[ Pobierz całość w formacie PDF ]
and 3 =123 . Since each i with id"n is nonzero in Z2[1,,n], we have
an n dimensional bundle whose first n Stiefel-Whitney classes are all nonzero.
The same reasoning applies in the complex case to show that then fold Cartesian
product of the canonical line bundle over CP " has its first n Chern classes nonzero.
In this example we see that the wi s and ci s can be identified with elementary
symmetric functions, and in fact this can be done in general using the splitting princi-
ple. Given an n dimensional vector bundle E B we know that the pullback toF(E)
!
Stiefel-Whitney and Chern Classes Section 3.1 69
splits as a sum L1 " "Ln F(E). Letting i = w1(Li), we see that w(E) pulls
!
back to w(L1 " "Ln)=(1 +1)(1 +n)= 1 +1 + +n, so wi(E) pulls
back toi. Thus we have embeddedH"(B; Z2)in a larger ringH"(F(E); Z2)such that
wi(E)becomes theith elementary symmetric polynomial in the elements1,,n
of H"(F(E); Z2).
Besides the evident formal similarity between Stiefel-Whitney and Chern classes
there is also a direct relation:
Proposition 3.8. Regarding an n dimensional complex vector bundle E B as a
!
2n dimensional real vector bundle, then w2i+1(E)= 0 and w2i(E) is the image of
ci(E) under the coefficient homomorphism H2i(B; Z) H2i(B; Z2).
!
For example, since the canonical complex line bundle over CP " hasc1 a generator
ofH2(CP "; Z), the same is true for its restriction overS2 = CP1 , so by the proposition
this 2 dimensional real vector bundle E S2 has w2(E)`" 0.
!
Proof: The bundleEhas two projectivizations RP(E)and CP(E), consisting of all the
real and all the complex lines in fibers ofE, respectively. There is a natural projection
p: RP(E) CP(E)sending each real line to the complex line containing it, since a real
!
line is all the real scalar multiples of any nonzero vector in it and a complex line is all
the complex scalar multiples. This projection p fits into a commutative diagram
R
( )
RP2n 1---! --P-g- RP "
--- RP E --(-)!
--- -- -
p
C
( )
CPn 1 ---! --P-g- CP "
--- CP E --(-)!
--- -- -
where the left column is the restriction of p to a fiber of E and the maps RP(g)
and CP(g) are obtained by projectivizing, over R and C, a map g:E C " which
!
is a C linear injection on fibers. It is easy to see that all three vertical maps in
this diagram are fiber bundles with fiber RP1 , the real lines in a complex line. The
Leray-Hirsch theorem applies to the bundle RP " CP " , with Z2 coefficients, so if
!
is the standard generator of H2(CP "; Z), the Z2 reduction "H2(CP "; Z2) pulls
back to a generator of H2(RP "; Z2), namely the square 2 of the generator "
H1(RP "; Z2). Hence the Z2 reduction xC(E) = CP(g)"() " H2(CP(E); Z2) of the
basic class xC(E)= CP(g)"() pulls back to the square of the basic class xR(E)=
RP(g)"()"H1(RP(E); Z2). Consequently the Z2 reduction of the defining relation
for the Chern classes of E, which is xC(E)n +c1(E)xC(E)n-1 + +cn(E) 1 = 0,
pulls back to the relationxR(E)2n+c1(E)xR(E)2n-2 ++cn(E) 1 = 0, which is the
defining relation for the Stiefel-Whitney classes of E. This means that w2i+1(E)= 0
and w2i(E)=ci(E).
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70 Chapter 3 Characteristic Classes
Cohomology of Grassmannians
From Example 3.7 and naturality it follows that the universal bundle En Gn
!
must also have all its Stiefel-Whitney classes w1(En),,wn(En) nonzero. In fact a
much stronger statement is true. Let f:(RP ")n Gn be the classifying map for the
!
n fold Cartesion product (E1)n of the canonical line bundle E1 , and for notational
simplicity let wi =wi(En). Then the composition
f"
Z2[w1,,wn] H"(Gn; Z2) H" (RP ")n; Z2 H" Z2[1,,n]
!
