorphism bundle

(r

hT*: (V**E*),

-

+ H(B).

Precomposing this map with the Taylor homomorphism,

%(El,

(V**E*),

7

(cf. sec. 6.21) yields the homomorphism s,:

Examples: 1.

Yo(E),+ H ( B ) .

L e t f E Y ( E ) ,be given by

where ?!' E ( W E * ) , . Then s E ( f )= he(!?'). 2.

Let [ be a Z-bundle with Z = 0 ; thus E

by Then s,(tr

fb)= t r e x p v , 0

. Definef

= LF

E

Y(L,),

VEL,.

exp) E H ( B ) ; it is (in general) a non-homomogeneous class.

§5. Examples I n this article we continue the notation conventions of article 3. 8.15. Dual Z-bundles. Canonical isomorphisms, N

+: FzJI A (FZ)q,P,

xE

B,

are defined by

*: w:

@

.**

@ w; @ w1@

-**

@ w, t+ w1 @

*.*

@ W @ @ w: @

*.. 0w;,

W?EF,*, W ~ E F ~ .

Thus they induce isomorphisms

*: Sec

[p,q

-% Sec([*)"p

'LI

and

t:A(B; [vsq) LA(B; ( [ * ) q J ' ) .

Now define a Z-bundle, ([*, Zp), as follows: f* = ( M * , T,B, F*) is a dual bundle for (, and ZC* = (*ul, ..., *urn). This Z-bundle is called the dual Z-bundle for ([, ZC). Next, let G and G, be the structure groups of 4 and 5". T h e isomorphism GL(F)5 GL(F*) given by p t-+ (p*)-l restricts to an isomorphism G % G, . Its derivative is the Lie algebra isomorphism p I+ -p* between the Lie algebras E and E , of G and G, . I n the same way, a canonical isomorphism, tE ( ( * ) E , , of the Lie algebra bundles is defined. Now let V be a 2-connection in 5. Then the dual connection V* is a Z-connection in (*, as follows from the equation V*(*U)

= *(Vo),

0

E

Sec [p%'J,

On the other hand, Example 2, sec. 7.16, shows that the curvatures of V* and V are related by R,*(x;h, K)

=

-[R,(x; h, k))*.

Thus the canonical isomorphism t E5 ( f * ) E , maps Rt to R,* . Finally, notice that the canonical isomorphisms, E 5 E , , f E 5 ( [ * ) E , , induce isomorphisms N

(VE*), A (V(E,)*),

and 394

Sec V [ g --%Sec V([z,)*.

5. Examples

395

Since R, is carried to R r r ,we obtain the commutative diagram

8.16. Whitney sums. Consider a second 2-bundle (9,2,) = ( N , 5, B, H ) ) over the same base B , where Z, = { T ~ ..., , r l } . Let K C GL(H)be the corresponding structure group, with Lie algebra L. T h e cross-sections ui of Z6and the cross-sections r j may be regarded as homogeneous elements of Sec[@(E @ 9) @ ( f @ 9)*]. Moreover, the projection operators, (q

and

pp:t@1+5

p,:t@q-rl,

may be regarded as elements of Sec(4 @ 9)lJ. Set &@,

= (01

,

.A ,

o m , 71

,

**a,

72,

pp ,p,);

then (5 @ q, Zto,) is a 2-bundle. It is called the Whitney sum of ( E , Z6) and (7,&J. Its structure group is G x K C GL(F @ H ) and the corresponding Lie algebra is E @ L C L F O H . T h e canonical isomorphism VE* @ VL*

restricts to an isomorphism (VE*), @ (VL*),

V(E@L)*

-

-

(V(E @ L)*), .

Proposition VI: T h e characteristic homomorphisms hE, h, , and h f e n of the Z-bundles (, q , and 8 @ 7 are related by the commutative diagram

396

VIII. Characteristic Homomorphism for Z-bundles

Proof: Observe first that the Lie algebra bundle associated with the C-bundle 5 @ 7 is given by

( E 0&I.

=

This induces an isomorphism Sec(VEE*)Oe Sec(VrlZ)

EE

0r),

z% -

Sec

W E 0rl)&

(where Oedenotes the tensor product over Y ( B ; from the definitions that the diagram,

r)).It follows directly

N -

(VE*), 0(VL*), A V(E OL)? X * @ X * ~

sec(vE3 Oe Sec(Vrl3

- lX* == -

Sec V(5 O rl)&L

,

commutes. Fix C-connections V, in 5 and V, in 7. Then V, @ V, is a Z-connection in 4 @ 7 ;its curvature is given by (cf. Example 3, sec. 7.16)

I t is now easy to check that the diagram,

A(B;r)OBA(B;r) (multiplication)

commutes. (In fact, since all maps are homomorphisms, it is sufficient to check commutativity on functions, on Sec [$ @ 1 and on 1 @ Sec 7;. Now combine the two diagrams above and pass to cohomology, to complete the proof.

Q.E.D.

