Vector Bundles and K-Theory

99 Pages·2001·0.656 MB·English

by  Hatcher A.

Version1.3,July2001
Allen Hatcher
Copyright(cid:13)c2001byAllenHatcher
Paperorelectroniccopiesfornoncommercialusemaybemadefreelywithoutexplicitpermissionfromtheauthor.
Allotherrightsreserved.
Table of Contents
Chapter 1. Vector Bundles
1.1. Basic Definitions and Constructions . . . . . . . . . . . . 1
Sections3. DirectSums5. PullbackBundles5. InnerProducts7.
Subbundles8. TensorProducts9. AssociatedBundles11.
1.2. Classifying Vector Bundles . . . . . . . . . . . . . . . . . 12
TheUniversalBundle12. VectorBundlesoverSpheres16.
OrientableVectorBundles21. ACellStructureonGrassmannManifolds22.
Appendix: Paracompactness24.
Chapter 2. Complex K-Theory
2.1. The Functor K(X) . . . . . . . . . . . . . . . . . . . . . . . 28
RingStructure31. CohomologicalProperties32.
2.2. Bott Periodicity . . . . . . . . . . . . . . . . . . . . . . . . 39
ClutchingFunctions38. LinearClutchingFunctions43.
ConclusionoftheProof45.
2.3. Adams’ Hopf Invariant One Theorem . . . . . . . . . . . 48
AdamsOperations51. TheSplittingPrinciple55.
2.4. Further Calculations . . . . . . . . . . . . . . . . . . . . . 61
TheThomIsomorphism61.
Chapter 3. Characteristic Classes
3.1. Stiefel-Whitney and Chern Classes . . . . . . . . . . . . 64
AxiomsandConstruction65. CohomologyofGrassmannians70.
Applicationsof w and c 73.
1 1
3.2. The Chern Character . . . . . . . . . . . . . . . . . . . . . 74
TheJ–Homomorphism77.
3.3. Euler and Pontryagin Classes . . . . . . . . . . . . . . . . 84
TheEulerClass88. PontryaginClasses91.
1. Basic Definitions and Constructions
Vectorbundlesarespecialsortsoffiberbundleswithadditionalalgebraicstruc-
!
ture. Here is the basic definition. An n dimensional vector bundle is a map p:E B
together with a real vector space structure on p−1(cid:132)b(cid:133) for each b 2 B, such that the
following local triviality condition is satisfied: There is a cover of B by open sets
U for each of which there exists a homeomorphism h :p−1(cid:132)U (cid:133)!U (cid:2)Rn taking
(cid:11) (cid:11) (cid:11) (cid:11)
p−1(cid:132)b(cid:133) to fbg(cid:2)Rn by a vector space isomorphism for each b 2 U . Such an h is
(cid:11) (cid:11)
calledalocaltrivializationofthevectorbundle. Thespace B iscalledthebasespace,
E is the total space, and the vector spaces p−1(cid:132)b(cid:133) are the fibers. Often one abbrevi-
ates terminology by just calling the vector bundle E, letting the rest of the data be
implicit. Wecouldequallywelltake C inplaceof R asthescalarfieldhere,obtaining
thenotionofacomplexvectorbundle.
Ifwemodifythedefinitionbydroppingallreferencestovectorspacesandreplace
Rn by an arbitrary space F, then we have the definition of a fiber bundle: a map
!
p:E B such that there is a cover of B by open sets U for each of which there
(cid:11)
exists a homeomorphism h :p−1(cid:132)U (cid:133)!U (cid:2)F taking p−1(cid:132)b(cid:133) to fbg(cid:2)F for each
(cid:11) (cid:11) (cid:11)
b2U .
(cid:11)
Herearesomeexamplesofvectorbundles:
(1) The product or trivial bundle E (cid:131) B(cid:2)Rn with p the projection onto the first
factor.
(2) Ifwelet E bethequotientspaceof I(cid:2)R undertheidentifications (cid:132)0;t(cid:133)(cid:24)(cid:132)1;−t(cid:133),
! !
thentheprojection I(cid:2)R I inducesamap p:E S1 whichisa1 dimensionalvector
bundle,orlinebundle. Since E ishomeomorphictoaM¨obiusbandwithitsboundary
circledeleted,wecallthisbundletheMo¨biusbundle.