!
sendswi toi, theith elementric symmetric polynomial. It is a classical algebraic re-
sult that the polynomials i are algebraically independent in Z2[1,,n]. Proofs
of this can be found in [van der Waerden, 26] or [Lang, p. 134] for example. Thus
the composition Z2[w1,,wn] Z2[1,,n] is injective, hence also the map
!
Z2[w1,,wn] H"(Gn; Z2). In other words, the classes wi(En) generate a poly-
!
nomial subalgebra Z2[w1,,wn]"H"(Gn; Z2). This subalgebra is in fact equal to
H"(Gn; Z2), and the corresponding statement for Chern classes holds as well:
Theorem 3.9. H"(Gn; Z2) is the polynomial ring Z2[w1,,wn] on the Stiefel-
Whitney classes wi = wi(En) of the universal bundle En Gn. Similarly, in the
!
complex caseH"(Gn(C "); Z)H" Z[c1,,cn]whereci =ci(En(C "))for the univer-
sal bundle En(C ") Gn(C ").
!
The proof we give here for this basic result will be a fairly quick application of the
CW structure on Gn constructed at the end of 1.2. A different proof will be given
in 3.3 where we also compute the cohomology of Gn with Z coefficients, which is
somewhat more subtle.
Proof: Consider a map f:(RP ")n Gn which pulls En back to the bundle (E1)n
!
considered above. We have noted that the image of f" contains the symmetric poly-
nomials in Z2[1,,n]H"H"((RP ")n; Z2). The opposite inclusion holds as well,
since if:(RP ")n (RP ")n is an arbitrary permutation of the factors, then pulls
!
(E1)n back to itself, sof f, which means thatf" ="f" , so the image off" is
invariant under" :H"((RP ")n; Z2) H"((RP ")n; Z2), but the latter map is just the
!
same permutation of the variables i.
To finish the proof in the real case it remains to see thatf" is injective. It suffices
to find a CW structure on Gn in which the r cells are in one-to-one correspondence
r
r
with monomialsw11 wnn of dimensionr =r1 + 2r2 ++nrn, since the number
of r cells in a CW complex X is an upper bound on the dimension of Hr(X; Z2) as a
Z2 vector space, and a surjective linear map between finite-dimensional vector spaces
is injective if the dimension of the domain is not greater than the dimension of the
range.
r
r
Monomialsw11 wnn of dimensionr correspond ton tuples(r1,,rn)with
r =r1 + 2r2 ++nrn. Such n tuples in turn correspond to partitions of r into at
Stiefel-Whitney and Chern Classes Section 3.1 71
most n integers, via the correspondence
(r1,,rn) rn d"rn+rn-1 d" d"rn+rn-1 ++r1.
!
!
Such a partition becomes the sequence1 - 1 d"2 - 2 d" d"n-n, corresponding
to the strictly increasing sequence 0
we have:
(r1,r2,r3) (1 - 1,2 -2,3 -3) (1,2,3) dimension
1 0 0 0 0 0 0 1 2 3 0
w1 1 0 0 0 0 1 1 2 4 1
w2 0 1 0 0 1 1 1 3 4 2
2
w1 2 0 0 0 0 2 1 2 5 2
w3 0 0 1 1 1 1 2 3 4 3
w1w2 1 1 0 0 1 2 1 3 5 3
3
w1 3 0 0 0 0 3 1 2 6 3
The cell structure on Gn constructed in 1.2 has one cell of dimension (1 - 1)+
(2 - 2)+ +(n -n) for each increasing sequence 0
we are done in the real case.