Corollary: Suppose and 1) are real Z-bundles. If f E Y 0 ( E ) ,and g E Y0(L),, then f x g E Y o ( E@ L ) , , and S,@,(f

x g) = s d f ) s,(g).

8.17. Z-substructures. Assume that two Z-structures, Z and 2 are given in a vector bundle 5. Then 2 is called a 2-substructure of Z if 2 C 2. If 2 is a Z-substructure of 2, then the corresponding Lie groups

5 . Examples

e

397

e.

G C GL(F) and C GL(F) satisfy G C It follows that E C I? (where E and f? are the corresponding Lie algebras). T h e inclusion map j : E .+ induces a homomorphism

j ; : (VE*), +-

(VB*), .

Moreover, there is an obvious inclusion map j , : associated Lie algebra bundles. Proposition VII: h,: ( VE*)I

Let --f

2 be a

H(B; r)

t E-+5s between

-

the

Z-substructure of Z and let and

&: ( VB*)I

r)

denote the corresponding characteristic homomorphisms. Then the diagram, (VE*),

-

(VB*),

commutes. Proof: Choose a Z-connection, V, for (6, Z(), Then V is also a Z-connection for (5, 20. T h e corresponding curvatures R and I? satisfy

This gives the commutative diagram,

(VE*),

x*

and the proposition follows.

Sec(Vt;)

9

Q.E.D.

VIII. Characteristic Homomorphism for Cbundles

398

Corollary I: s,:

Assume that [ is a real bundle. T h e homomorphisms,

$(I?),

+H(B)

and

are related by f,(f) = sf( j * f ) , f g %(I?),

f,: %(I?),

--f

H(B),

.

Corollary 11: T h e characteristic subalgebra for (4, 2,) contains the characteristic subalgebra for (4, 2,).In particular, the characteristic subalgebra for (6, 0)([considered as a GL(F)-bundle) is contained in the where C, is an arbitrary 2-structure characteristic algebra for (t,ZE), in (. Example: Let 4 = ( M , , T , , B, F ) and = (M,, , n,,, B, H ) be vector bundles and regard these bundles as Z-bundles with Z, = 0 and Z, = 0 . Then the Whitney sum is a Z-bundle with CCmn = (pc ,p,,) (cf. Example 5 , sec. 8.3). Denote its characteristic homomorphism by

hcm

-

On the other hand, 6 0.1 may be considered as a Z-bundle with ZEOn = 0 . T h e corresponding characteristic homomorphism will be denoted by hco, : ( V(LZOH)),-+ H ( B ; I‘). Applying Proposition VII (with E = LF ,L = LH) we obtain the commutative diagram,

where j : LF @ LH ---f LFeH denotes the inclusion map. Combining this with the commutative diagram of Proposition VI, sec. 8.16, we obtain the commutative diagram

SG. c-bundles with compact carrier 8.18. Bundles with compact carrier. A vector bundle with compact carrier (or compact support) is a pair (5,a ) , where

5 = ( M , 71, B, F ) is a smooth vector bundle. (2) 0 x F 5 5 lo is a trivializing map, and (1)

a:

(3) 0 is an open subset of B such that B - 0 is compact. Any open subset U C 0 such that B - U is compact will be called a

complement for (5,a). Note that a vector bundle with compact carrier is a vector bundle which is trivial off some compact set, together with an explicit trivialization. Suppose (?, (s) is a second compactly supported bundle with base B and typical fibre H. A homomorphism v: (5,a ) --+ (7,(s) of compactly supported bundles is a bundle map y : 5 --+ 7 with the following properties: (1) T h e induced map $: B -+ is proper. (2) There is a commutative diagram,

where U and V are complements for (5,a ) and (7, (s), and y : F + H is a linear isomorphism. If v is a bundle isomorphism (respectively, a strong bundle isomorphism), we say the pairs (5,a ) and ( ~ , / 3 ) are isomorphic (respectively, strongly isomorphic). More generally, a 22-bundle with compact carrier is a triple, a), where ( 1 ) (5, Z6) is a Z-bundle and (5,a ) is a bundle with compact support and (2) the cross-sections a#ui (uiE Zc) are constant in some complement, U , for (5,a).

(5,Zc

9

A homomorphism of compactly supported Z-bundles is a bundle map which is simultaneously a homomorphism of Z-bundles and a homomorphism of compactly supported bundles. 399

400

VIII. Characteristic Homomorphism for C-bundles

Henceforth ((, Z t , a ) is a fixed compactly supported Z-bundle, where is the 2-bundle described at the start of article 3. (f, A compact 2-connection in (4, Zc,a ) is a Z-connection, V, such that, in some complement, U , for ( l ,Z t , a), ax

Q

v =6

0

ax,

where 6 is the standard connection in A( U ;F ) . T h e following proposition is obvious. Proposition VIII: (1) has compact support,

The curvature, R,of a compact Z-connection R E A@;

5E).