(3) The tangent bundle of the unit sphere Sn in Rn(cid:130)1, a vector bundle p:E!Sn
where E (cid:131) f(cid:132)x;v(cid:133) 2 Sn(cid:2)Rn(cid:130)1 j x ? vg and we think of v as a tangent vector to
!
Sn by translating it so that its tail is at the head of x, on Sn. The map p:E Sn
2 Chapter 1 Vector Bundles
sends (cid:132)x;v(cid:133) to x. To construct local trivializations, choose any point b 2 Sn and
let U (cid:26) Sn be the open hemisphere containing b and bounded by the hyperplane
b
through the origin orthogonal to b. Define h :p−1(cid:132)U (cid:133)!U (cid:2)p−1(cid:132)b(cid:133) (cid:25) U (cid:2)Rn by
b b b b
h (cid:132)x;v(cid:133) (cid:131) (cid:132)x;(cid:25) (cid:132)v(cid:133)(cid:133) where (cid:25) is orthogonal projection onto the tangent plane
b b b
p−1(cid:132)b(cid:133). Then h is a local trivialization since (cid:25) restricts to an isomorphism of
b b
p−1(cid:132)x(cid:133) onto p−1(cid:132)b(cid:133) foreach x2U .
b
(4) The normal bundle to Sn in Rn(cid:130)1, a line bundle p:E!Sn with E consisting of
pairs (cid:132)x;v(cid:133) 2 Sn(cid:2)Rn(cid:130)1 such that v is perpendicular to the tangent plane to Sn at
!
x,i.e., v (cid:131)tx forsome t2R. Themap p:E Sn isagaingivenby p(cid:132)x;v(cid:133)(cid:131)x. As
in the previous example, local trivializations h :p−1(cid:132)U (cid:133)!U (cid:2)R can be obtained
b b b
byorthogonalprojectionofthefibers p−1(cid:132)x(cid:133) onto p−1(cid:132)b(cid:133) for x2U .
b
!
(5) The canonical line bundle p:E RPn. Thinking of RPn as the space of lines in
Rn(cid:130)1 through the origin, E is the subspace of RPn(cid:2)Rn(cid:130)1 consisting of pairs (cid:132)‘;v(cid:133)
with v 2‘,and p(cid:132)‘;v(cid:133)(cid:131)‘. Againlocaltrivializationscanbedefinedbyorthogonal
projection. Wecouldalsotake n(cid:131)1 andgetthecanonicallinebundle E!RP1.
(6) Theorthogonalcomplement E? (cid:131)f(cid:132)‘;v(cid:133)2RPn(cid:2)Rn(cid:130)1 jv ?‘g ofthecanonical
line bundle. The projection p:E?!RPn, p(cid:132)‘;v(cid:133) (cid:131) ‘, is a vector bundle with fibers
?
the orthogonal subspaces ‘ , of dimension n. Local trivializations can be obtained
oncemorebyorthogonalprojection.
! !
Anisomorphismbetweenvectorbundlesp :E B andp :E B overthesame
1 1 2 2
base space B is a homeomorphism h:E !E taking each fiber p−1(cid:132)b(cid:133) to the cor-
1 2 1
responding fiber p−1(cid:132)b(cid:133) by a linear isomorphism. Thus an isomorphism preserves
2
allthestructureofavectorbundle,soisomorphicbundlesareoftenregardedasthe
same. Weusethenotation E (cid:25)E toindicatethat E and E areisomorphic.
1 2 1 2
Forexample,thenormalbundleof Sn in Rn(cid:130)1 isisomorphictotheproductbun-
dle Sn(cid:2)R bythemap (cid:132)x;tx(cid:133),(cid:132)x;t(cid:133). Thetangentbundleto S1 isalsoisomorphic
tothetrivialbundle S1(cid:2)R,via (cid:132)ei(cid:18);itei(cid:18)(cid:133),(cid:132)ei(cid:18);t(cid:133),for ei(cid:18) 2S1 and t2R.
Asafurtherexample,theM¨obiusbundlein(2)aboveisisomorphictothecanon-
ical line bundle over RP1 (cid:25)S1. Namely, RP1 is swept out by a line rotating through
anangleof (cid:25),sothevectorsintheselinessweepoutarectangle (cid:134)0;(cid:25)(cid:135)(cid:2)R withthe
twoends f0g(cid:2)R and f(cid:25)g(cid:2)R identified. Theidentificationis (cid:132)0;x(cid:133)(cid:24)(cid:132)(cid:25);−x(cid:133) since
rotatingavectorthroughanangleof (cid:25) producesitsnegative.