The complex case is entirely similar, keeping in mind that ci has dimension 2i [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl rafalstec.xlx.pl
and 3 =123 . Since each i with id"n is nonzero in Z2[1,,n], we have
an n dimensional bundle whose first n Stiefel-Whitney classes are all nonzero.
The same reasoning applies in the complex case to show that then fold Cartesian
product of the canonical line bundle over CP " has its first n Chern classes nonzero.
In this example we see that the wi s and ci s can be identified with elementary
symmetric functions, and in fact this can be done in general using the splitting princi-
ple. Given an n dimensional vector bundle E B we know that the pullback toF(E)
!
Stiefel-Whitney and Chern Classes Section 3.1 69
splits as a sum L1 " "Ln F(E). Letting i = w1(Li), we see that w(E) pulls
!
back to w(L1 " "Ln)=(1 +1)(1 +n)= 1 +1 + +n, so wi(E) pulls
back toi. Thus we have embeddedH"(B; Z2)in a larger ringH"(F(E); Z2)such that
wi(E)becomes theith elementary symmetric polynomial in the elements1,,n
of H"(F(E); Z2).
Besides the evident formal similarity between Stiefel-Whitney and Chern classes
there is also a direct relation:
Proposition 3.8. Regarding an n dimensional complex vector bundle E B as a
!
2n dimensional real vector bundle, then w2i+1(E)= 0 and w2i(E) is the image of
ci(E) under the coefficient homomorphism H2i(B; Z) H2i(B; Z2).
!
For example, since the canonical complex line bundle over CP " hasc1 a generator
ofH2(CP "; Z), the same is true for its restriction overS2 = CP1 , so by the proposition
this 2 dimensional real vector bundle E S2 has w2(E)`" 0.
!
Proof: The bundleEhas two projectivizations RP(E)and CP(E), consisting of all the
real and all the complex lines in fibers ofE, respectively. There is a natural projection
p: RP(E) CP(E)sending each real line to the complex line containing it, since a real
!
line is all the real scalar multiples of any nonzero vector in it and a complex line is all
the complex scalar multiples. This projection p fits into a commutative diagram
R
( )
RP2n 1---! --P-g- RP "
--- RP E --(-)!
--- -- -
p
C
( )
CPn 1 ---! --P-g- CP "
--- CP E --(-)!
--- -- -
where the left column is the restriction of p to a fiber of E and the maps RP(g)
and CP(g) are obtained by projectivizing, over R and C, a map g:E C " which
!
is a C linear injection on fibers. It is easy to see that all three vertical maps in
this diagram are fiber bundles with fiber RP1 , the real lines in a complex line. The
Leray-Hirsch theorem applies to the bundle RP " CP " , with Z2 coefficients, so if
!
is the standard generator of H2(CP "; Z), the Z2 reduction "H2(CP "; Z2) pulls
back to a generator of H2(RP "; Z2), namely the square 2 of the generator "
H1(RP "; Z2). Hence the Z2 reduction xC(E) = CP(g)"() " H2(CP(E); Z2) of the
basic class xC(E)= CP(g)"() pulls back to the square of the basic class xR(E)=
RP(g)"()"H1(RP(E); Z2). Consequently the Z2 reduction of the defining relation
for the Chern classes of E, which is xC(E)n +c1(E)xC(E)n-1 + +cn(E) 1 = 0,
pulls back to the relationxR(E)2n+c1(E)xR(E)2n-2 ++cn(E) 1 = 0, which is the
defining relation for the Stiefel-Whitney classes of E. This means that w2i+1(E)= 0
and w2i(E)=ci(E).
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70 Chapter 3 Characteristic Classes
Cohomology of Grassmannians
From Example 3.7 and naturality it follows that the universal bundle En Gn
!
must also have all its Stiefel-Whitney classes w1(En),,wn(En) nonzero. In fact a
much stronger statement is true. Let f:(RP ")n Gn be the classifying map for the
!
n fold Cartesion product (E1)n of the canonical line bundle E1 , and for notational
simplicity let wi =wi(En). Then the composition
f"
Z2[w1,,wn] H"(Gn; Z2) H" (RP ")n; Z2 H" Z2[1,,n]
!