(2) If Q is a compact Z-connection in (7,Z, , p), and, if

v: ( t , 4, 4

+

(7,&l

9

8)

is a homomorphism of compactly supported Z-bundles, then the pullback of 9 to ( via is a compact Z-connection. 8.19. Compact characteristic homomorphism. Recall from sec. 8.1 1 that each Z-connection in (5, 2,) determines a homomorphism,

r),

y:: (VE*), -+ A ( B ;

satisfying 6 o y: = 0. It is immediate from Proposition VIII that, if V is a compact Z-connection in (4, Z t , a), then y:( V+E*),

c A@; r).

Hence 7: determines a homomorphism hi: ( V+E*),

Moreover, the diagram,

-+

Hc(B;r),

6. Z-bundles with compact carrier

401

commutes, where A, is induced by the inclusion A: A,(B) + A(B).h," is called the compact characteristic homomorphism for (5,Zt , a). I n analogy with Theorems I11 and IV of sec. 8.13, we have Theorem V: The homomorphism h: is independent of the choice of compact 2-connection. Thus it is an invariant of the compactly supported Z-bundle {[, Z6,a ) .

be ) a homomorphism Theorem VI: Let q: (4, Z f , a ) -+ (7,2,,,,l? of compactly supported Z-bundles, inducing cpB : B --+ B between the base manifolds. Then the diagram,

commutes. Proof of Theorem V: A compactly supported S1,Z r x s l a, x L ) , is given by

(5 x

6 x S'

=

( M x S1,nC x

L,

2-bundle,

B x S',F),

with Zrxxl= (4., ..., Zk).Here Gi(x, Z) = (ui(x), z),

while, since [ lo x S1= a

(5 x

x L :0 x

x E B,

z E S1,

S1)lOxs1,

-

s1=_ ('f

x Sl)(oxs'.

Choose compact Z-connections V, and V, in ([, Zc, a). T h e projection 4 x S1-+5 is a homomorphism of compactly supported Z-bundles; hence V, and V, pull back to compact Z-connections 6, and Vl in 5 x S1. Fix two points a , b E S1 and let f E Y ( F ) satisfy f ( a ) = 1 and f ( b ) = 0. Then p:

6 = f . 6,+ (1 - f) .6,

402

VIII. Characteristic Homomorphism for Z-bundles

is a compact Z-connection in inclusions,

(5 x S1, .Ztfxsl, 01 x

L).

Moreover, the

j a , j b : 5 - 5 x S', are homomorphisms of compactly supported Z-bundles, and satisfy jz o P

= V,

o

jz

and

jf

o

9 = V, o j f .

In particular if y i , 7: , y1 are the homomorphisms corresponding to V, , V, , and Q,we have the commutative diagrams,

and

as follows from Lemma 111, sec. 8.13 (ia , ib are the inclusions B+B x S1 opposite a and b). Finally, since i, and ib are properly homotopic, (i,): = (ib); (cf. sec. 5.10, volume I). It follows that the compact characteristic homomorphisms h t , hy and h,",,, , defined via V, V, , and Q,satisfy h;

=

("),"h;xsl

=

(ib)c#k;xsl = hl" .

Q.E.D. Proof of Theorem VI: This is an obvious consequence of Lemma 111, sec. 8.13, and Proposition VIII, sec. 8.18. Q.E.D.

s7. Associated principal bundles I n this article r = R. T h e notation conventions of article 3 arc continued; in particular (5,Zc) is a Z-bundle with structure group G whose Lie algebra is denoted by E . Moreover, 5 = ( M , r E B, , F ) ! Z; = (aI,...,om>,and .ZF = ( u 1 , ..., vm) is a fixed set of tensors over E corresponding to the ui under a Z-coordinate representation. 8.20. The associated principal bundle. The purpose of this and the T ,B, G) such that next section is to construct a principal bundle Pc = (P, 5 is the associated vector bundle (with respect to the natural representation of G in F , cf. sec. 5.6). A linear isomorphism 9:F % F, will be called admissible if

v(vi)= u i ( x ) ,

i

=

1,

...,m.

T h e set of admissible isomorphisms is denoted by G, action of G on the set G, by Tx'*=vxo*,

TXEGX,

. Define a right

*EG.

Now consider the disjoint union P = UrEBG, and let T :P -+ B be the natural projection. T h e right actions of G on each G, define a right action, T , of G on the set P. Finally, let (( U, , $,)} be a Z-coordinate representation for ( E , ZJ. : U, x G 3 n-l( U,) are given by Then G-equivariant bijections

They satisfy

We show now that v;' 0 pBis a diffeomorphism. I n fact, for v E F , x E U, n U, , we have