!
The zero section of a vector bundle p:E B is the union of the zero vectors in
all the fibers. This is a subspace of E which projects homeomorphically onto B by
p. Moreover, E deformationretractsontoitszerosectionviathehomotopy f (cid:132)v(cid:133)(cid:131)
t
(cid:132)1−t(cid:133)v givenbyscalarmultiplicationofvectors v 2E. Thusallvectorbundlesover
B havethesamehomotopytype.
Onecansometimesdistinguishnonisomorphicbundlesbylookingatthecomple-
!
mentofthezerosectionsinceanyvectorbundleisomorphism h:E E musttake
1 2
Basic Definitions and Constructions Section 1.1 3
thezerosectionof E ontothezerosectionof E ,hencethecomplementsofthezero
1 2
sectionsin E and E mustbehomeomorphic. Forexample,theM¨obiusbundleisnot
1 2
isomorphic to the product bundle S1(cid:2)R since the complement of the zero section
in the M¨obius bundle is connected while for the product bundle the complement of
thezerosectionisnotconnected. Thismethodfordistinguishingvectorbundlescan
alsobeusedwithmorerefinedtopologicalinvariantssuchas H inplaceof H .
n 0
We shall denote the set of isomorphism classes of n dimensional real vector
bundles over B by Vectn(cid:132)B(cid:133), and its complex analogue by VectnC(cid:132)B(cid:133). For those who
worry about set theory, we are using the term ‘set’ here in a naive sense. It follows
fromTheorem1.8laterinthechapterthat Vectn(cid:132)B(cid:133) and VectnC(cid:132)B(cid:133) areindeedsetsin
thestrictsensewhen B isparacompact.
Forexample, Vect1(cid:132)S1(cid:133) containsexactlytwoelements,theM¨obiusbundleandthe
product bundle. This will be a rather trivial application of later theory, but it might
beaninterestingexercisetoproveitnowdirectlyfromthedefinitions.
Sections
! !
Asectionofabundlep:E B isamaps:B E suchthatps (cid:131)11,orequivalently,
s(cid:132)b(cid:133) 2 p−1(cid:132)b(cid:133) for all b 2 B. We have already mentioned the zero section, which
is the section whose values are all zero. At the other extreme would be a section
whosevaluesareallnonzero. Notallvectorbundleshavesuchanonvanishingsection.
Considerforexamplethetangentbundleto Sn. Hereasectionisjustatangentvector
fieldto Sn. Oneofthestandardfirstapplicationsofhomologytheoryisthetheorem
that Sn has a nonvanishing vector field iff n is odd. From this it follows that the
tangentbundleof Sn isnotisomorphictothetrivialbundleif n isevenandnonzero,
since the trivial bundle obviously has a nonvanishing section, and an isomorphism
betweenvectorbundlestakesnonvanishingsectionstononvanishingsections.
!
Infact,an n dimensionalbundle p:E B isisomorphictothetrivialbundleiff
it has n sections s ;(cid:1)(cid:1)(cid:1);s such that s (cid:132)b(cid:133);(cid:1)(cid:1)(cid:1);s (cid:132)b(cid:133) are linearly independent in
1 n 1 n
each fiber p−1(cid:132)b(cid:133). For if one has such sections s , the map h:B(cid:2)Rn!E given by
P i
h(cid:132)b;t ;(cid:1)(cid:1)(cid:1);t (cid:133)(cid:131) t s (cid:132)b(cid:133) isalinearisomorphismineachfiber,andiscontinuous,
1 n i i i
ascanbeverifiedbycomposingwithalocaltrivialization p−1(cid:132)U(cid:133)!U(cid:2)Rn. Hence h
isanisomorphismbythefollowingusefultechnicalresult:
L !
emma 1.1. A continuous map h:E E between vector bundles over the same
1 2
base space B is an isomorphism if it takes each fiber p−1(cid:132)b(cid:133) to the corresponding
1
fiber p−1(cid:132)b(cid:133) byalinearisomorphism.