!
sendswi toi, theith elementric symmetric polynomial. It is a classical algebraic re-
sult that the polynomials i are algebraically independent in Z2[1,,n]. Proofs
of this can be found in [van der Waerden, 26] or [Lang, p. 134] for example. Thus
the composition Z2[w1,,wn] Z2[1,,n] is injective, hence also the map
!
Z2[w1,,wn] H"(Gn; Z2). In other words, the classes wi(En) generate a poly-
!
nomial subalgebra Z2[w1,,wn]"H"(Gn; Z2). This subalgebra is in fact equal to
H"(Gn; Z2), and the corresponding statement for Chern classes holds as well:
Theorem 3.9. H"(Gn; Z2) is the polynomial ring Z2[w1,,wn] on the Stiefel-
Whitney classes wi = wi(En) of the universal bundle En Gn. Similarly, in the
!
complex caseH"(Gn(C "); Z)H" Z[c1,,cn]whereci =ci(En(C "))for the univer-
sal bundle En(C ") Gn(C ").
!
The proof we give here for this basic result will be a fairly quick application of the
CW structure on Gn constructed at the end of 1.2. A different proof will be given
in 3.3 where we also compute the cohomology of Gn with Z coefficients, which is
somewhat more subtle.
Proof: Consider a map f:(RP ")n Gn which pulls En back to the bundle (E1)n
!
considered above. We have noted that the image of f" contains the symmetric poly-
nomials in Z2[1,,n]H"H"((RP ")n; Z2). The opposite inclusion holds as well,
since if:(RP ")n (RP ")n is an arbitrary permutation of the factors, then pulls
!
(E1)n back to itself, sof f, which means thatf" ="f" , so the image off" is
invariant under" :H"((RP ")n; Z2) H"((RP ")n; Z2), but the latter map is just the
!
same permutation of the variables i.
To finish the proof in the real case it remains to see thatf" is injective. It suffices
to find a CW structure on Gn in which the r cells are in one-to-one correspondence
r
r
with monomialsw11 wnn of dimensionr =r1 + 2r2 ++nrn, since the number
of r cells in a CW complex X is an upper bound on the dimension of Hr(X; Z2) as a
Z2 vector space, and a surjective linear map between finite-dimensional vector spaces
is injective if the dimension of the domain is not greater than the dimension of the
range.
r
r
Monomialsw11 wnn of dimensionr correspond ton tuples(r1,,rn)with
r =r1 + 2r2 ++nrn. Such n tuples in turn correspond to partitions of r into at
Stiefel-Whitney and Chern Classes Section 3.1 71
most n integers, via the correspondence
(r1,,rn) rn d"rn+rn-1 d" d"rn+rn-1 ++r1.
!
!
Such a partition becomes the sequence1 - 1 d"2 - 2 d" d"n-n, corresponding
to the strictly increasing sequence 0
we have:
(r1,r2,r3) (1 - 1,2 -2,3 -3) (1,2,3) dimension
1 0 0 0 0 0 0 1 2 3 0
w1 1 0 0 0 0 1 1 2 4 1
w2 0 1 0 0 1 1 1 3 4 2
2
w1 2 0 0 0 0 2 1 2 5 2
w3 0 0 1 1 1 1 2 3 4 3
w1w2 1 1 0 0 1 2 1 3 5 3
3
w1 3 0 0 0 0 3 1 2 6 3
The cell structure on Gn constructed in 1.2 has one cell of dimension (1 - 1)+
(2 - 2)+ +(n -n) for each increasing sequence 0
we are done in the real case.
The complex case is entirely similar, keeping in mind that ci has dimension 2i [ Pobierz całość w formacie PDF ]