404

VIII. Characteristic Homomorphism for Z-bundles

I t follows that the map x I-+ + :(; 0 1,5~,,)(v)is a smooth map from U, n U, to F. This implies that the correspondence,

defines a smooth map of U, n Up into GL(F). E G and G is a submanifold of GL(F),this map may be Since +? ;+ 0 regarded as a smooth map into G (cf. Proposition VI, sec. 3.10, volume I). It follows that y;' 0 yo is a diffeomorphism. Now Proposition X, sec. 1.13, volume I, applies and shows that PC= (P, n, 3, G ) is a smooth fibre bundle with coordinate representation {( U, , y,)). Since the maps y a are equivariant, gt is a smooth principal bundle with principal action, T , and principal coordinate representation {( U, , y,)}. It is called the principal bundle associated with the 2-bundle ( f , 2J. Next fix a basis ej ( j= 1, ..., r ) in F . Then the admissible maps are in 1-1 correspondence with the r-tuples,

of vectors in F, . These r-tuples are called frames in F, and so G, may be identified with a set of frames in F, . For this reason, is sometimes called the frame bundle of the Z-bundle (t,Zt). Examples: 1. If 2 = O , then G = GL(F), G, is the set of all frames in F, and Pt is the frame bundle of [ (cf. Example 3, sec. 5.1). 2. If ZE= (g), g a Riemannian metric, then G = O ( F ) and G, may be identified with the set of orthonormal frames of F, via an orthonormal basis of F . Pf is called the orthonormal frame bundle. 3. Zf = (g,A ) , where g is a Riemannian metric and A is an orienting determinant function in f . In this case G = S O ( F ) and G, consists of the positive orthonormal frames in F, .

8.21, The principal map, Let 9$ = ( P , n,3, G) be the associated principal bundle of the 2-bundle ( f , ZJ, and recall that given by [ = ( M , np, 3,F ) . Consider the trivial 2-bundle (7, Z,,) 7 = ( P x F , n p ,P, F ) , Z, = ( v l , ..., vm). Since the fibre, G,, of YE consists of linear maps F -+ F, , a set map q: P x F M is given by --f

7. Associated principal bundles

405

It makes the diagram,

P - B7l, commute. Use the local coordinate representations of sec. 8.20 to show that q is smooth, and hence a bundle map. It restricts to isomorphisms in the fibres and the induced maps, q: P x 8 p . q F -+ t P , q , satisfy q(z, vi) = ai(n-z),

Z E

P, i = 1, ..., m.

Thus q: (7,C,,) -+ ( E , Cc)is a homomorphism of C-bundles. I t follows that the pull-back of (6, Z;) to P, via n-, is trivial. T h e bundle map q factors over the projection P x F -+ P X, F to yield a strong isomorphism from the associated bundle ( P x F, p , B, F ) to 6. These bundles will be identified via this isomorphism; in particular q is then identified with the principal map (cf. sec. 5.3).

,

Examples: 1. Tangent bundle: Assume the tangent bundle of B is made into a C-bundle ( T ~ CR) , (possibly by setting ZB= 0 ) . Let (P, v , B, G) be the associated principal bundle. We shall show that the manifold P is parallelizable; i.e., the tangent bundle r p is trivial. I n fact, r p = H P @ V p, where H P is some horizontal subbundle. Since H P is the pull-back of rBto P,and P is the principal bundle associated with T ~ it,follows that H , is trivial. On the other hand, Corollary I to Proposition I, sec. 6.1, shows that the vertical subbundle V p is trivial. Hence so is 7,. 2. The associated Lie algebra bundle: Since the principal map is a homomorphism of 2-bundles it determines a bundle map,

between the associated Lie algebra bundles. On the other hand, using the adjoint action of G in E we obtain a vector bundle P x E over B (with fibre E ) and a projection

,

9: P x E -+ P

X~

E.

q E factors over 4 to yield a strong isomorphism P x ,E 2 tEof bundles; this isomorphism identifies q E with the principal map 8.

VIII. Characteristic Homomorphism for 2-bundles

406

= (P, T , B, G) be the prin8.22. Associated connections. Let 9& cipal bundle associated with (t,2,). Recall that the space of F-valued forms on P which are both horizontal and equivariant is called the space of basic forms, and is denoted by A,(P;F) (cf. sec. 6.6). On the other hand, the principal map q induces a linear map

q#: A(B; 6) -+ A(P;F )

(cf. sec. 7.3). It follows from the relations, @ E A ( B ) , u ~ S e 6, c

q#(@ A u) = T * @ A q#u,

and q(z

* 7,

Y) = q(z, .(Y)),

z E p,

7

E

G , Y EF,

that Im q# C A,(P; F). Exactly the same argument as that given for Proposition 111, sec. 6.3, establishes Proposition IX: With the hypotheses and notation above, q# is a linear isomorphism of A ( B ;t)onto A,(P; F ) ,

q#: A(B; t)

N

A,(P; F ) .

Using the isomorphism q#, we shall now construct a canonical bijection between 2-connections in (t,2,)and principal connections in 9,. Recall from sec. 6.12 that a principal connection, V , in 9, determines a covariant exterior derivative, Vg

=

H*06

in A(P; F ) . Since Q B is equivariant and H" 0 V g = Q b exterior derivative restricts to an operator in A,(P; F ) . Define an operator, V, : Sec [ -+ A1(B;t),by

v, Lemma IV:

= (q")-1

0

vg

V, is a .%connection in

0

, the covariant

q#.

[.