2
Proof: Thehypothesisimpliesthat h isone-to-oneandonto. Whatmustbechecked
isthat h−1 iscontinuous. Thisisalocalquestion, sowemayrestricttoanopenset
U (cid:26)B overwhich E and E aretrivial. Composingwithlocaltrivializationsreduces
1 2
!
tothecaseofanisomorphism h:U(cid:2)Rn U(cid:2)Rn oftheform h(cid:132)x;v(cid:133)(cid:131)(cid:132)x;g (cid:132)v(cid:133)(cid:133).
x
4 Chapter 1 Vector Bundles
Here g is an element of the group GL (cid:132)R(cid:133) of invertible linear transformations of
x n
Rn whichdependscontinuouslyon x. Thismeansthatif g isregardedasan n(cid:2)n
x
matrix,itsn2 entriesdependcontinuouslyonx. Theinversematrixg−1 alsodepends
x
continuously on x since its entries can be expressed algebraically in terms of the
entries of g , namely, g−1 is 1=(cid:132)detg (cid:133) times the classical adjoint matrix of g .
x x x x
Therefore h−1(cid:132)x;v(cid:133)(cid:131)(cid:132)x;g−1(cid:132)v(cid:133)(cid:133) iscontinuous. tu
x
As an example, the tangent bundle to S1 is trivial because it has the section
(cid:132)x ;x (cid:133) , (cid:132)−x ;x (cid:133) for (cid:132)x ;x (cid:133) 2 S1. In terms of complex numbers, if we set
1 2 2 1 1 2
z(cid:131)x (cid:130)ix thenthissectionis z,iz since iz(cid:131)−x (cid:130)ix .
1 2 2 1
There is an analogous construction using quaternions instead of complex num-
bers. Quaternionshavetheform z(cid:131)x (cid:130)ix (cid:130)jx (cid:130)kx ,andformadivisionalgebra
1 2 3 4
H via the multiplication rules i2 (cid:131) j2 (cid:131) k2 (cid:131) −1, ij (cid:131) k, jk (cid:131) i, ki (cid:131) j, ji (cid:131) −k,
kj (cid:131)−i, and ik(cid:131)−j. If we identify H with R4 via the coordinates (cid:132)x ;x ;x ;x (cid:133),
1 2 3 4
then the unit sphere is S3 and we can define three sections of its tangent bundle by
theformulas
z,iz or (cid:132)x ;x ;x ;x (cid:133),(cid:132)−x ;x ;−x ;x (cid:133)
1 2 3 4 2 1 4 3
z,jz or (cid:132)x ;x ;x ;x (cid:133),(cid:132)−x ;x ;x ;−x (cid:133)
1 2 3 4 3 4 1 2
z,kz or (cid:132)x ;x ;x ;x (cid:133),(cid:132)−x ;−x ;x ;x (cid:133)
1 2 3 4 4 3 2 1
Itiseasytocheckthatthethreevectorsinthelastcolumnareorthogonaltoeachother
and to (cid:132)x ;x ;x ;x (cid:133), so we have three linearly independent nonvanishing tangent
1 2 3 4
vectorfieldson S3,andhencethetangentbundleto S3 istrivial.
Theunderlyingreasonwhythisworksisthatquaternionmultiplicationsatisfies
jzwj(cid:131)jzjjwj,where j(cid:1)j istheusualnormofvectorsin R4. Thusmultiplicationbya
quaternionintheunitsphere S3 isanisometryof H. Thequaternions 1;i;j;k form
thestandardorthonormalbasisfor R4,sowhenwemultiplythembyanarbitraryunit
quaternion z2S3 wegetaneworthonormalbasis z;iz;jz;kz.
The same constructions work for the Cayley octonions, a division algebra struc-
ture on R8. Thinking of R8 as H(cid:2)H, multiplication of octonions is defined by
(cid:132)z ;z (cid:133)(cid:132)w ;w (cid:133)(cid:131)(cid:132)z w −w z ;z w (cid:130)w z (cid:133) andsatisfiesthekeyproperty jzwj(cid:131)
1 2 1 2 1 1 2 2 2 1 2 1
jzjjwj. This leads to the construction of seven orthogonal tangent vector fields on
the unit sphere S7, so the tangent bundle to S7 is also trivial. As we shall show in
x2.3,theonlysphereswithtrivialtangentbundleare S1, S3,and S7.
One final general remark before continuing with our next topic: Another way of
characterizingthetrivialbundle E (cid:25)B(cid:2)Rn istosaythatthereisacontinuousprojec-
!
tionmap E Rn whichisalinearisomorphismoneachfiber,sincesuchaprojection
!
togetherwiththebundleprojection E B givesanisomorphism E (cid:25)B(cid:2)Rn.
Basic Definitions and Constructions Section 1.1 5
Direct Sums
Asapreliminarytodefiningadirectsumoperationonvectorbundles, wemake
twosimpleobservations:
(a) Given a vector bundle p:E!B and a subspace A (cid:26) B, then p:p−1(cid:132)A(cid:133)!A is
clearlyavectorbundle. WecallthistherestrictionofEover A .
! ! !
(b) Given vector bundles p :E B and p :E B , then p (cid:2)p :E (cid:2)E B (cid:2)B
1 1 1 2 2 2 1 2 1 2 1 2
is also a vector bundle, with fibers the products p−1(cid:132)b (cid:133)(cid:2)p−1(cid:132)b (cid:133). For if we have
1 1 2 2
local trivializations h :p−1(cid:132)U (cid:133)!U (cid:2)Rn and h :p−1(cid:132)U (cid:133)!U (cid:2)Rm for E and
(cid:11) 1 (cid:11) (cid:11) (cid:12) 2 (cid:12) (cid:12) 1
E ,then h (cid:2)h isalocaltrivializationfor E (cid:2)E .
2 (cid:11) (cid:12) 1 2
! !
Now suppose we are given two vector bundles p :E B and p :E B over
1 1 2 2
thesamebasespace B. Therestrictionoftheproduct E (cid:2)E overthediagonal B (cid:131)
1 2
f(cid:132)b;b(cid:133)2B(cid:2)Bg isthenavectorbundle,calledthedirectsum E (cid:8)E !B. Thus
1 2
E (cid:8)E (cid:131)f(cid:132)v ;v (cid:133)2E (cid:2)E jp (cid:132)v (cid:133)(cid:131)p (cid:132)v (cid:133)g
1 2 1 2 1 2 1 1 2 2
The fiber of E (cid:8)E over a point b 2 B is the product, or direct sum, of the vector
1 2
spaces p−1(cid:132)b(cid:133) and p−1(cid:132)b(cid:133).
1 2
The direct sum of two trivial bundles is again a trivial bundle, clearly, but the
direct sum of nontrivial bundles can also be trivial. For example, the direct sum of
the tangent and normal bundles to Sn in Rn(cid:130)1 is the trivial bundle Sn(cid:2)Rn(cid:130)1 since
elementsofthedirectsumaretriples (cid:132)x;v;tx(cid:133)2Sn(cid:2)Rn(cid:130)1(cid:2)Rn(cid:130)1 with x ?v,and
themap (cid:132)x;v;tx(cid:133),(cid:132)x;v(cid:130)tx(cid:133) givesanisomorphismofthedirectsumbundlewith
Sn(cid:2)Rn(cid:130)1. Sothetangentbundleto Sn isstablytrivial: itbecomestrivialaftertaking
thedirectsumwithatrivialbundle.
Asanotherexample,thedirectsum E(cid:8)E? ofthecanonicallinebundle E!RPn
with its orthogonal complement, defined in example (6) above, is isomorphic to the
trivial bundle RPn(cid:2)Rn(cid:130)1 via the map (cid:132)‘;v;w(cid:133),(cid:132)‘;v(cid:130)w(cid:133) for v 2‘ and w ?‘.
Specializing to the case n(cid:131)1, both E and E? are isomorphic to the M¨obius bundle
over RP1 (cid:131)S1,sothedirectsumoftheM¨obiusbundlewithitselfisthetrivialbundle.
Thisisjustsayingthatifonetakesaslab I(cid:2)R2 andgluesthetwofaces f0g(cid:2)R2 and
f1g(cid:2)R2 to each other via a 180 degree rotation of R2, the resulting vector bundle
over S1 is the same as if the gluing were by the identity map. In effect, one can
gradually decrease the angle of rotation of the gluing map from 180 degrees to 0
withoutchangingthevectorbundle.
Pullback Bundles
!