Proof: T h e relation Vg(f

*

@) = Sf

A

Q,

+f

*

Vgp@,

f € g ( P ;F ) ,

@ E A,(P;

F)

(cf. Proposition VII, sec. 6.12) implies that V, is a linear connection. T o show that Q, is a Z-connection, consider the induced map, q: P x FpsQ-+ [ P J ,

407

7. Associated principal bundles

of tensor bundles. T h e covariant exterior derivative, V p , in and the induced connection, 0, , in EPvq satisfy q# 0 v,

=

v,

0

A(P;F p - q )

q#.

Since q#(ai) is the constant cross-section, x t+ vi , it follows that q#(v,oi)= H*(6v,)

= 0.

Hence V, is a Z-connection. Proposition

Q.E.D.

X: T h e set map,

A: {principal connections in P } -+ {Z-connectionsin

t},

defined by A: V ct V, , is a bijection.

Proof: We construct the inverse map. Let 0 be the pull-back, via 7r, of a Z-connection, V, , in 4 to the trivial Z-bundle ( ( P x F , T P , p , F ) , ZF).

According to Example 5, sec. 8.6, we can write

ti = 6 + w ,

wE

AyP;E)

(recall that E C LF is the Lie algebra of G). We show now that w is a connection form for 9.Let Z , be the fundamental field on P generated by h E E. Then, for a E Sec [,

0 = i(2Jq#(Vu) = i(Z,) ti(q%)

= Z,(q”u)

+ (i(Z,)w)(q#u).

Since q#a is equivariant. Proposition VII, sec. 3.15, yields Z,(q#o)

=

--h(q”rJ).

These relations show that i(h)w = h. Similarly, w is equivariant and thus it is a connection form. Finally, let V be the unique principal connection in B with connection form w (cf. Proposition VI, sec. 6.10). T h e correspondence V, M V defines a set map p:

{C-connections in (}

4

{principal connections in P},

and it has to be shown that A and p are inverse.

VIII. Characteristic Homomorphism for Z-bundles

408

First, fix a Z-connection V, in [ with pull-back 0 to P x F. Let H* and w be the horizontal projection and connection form associated with p(V,). Then H*w = 0, and so (H*S) 0 q# = H* 0 (6 = H*

o

+w)

o

q”

Q o q# = H* o q # o V, = q# o Q , ,

I t follows that hp(V,) = V, . On the other hand, fix a principal connection V in 9 with horizontal projection, H * , and connection form, u. Let V, = A( V ) ,and let w1 be the connection form of p(V,). Then

Applying H* to both sides, we find that H*w, q# 0

whence H*w,

= 0.

This implies that

= 0, w1 = w

and so pA( V ) = V

Q.E.D. Definition: A principal connection, V , and a Z-connection, V, , are called associated if A( V ) = V E(or, equivalently, if p(V,) = V ) . 8.23. Curvature. Recall from Example 2, sec. 8.21, that the principal map for EE = P x G E is the bundle map qE which is the restriction to P x E of 8: P x LF-+L p (cf. sec. 8.4). T h u s (cf. Lemma 111, sec. 7.7) (q#EWq#Qj)= 4”(W@))l

y E 4 B ; t,),

Qj 6 A(B;

5).

Now let V be the principal connection in 9 associated with a 2-connection, V. Their curvatures,

LI E A2(P;E )

and

R E Az(B;t

are, respectively, E-valued and [,-valued sec. 8.7). Proposition XI: related by

~),

2-forms (cf. sec. 6.14 and

T h e curvatures of associated connections are

l2 = q,“R.

409

7. Associated principal bundles

Proof: Let V, be the covariant exterior derivative with respect to the connection V. If 7 E Y ( P ;E ) is equivariant, then G(T) = v$(T)

(cf. the corollary to Proposition XII, sec. 6.14). Hence, for

0

E

Sec 6,

G(q#u) = V$(q#(T) = q#V% =

q#(R(u)) = qfR(q".),

and so SZ = qgR.

Q.E.D.

8.24. Weil and characteristic homomorphisms. Theorem VII: T h e characteristic homomorphism for $. and the Weil homomorphism for Pt coincide.

Proof: qE induces bundle maps, qE:P x V"E*

+

V*[g,

and, since q was a homomorphism of Z-bundles, the diagram, q ( P ; VE*)

Sec V[X

,

commutes (cf. sec. 8.10). Moreover X, : (WE*), -+ q ( P ; W E * ) simply identifies ( W E * ) , with the constant functions. I t induces a homoNext, consider a principal connection V in 9. morphism, y g : VE* A(P) ---f

(cf. sec. 6.17), defined via the curvature SZ of V . Extend y, to a homomorphism, 9 ( P ; VE*) -+ A ( P ) , in the obvious way, and then restrict this to a homomorphism y9:

%(P; VE*) 4A,(P)

(basic means with respect to the representation of G in VE*).

410

VIII. Characteristic Homomorphism for Z-bundles

On the other hand, in sec. 8.11 we defined (for a C-connection V,) a homomorphism y p : Sec V[f

--f

A(B).