Next we describe a procedure for using a map f:A B to transform vector
!
bundles over B into vector bundles over A. Given a vector bundle p:E B, let
6 Chapter 1 Vector Bundles
f(cid:3)(cid:132)E(cid:133)(cid:131)f(cid:132)a;v(cid:133)2A(cid:2)E jf(cid:132)a(cid:133)(cid:131)p(cid:132)v(cid:133)g. Thissubspaceof A(cid:2)E fitsintothecommu-
tativediagramattherightwhere(cid:25)(cid:132)a;v(cid:133)(cid:131)aandfe(cid:132)a;v(cid:133)(cid:131)v. Itis »
(cid:3) ! f⁄(E)¡¡¡¡f¡!E
nothardtoseethat(cid:25):f (cid:132)E(cid:133) Aisalsoavectorbundlewithfibers
…¡¡! ¡¡!p
of the same dimension as in E. For example, we could say that ¡¡¡¡f¡!
f(cid:3)(cid:132)E(cid:133) is the restriction of the vector bundle 11(cid:2)p:A(cid:2)E!A(cid:2)B A B
overthegraphof f, f(cid:132)a;f(cid:132)a(cid:133)(cid:133)2A(cid:2)Bg,whichweidentifywith A viatheprojection
, (cid:3)
(cid:132)a;f(cid:132)a(cid:133)(cid:133) a. Thevectorbundle f (cid:132)E(cid:133) iscalledthepullback orinduced bundle.
As a trivial example, if f is the inclusion of a subspace A (cid:26) B, then f(cid:3)(cid:132)E(cid:133) is
isomorphic to the restriction p−1(cid:132)A(cid:133) via the map (cid:132)a;v(cid:133),v, since the condition
f(cid:132)a(cid:133) (cid:131) p(cid:132)v(cid:133) just says that v 2 p−1(cid:132)a(cid:133). So restriction over subspaces is a special
caseofpullback.
Aninterestingexamplewhichissmallenoughtobevisualizedcompletelyisthe
! !
pullback of the M¨obius bundle E S1 by the two-to-one covering map f:S1 S1,
f(cid:132)z(cid:133) (cid:131) z2. In this case the pullback f(cid:3)(cid:132)E(cid:133) is a two-sheeted covering space of E
whichcanbethoughtofasacoatofpaintappliedto‘bothsides’oftheM¨obiusbundle.
(cid:3)
Since E hasonehalf-twist, f (cid:132)E(cid:133) hastwohalf-twists,henceisthetrivialbundle. More
generally,if E isthepullbackoftheM¨obiusbundlebythemap z,zn,then E is
n n
thetrivialbundlefor n evenandtheM¨obiusbundlefor n odd.
Someelementarypropertiesofpullbacks,whoseproofsareone-minuteexercises
indefinition-chasing,are:
(i) (cid:132)fg(cid:133)(cid:3)(cid:132)E(cid:133)(cid:25)g(cid:3)(cid:132)f(cid:3)(cid:132)E(cid:133)(cid:133).
(ii) If E (cid:25)E then f(cid:3)(cid:132)E (cid:133)(cid:25)f(cid:3)(cid:132)E (cid:133).
1 2 1 2
(iii) f(cid:3)(cid:132)E (cid:8)E (cid:133)(cid:25)f(cid:3)(cid:132)E (cid:133)(cid:8)f(cid:3)(cid:132)E (cid:133).
1 2 1 2
Nowwecometoourfirstimportantresult:
T ! !
heorem 1.2. Given a vector bundle p:E B and homotopic maps f ;f :A B,
0 1
(cid:3) (cid:3)
thentheinducedbundles f (cid:132)E(cid:133) and f (cid:132)E(cid:133) areisomorphicif A isparacompact.
0 1
Allthespacesoneordinarilyencountersinalgebraicandgeometrictopologyare
paracompact,forexamplecompactHausdorffspacesandCWcomplexes;seetheAp-
pendixtothischapterformoreinformationaboutthis.
Proof: Let F:A(cid:2)I!B beahomotopyfrom f to f . Therestrictionsof F(cid:3)(cid:132)E(cid:133) over
0 1
A(cid:2)f0g and A(cid:2)f1g arethen f(cid:3)(cid:132)E(cid:133) and f(cid:3)(cid:132)E(cid:133). Sothetheoremwillfollowfrom: tu
0 1
Proposition 1.3. The restrictions of a vector bundle E!X(cid:2)I over X(cid:2)f0g and
X(cid:2)f1g areisomorphicif X isparacompact.
Proof: Weneedtwopreliminaryfacts:
!
(1) A vector bundle p:E X(cid:2)(cid:134)a;b(cid:135) is trivial if its restrictions over X(cid:2)(cid:134)a;c(cid:135) and
X(cid:2)(cid:134)c;b(cid:135) are both trivial for some c 2 (cid:132)a;b(cid:133). To see this, let these restrictions
be E (cid:131) p−1(cid:132)X(cid:2)(cid:134)a;c(cid:135)(cid:133) and E (cid:131) p−1(cid:132)X(cid:2)(cid:134)c;b(cid:135)(cid:133), and let h :E !X(cid:2)(cid:134)a;c(cid:135)(cid:2)Rn
1 2 1 1
Basic Definitions and Constructions Section 1.1 7
!
and h :E X(cid:2)(cid:134)c;b(cid:135)(cid:2)Rn beisomorphisms. Theseisomorphismsmaynotagreeon
2 2
p−1(cid:132)X(cid:2)fcg(cid:133),buttheycanbemadetoagreebyreplacing h byitscompositionwith
2
!
theisomorphism X(cid:2)(cid:134)c;b(cid:135)(cid:2)Rn X(cid:2)(cid:134)c;b(cid:135)(cid:2)Rn whichoneachslice X(cid:2)fxg(cid:2)Rn is
givenby h h−1:X(cid:2)fcg(cid:2)Rn!X(cid:2)fcg(cid:2)Rn. Once h and h agreeon E \E ,they
1 2 1 2 1 2
defineatrivializationof E.
!
(2) Foravectorbundle p:E X(cid:2)I,thereexistsanopencover fU g of X sothateach
(cid:11)
restriction p−1(cid:132)U (cid:2)I(cid:133)!U (cid:2)I istrivial. Thisisbecauseforeach x2X wecanfind
(cid:11) (cid:11)
openneighborhoods U ;(cid:1)(cid:1)(cid:1);U in X andapartition 0(cid:131)t <t <(cid:1)(cid:1)(cid:1)<t (cid:131)1 of
x;1 x;k 0 1 k
(cid:134)0;1(cid:135) suchthatthebundleistrivialover Ux;i(cid:2)(cid:134)ti−1;ti(cid:135),usingcompactnessof (cid:134)0;1(cid:135).
Thenby(1)thebundleistrivialover U (cid:2)I where U (cid:131)U \(cid:1)(cid:1)(cid:1)\U .
(cid:11) (cid:11) x;1 x;k
Nowweprovetheproposition. By(2),wecanchooseanopencover fU g of X so
(cid:11)
that E istrivialovereach U (cid:2)I. Lemma1.19intheAppendixtothischapterasserts
(cid:11)
that there is a countable cover fVkgk(cid:21)1 of X and a partition of unity f’kg with ’k
supportedin V ,suchthateach V isadisjointunionofopensetseachcontainedin
k k
some U . Thismeansthat E istrivialovereach V (cid:2)I.
(cid:11) k
For k (cid:21) 0, let (cid:131) ’ (cid:130)(cid:1)(cid:1)(cid:1)(cid:130)’ , with (cid:131) 0. Let X be the graph of ,
k 1 k 0 k k
!
so X (cid:131) f(cid:132)x; (cid:132)x(cid:133)(cid:133) 2 X(cid:2)Ig, and let p :E X be the restriction of the bun-
k k k k k
dle E over X . Choosing a trivialization of E over V (cid:2)I, the natural projection
k k
! !
homeomorphism Xk Xk−1 liftstoanisomorphism hk:Ek Ek−1 whichistheiden-
tity outside p−1(cid:132)V (cid:133). The infinite composition h (cid:131) h h (cid:1)(cid:1)(cid:1) is then a well-defined
k k 1 2
isomorphism from the restriction of E over X(cid:2)f0g to the restriction over X(cid:2)f1g
sinceneareachpoint x 2X onlyfinitelymany ’ ’sarenonzero,whichimpliesthat
i
forlargeenough k, h (cid:131)11 overaneighborhoodof x. tu
k
C !
orollary1.4. Ahomotopyequivalence f:A B ofparacompactspacesinducesa
bijection f(cid:3):Vectn(cid:132)B(cid:133)!Vectn(cid:132)A(cid:133). In particular, every vector bundle over a con-
tractibleparacompactbaseistrivial.