If V and V, are associated, then it follows at once from Proposition XI, sec. 8.23, that the diagram (note that ,v’,(P; VE*) = Y B ( P ;VE*)), Y;(P; VE*)

\

A&’)

§8. Characteristic homomorphism for associated vector bundles 8.25. Representations. I n this article r = R. 9 = (P, 7 ~ B, , G) denotes a principal bundle, and E is the Lie algebra of the Lie group G. Further, @: G + GL( W ) denotes a representation of G and (vl , ..., vm) denotes a set of tensors over W such that vi E W p i * q i ,and such that each vi is left fixed by G. We let K denote the subgroup of GL( W ) consisting of those transformations which fix each vi . T h e Lie algebra of K is denoted by F. Thus @ is a homomorphism from G to K and 0': E --+ F is its derivative. @' induces the homomorphisms (cf. sec. 6.25) (@')": V E *

and

+- VF*

(a');: ( V E * ) , c

(VF*),

,

On the other hand, recall from sec. 5.6 that @ determines an associated vector bundle, E=(Px.W,p,,B,W), and a principal map q: P x W + P x W. Denote P x W simply by M . Since the vi are G-invariant, there are unique cross-sections uiE Sec e p d $ q i such that qfui = vi (i = 1, ..., m). It is evident from the construction that (t,Z,) is a Z-bundle, with Z, = (ul,..., .),u Now, in view of sec. 8.20, (t,Z,) determines an associated principal bundle, 9,= (P, , r t B, K ) , I

whose fibre, K, , at x E B consists of the admissible isomorphisms from W to W, . But by definition the linear maps, 9.:

-

-

w --+

W&) ,

z

E

P,

carry vi to ui(v(z)).Thus each qz is admissible. It follows that maps, lp,:

G, -+ K , ,

x E B,

are defined by ~ , ( z )= qz , z E G, . A simple calculation using appropriate coordinate representations shows that the maps V , together define a smooth fibre preserving map lp:

P - t P,, 41 1

412

VIII. Characteristic Homomorphism for Zbundles

which satisfies z E P, a E G ,

q(z * a ) = p(z) * @(a),

and which induces the identity map in B . Thus cp is a reduction of the structure group of 8, from K to G (cf. Example 5 , sec. 5.5). Moreover, if qz : P, x W + M is the principal map, then the diagram, Pxr P x w

commutes. Now, exactly as in sec. 8.22, we show that a principal connection, V , , in 9determines a Z-connection, 0,, in (5, 2,).I n fact, the argument of Proposition 111, sec. 6.3, shows that q x is an isomorphism from A ( B ; 5) to A,(P; W ) .On the other hand (cf. Proposition VII, sec. 6.12), the covariant exterior derivative for V p restricts to an operator, V, in A,(P; W ) .Thus we set v, = (49-1 0 v 0 q# and argue as in Lemma IV, sec. 8.22, that V, is in fact a Z-connection. Next, let V p t be the principal connection in 8, determined by V, (cf. sec. 8.22). Then it is a straightforward consequence of the definitions that dp V p = V p , dp. 0

0

Thus Lemma VI, sec. 6.25, applies, and shows that the curvatures of V , and Q p e of V p eare related by

Qp

(@')*(QP)

= V*(QP,).

Finally, use the commutative diagram above to obtain the commutative diagram,

PxF

SF

where f F is the associated Lie algebra bundle for (6, Z,). According to Proposition XI, sec. 8.23, the curvature, R, of V, satisfies (42)s( R ) = Q*,

'

8. Characteristic homomorphism for associated vector bundles

413

It follows that 52, and R are related by the equation

(@’I* P P ) = (4F)+ (W

(8.3)

Now a simple calculation (as in the proofs of Theorem 111, sec. 6.25, and Theorem VII, sec. 8.24) gives the commutative diagram,

(V$*),

r:

A(B),

6

where y, and are the homomorphisms determined respectively by 52 and R as described in secs. 6.17 and 8.11. Passing to cohomology we obtain the formula h,

=@ I/

0

(a’):.

(8.4)

It, in turn, yields the relation sp = Sp

0

(@’)*.

8.26. Examples: 1. Exterior algebra bundles: Let 5 be any real vector bundle with associated principal bundle B = (P,T , B, GL( W)). Let @ denote the natural representation of GL( W) in A W. Then

@’(d = QJ).

v ELW

9

where O(rp) is the unique derivation in A W that extends y . Since the bundle 4 = (P x cL(W) A W,p, B, A W) is simply the exterior algebra bundle A[ and since h@ = h, , formula (8.4) gives the cornmutative diagram ( v1

VIII. Characteristic Homomorphism for Z-bundles

414

2. Homogeneous spaces: Let H be a closed subgroup of a Lie group G and let @ be a representation of H in a vector space W. Consider the principal bundle 9 = (G, T,G/H, H ) and let

6

z=

(G

X H w,p,

w,

be the associated vector bundle (with respect to @). Let E and F denote the Lie algebras of G and H. Assume that there is a stable decomposition E = F OF, under the action of H. This decomposition determines a G-invariant principal connection in B (cf. sec. 6.30) whose curvature, Q, satisfies Q(e;

h, k )