Proof: If g is a homotopy inverse of f then we have f(cid:3)g(cid:3) (cid:131) 11(cid:3) (cid:131) 11 and g(cid:3)f(cid:3) (cid:131)
11(cid:3) (cid:131)11. tu
Theorem 1.2 holds for fiber bundles as well as vector bundles, with the same
proof.
Inner Products
An inner product on a vector bundle p:E!B is a map h;i:E(cid:8)E!R which
restricts in each fiber to an inner product, i.e., a positive definite symmetric bilinear
form.
P !
roposition 1.5. An inner product exists for a vector bundle p:E B if B is para-
compact.
8 Chapter 1 Vector Bundles
!
Proof: An inner product for p:E B can be constructed by first using local trivial-
izations h :p−1(cid:132)U (cid:133)!U (cid:2)Rn,topullbackthestandardinnerproductin Rn toan
(cid:11) (cid:11) (cid:11) P
innerproduct h(cid:1);(cid:1)i on p−1(cid:132)U (cid:133),thensetting hv;wi(cid:131) ’ p(cid:132)v(cid:133)hv;wi where
(cid:11) (cid:11) (cid:12) (cid:12) (cid:11)(cid:132)(cid:12)(cid:133)
f’ g isapartitionofunitywiththesupportof ’ containedin U . tu
(cid:12) (cid:12) (cid:11)(cid:132)(cid:12)(cid:133)
InthecaseofcomplexvectorbundlesonecanconstructHermitianinnerproducts
inthesameway.
Having an inner product on a vector bundle E, lengths of vectors are defined,
!
and so we can speak of the associated unit sphere bundle S(cid:132)E(cid:133) B, a fiber bundle
with fibers the spheres consisting of all vectors of length 1 in fibers of E. Similarly
!
there is a disk bundle D(cid:132)E(cid:133) B with fibers the disks of vectors of length less than
or equal to 1. It is possible to describe S(cid:132)E(cid:133) without reference to an inner product,
as the quotient of the complement of the zero section in E obtained by identifying
each nonzero vector with all positive scalar multiples of itself. It follows that D(cid:132)E(cid:133)
canalsobedefinedwithoutinvokingametric,namelyasthemappingcylinderofthe
!
projection S(cid:132)E(cid:133) B.
!
Thecanonicallinebundle E RPn hasasitsunitspherebundle S(cid:132)E(cid:133) thespace
of unit vectors in lines through the origin in Rn(cid:130)1. Since each unit vector uniquely
determines the line containing it, S(cid:132)E(cid:133) is the same as the space of unit vectors in
Rn(cid:130)1,i.e., Sn. Itfollowsthatcanonicallinebundleisnontrivialif n>0 sinceforthe
trivialbundle RPn(cid:2)R theunitspherebundleis RPn(cid:2)S0,whichisnothomeomorphic
to Sn.
!
Similarly,inthecomplexcasethecanonicallinebundle E CPn has S(cid:132)E(cid:133) equal
totheunitsphere S2n(cid:130)1 in Cn(cid:130)1. Againif n>0 thisisnothomeomorphictotheunit
spherebundleofthetrivialbundle,whichis CPn(cid:2)S1,sothecanonicallinebundleis
nontrivial.
Subbundles
!
Avectorsubbundleofavectorbundle p:E B hasthenaturaldefinition: asub-
spaceE (cid:26)E intersectingeachfiberofE inavectorsubspace,suchthattherestriction
0
!
p:E B isavectorbundle.
0
Proposition1.6. If E!B isavectorbundleoveraparacompactbase B and E (cid:26)E
0
isavectorsubbundle,thenthereisavectorsubbundle E? (cid:26)E suchthat E (cid:8)E? (cid:25)E.
0 0 0
Proof: With respect to a chosen inner product on E, let E? be the subspace of E
0
which in each fiber consists of all vectors orthogonal to vectors in E . We claim
0
?! (cid:8) ?
that the natural projection E B is a vector bundle. If this is so, then E E is
0 0 0
isomorphicto E viathemap (cid:132)v;w(cid:133),v(cid:130)w,usingLemma1.1.
?
To see that E satisfies the local triviality condition for a vector bundle, note
0
first that we may assume E is the product B(cid:2)Rn since the question is local in B.