=

-p([h, kl),

h, k €F1,

where p : E -+ F is the projection with kernel F , (cf. sec. 6.31). I n view of sec. 8.25 this principal connection determines a linear connection, V, , in f. Moreover, because the principal connection was G-invariant, it follows that the natural action of G on G x, W is by connection preserving bundle maps. Now consider the curvature, R,of V, . It takes values in the bundle G x, L , , and is invariant under the action of G. T h u s R is com) *L , . Moreover, pletely determined by I?(.?), and R(?)E A ~ T ~ ( G / H@ formula (8.3) of sec. 8.25 shows that R(a; (d7r)h, (d7r)k) = - @ ' ( p ( [ h , k ] ) ) ,

h, k E F l .

It follows from this that the homomorphism h, can be described as follows: Let r E ( v PLX,),. Then h,(l') is represented by the (uniquely determined) differential form Y E A?(G/ H ) that satisfies (r*Y)(e; h,

I

***,

hP)

Now suppose that G and H a r e compact and connected. Then GIH is orientable. Assume that dim G / H = 2p. Then we can form the integral

Let ACiHbe the unique G-invariant 2p-form on GIH which satisfies

8. Characteristic homomorphism for associated vector bundles

415

(cf. sec. 2.14). Since Y i s also G-invariant, it follows that

Thus to compute the integral it is sufficient to have an explicit formula for ( X * & l H ) ( 4 In sec. 9.15 such an explicit formula will be established in the case rank H = rank G. By contrast, in volume I11 it will be shown that if rank H < rank G, then h , ( r ) = 0, for every r E (VpL$), .

JG,H

Problems 1. Tensor product. (i) Let F and W be finite dimensional vector spaces. Show that ('p, 4) t+ 'p @J 4 is a representation of GL(F) x GL( W ) in F Q W. Construct a finite set, Z, of tensors over F Q W such that the subgroup of GL(F Q W )leaving Z fixed is GL(F) x GL(W ) . (ii) Represent the tensor product of two vector bundles 4 and 7 as a Z-bundle. (iii) Find an analogue of the results in sec. 8.16 and the example of sec. 8.17 for tensor products. In particular, if ( f , Zt) and (7, Z,,) are Z-bundles, define a Z-bundle ( f Q 7, Zc0,,). 2. Let f be a complex vector bundle of rank Y. Show that A r t is trivial if and only if the structure group of f can be reduced to SU(Y). Interpret this in terms of 2-bundles.

3. Extend the notion of Z-bundles to sets Zt with infinitely many elements. Show that the structure group for such a bundle is the same as the structure group for some finite substructure. 4. Let (5,2 ) be a Z-bundle with structure group G and let (? 3 G denote the group which fixes the set 2 (but not necessarily pointwise). Show that Go = GO. Construct a Z-bundle ( f , Z )such that (? is the structure group of

(5,Q.

5. Compact carrier. Let (t,Z t , a) be a Z-bundle with compact carrier and compact Z-connection. Show that (4, Zt)determines an associated principal bundle, 9, with compact carrier. Extend the results of article 7 to this case. I n particular, show that h& = h: (cf. problem 15, Chap. VI). 6. Let ( M , r, B , F ) be a smooth bundle. Define the notion of a Z-bundle over M with fibre-compact carrier. Show that such a bundle determines a canonical homomorphism h,F: ( V+E*), -+ H,(M).

7. Manifold algebras. A real manifold algebra is an algebra, A, over R such that the derivations in A form a finitely generated projective A-module, Der A. 416

417

Problems

(i) Define substitution operator, Lie derivative and exterior derivative in A,(Der A)*. Set H(A,(Der A)*, 6) = H ( A ) . (ii) Let M be a finitely generated projective A-module. Define a linear connection in M as a map,

OAM, which satisfies V(x + y ) = Vx + Vy and V( f * x) V: M

--f

(Der A)*

+

= 6f Q x f * Vx. Show that M always admits a linear connection. (iii) Define the curvature of a linear connection. Establish the Bianchi identity. (iv) Show that V induces linear connections in the modules ( @ p M * ) @ (@*M),AM, VPM and obtain their curvatures.

(v) Let aiE ( O M * ) @ ( O M ) ( i = 1, ..., m ) satisfy Vai = 0. Set Z = (a1,..., a,). Let L M act by derivations in this algebra and let E be the submodule of endomorphismsg, such that q(ai) = 0 (i = 1, ..., m). Show that 9:LM--+(Der A)* OALM restricts to a map 9 : E -+ (Der A)* QA E. (vi) Construct a characteristic homomorphism (V,E*),,,,,=, -+ H ( A ) and prove it is independent of the connection. (vii) If the homomorphism in (vi) is nonzero (with Z = a),conclude that M is not free. 8. (i) Define the odd characteristic homomorphism for 2-bundles of compact carrier over B x R which are trivial as Z-bundles (cf. problem 16, Chap. VI). Show that it coincides with the map 0 h: of problem 16, Chap. VI. (ii) Convert problem 17, Chap. VI, to a theorem on Z-bundles. 9. The ring V,(B). T h e isomorphism class of a 2-bundle (4, Z7,) over B is the collection of all vector bundles over B which are strongly isomorphic to (5,Zf).Denote the set of isomorphism classes of Z-bundles by Vect2(B). Let .Fz(B) be the free abelian group with the elements of Vect,(B) as basis. Consider the factor group generated by elements and denote this factor of the form [ f , Z4 + [T, Z,,] - [ E @ 7, ZBOn] group by Vz(B).(Note that we get two separate groups, depending on whether the field I' is [w or C.)

(i) Show that every element of V,(B) can be represented in the form [4, Z,l - [T, Z?l.

VIII. Characteristic Homomorphism for Z-bundles

418

(ii) Show that the tensor product of problem 1, (iii) makes V,(B) into a commutative ring with identity. B, induces a ring homo(iii) Show that a smooth map rp: B, morphism rp*: Vz(Bl)+- V,(B,) which depends only on the homotopy class of v. (iv) Define the rings Vg(B) corresponding to 2-bundles with compact support. Obtain an analogue of (iii) for proper maps and proper homotopies. If B is compact, prove that V:(B) = V,(B). (v) Let ( M , T , B, F ) be a smooth bundle. Define the ring V:(M) corresponding to Z-bundles with fibre-compact carrier. --f

10, The ring Y ( B ) , (i) Repeat the construction of problem 9 for ij = 5 @ 7 ordinary vector bundles to obtain a ring V ( B )in which 4 and f i j = f @ 7, where f and 7 are vector bundles representing f and i j .

+

-

(ii) Similarly obtain rings V c ( B )and V F ( M )as in problem 9, (iv) and problem 9, (v). If the field is C then V c ( B )is denoted by K(B). (iii) Show that the map ( f , Z,) I-+ f defines surjective ring homomorphisms V,(B) + V ( B ) ,Vg(B)3 V c ( B )and V:(M) -+ V F ( M ) . (iv) Show that f tt rank f determines surjective ring homo) Z, and V F ( M -+ ) Z. Construct left morphisms V ( B )+ Z, V C ( B + inverses for these homomorphisms. Denote their kernels respectively by a ( B ) ,PC(B),and a F ( M ) . 11.

Represent quaternionic vector bundles as real Z-bundles.

12.

Let

(5,ZE)be

a 2-bundle with total space M .

(i) Find necessary and sufficient conditions on a horizontal subbundle, H , , for M so that the corresponding general connection is a Z-connection. (ii) Consider the principal map q: P , x F + M . Assume that V , is a principal connection for P , with horizontal bundle HPP . Show that z E P , , y E F, are the fibres of a horizontal the spaces (dq)(z,u)(Hs(P,)), bundle for f . Show that the corresponding general connection is the Z-connection in ( associated with V p E. 13. Construct Z-bundles (t,20 and (5,Z6)with the same underlying vector bundle, 5, but such that ( f , Z, u 2,)is not a Z-bundle.

Problems

419

14. Let @ be a tensor field on a connected manifold M whose covariant derivative (with respect to some linear connection) is zero (cf. , as a Z-bundle. problem 8, Chap. VII). Interpret ( T ~ {@})

di

15. =

Let 4 be a vector bundle over B and suppose d E Sec L , satisfies 0, x E B. Show that the following are equivalent:

(i) ( E , { d } ) is a Z-bundle. (ii) T h e spaces H(F, , d,) all have the same dimension. (iii) T h e spaces ker d, are the fibres of a subbundle, 5, of H(F,), define a strong bundle map of the projections, ker d, a vector bundle K . --f

5, and 5 onto

16. Let 6 be a vector bundle and assume ( , ) is a skew-symmetric nondegenerate bilinear form in 6 (i.e., ( , ) restricts to scalar products in each fibre). Prove that (5,{( , )}) is a Z-bundle.

17.

GIH.

Extend the results of Example 2, sec. 8.26, to Z-bundles over

18. Let V, and V, be two Z-connections in a Z-bundle ([,Z,), with Lie algebra E . Let r E ( W E * ) , , and write (as in sec. 6.20) ((Q - (y;)”(r) as an explicit coboundary, giving a “vector bundle” proof. 19. Let (5,Z,) be a Z-bundle which admits a Z-coordinate repre1 elements sentation with p elements. Prove that the product of any p in he(V+E*), is zero.

+

20. Let E be a vector bundle, and assume cp E Sec L , . Suppose f ( t ) is a polynomial with constant coefficients such that f(cpz) = 0 for all x.

(i) Prove that there are unique cross-sections cps , cpN E Sec L , such that (a) each (cps)z is semisimple, (b) each ( c p N ) , is nilpotent, (c) = cps cpN , and (d) ps 0 cpN = cpN o cps . Prove that cps is 0-deformable. Establish a converse to (i). (ii)

+