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with basis 1,L,···,Ln-1 . In order to describe the multiplication in K"(P(E)) one
therefore needs only a relation expressing Ln in terms of lower powers of L. Such
a relation can be found as follows. The pullback of E over P(E) splits as L•"E for
some bundle E of dimension n- 1, and the desired relation will be »n(E )= 0. To
compute »n(E )= 0 we use the formula »t(E)=»t(L)»t(E ) in K"(P(E))[t], where
to simplify notation we let  E also denote the pullback ofE overP(E). The equation
»t(E) = »t(L)»t(E ) can be rewritten as »t(E ) = »t(E)»t(L)-1 where »t(L)-1 =
(-1)iLiti since »t(L) = 1 +Lt. Equating coefficients of tn in the two sides of
i
»t(E )=»t(E)»t(L)-1 , we get»n(E )= (-1)n-i»i(E)Ln-i. The relation»n(E )= 0
i
can be written as (-1)i»i(E)Ln-i = 0, with the coefficient of Ln equal to 1, as
i
desired. The result can be stated in the following form:
Proposition 2.25. For an n dimensional vector bundle E X the ring K(P(E)) is
’!
isomorphic to the quotient ring K"(X)[L]/ (-1)i»i(E)Ln-i .
i
n
For example whenXis a point we haveP(E)= CPn-1 and»i(E)= Ck fork= ,
i
so the polynomial (-1)i»i(E)Ln-i becomes(L-1)n and we see that the proposition
i
generalizes the isomorphism K"(CPn-1)H" Z[L]/(L- 1)n).
Appendix: Finite Cell Complexes
As we mentioned in the remarks following Proposition 2.21 it is convenient for
purposes of the splitting principal to work with spaces slightly more general than finite
Further Calculations Section 2.4 61
CW complexes. By a finite cell complex we mean a space which has a finite filtration
X0 ‚"X1 ‚" ··· ‚"Xk =X where X0 is a finite discrete set and Xi+1 is obtained from
Xi by attaching a cell eni via a map Õi:Sni-1 Xi. Thus Xi+1 is the quotient space
’!
of the disjoint union of Xi and a disk Dni under the identifications x
x""Dni =Sni-1 .
Proposition 2.26. If p:E B is a fiber bundle whose fiber F and base B are both
’!
finite cell complexes, then E is also a finite cell complex, whose cells are products of
cells in B with cells in F.
Proof: SupposeB is obtained from a subcomplexB by attaching a cellen. By induc-
tion on the number of cells ofB we may assume thatp-1(B )is a finite cell complex.
If ¦ :Dn B is a characteristic map for en then the pullback bundle ¦"(E) Dn is
’! ’!
a product since Dn is contractible. Since F is a finite cell complex, this means that
we may obtain ¦"(E)from its restriction overSn-1 by attaching cells. Hence we may
obtain E from p-1(B ) by attaching cells.
4. Further Calculations
In this section we give computations of the K theory of some other interesting
spaces.
The Thom Isomorphism
The relative form of the Leray-Hirsch theorem for disk bundles is a useful tech-
nical result known as the Thom isomorphism:
Proposition 2.27. Let p:E B be a fiber bundle with fibers Dn and with base B a
’!
finite cell complex, and let E B be the sphere subbundle with fibers the boundary
’!
spheres of the fibers of E. If there is a class c " K"(E,E ) which restricts to a
generator of K"(Dn,Sn) H" Z in each fiber, then the map ¦ :K"(B) K"(E,E ),
’!
¦(b)=p"(b)·c, is an isomorphism.
The classc is called a Thom class for the bundle. As we will show below, the unit
disk bundle in every complex vector bundle has a Thom class.
Proof: LetE Bbe the bundle with fiberSn obtained as a quotient ofEby collapsing
’!
each fiber of the subbundle E to a point. The union of these points is a copy of B
in E forming a section of E . The long exact sequence for the pair (E,B) then splits,
giving an isomorphismK"(E)H"K"(E,B)•"K"(B). Under this isomorphism the class
c"K"(E,E )=K"(E,B) corresponds to a class c"K"(E), which, together with the
62 Chapter 2 Complex K Theory
element 1 "K"(E), allows us to define the left-hand ¦ in the following commutative
diagram, where " is a point.
H"
" "
( ) ( ) ( ) ( •"K" B —"K"
K" B —"K" Sn -’!
- K" B —"K" Sn, ) ( ) ( )
-
-
-
•"
¦ ¦ ¦
H"
( ) ( ) ( )
K" E -------- K" E B •"K" B
-------’! ,
The Leray-Hirsch theorem implies that the left-hand ¦ is an isomorphism, hence both
¦  s on the right-hand side of the diagram are isomorphisms as well.
Example 2.28. For a complex vector bundleE X withX compact Hausdorff we will
’!
now show how to find a Thom classU"K(D(E),S(E)), whereD(E)andS(E)are the
unit disk and sphere bundles in E. We can also regard U as an element of K(T(E))
where the Thom spaceT(E)is the quotientD(E)/S(E). SinceX is compact,T(E)can
also be described as the one-point compactification of E. We may view T(E) as the
quotient P(E•"1)/P(E) since in each fiber Cn of E we obtain P(Cn•"C)= CPn from
P(Cn)= CPn-1 by attaching the 2n cell Cn×{1}, so the quotient P(Cn•"C)/P(Cn)
is S2n, which is the part of T(E) coming from this fiber Cn. From Example 2.24 we
know thatK"(P(E•"1))is the freeK"(X) module with basis 1,L,···,Ln, whereLis
the canonical line bundle over P(E•"1). Restricting to P(E)‚"P(E•"1),K"(P(E)) is
the free K"(X) module with basis the restrictions of 1,L,···,Ln-1 to P(E). So we
have a short exact sequence
Á
0 K "(T(E)) K"(P(E•"1)) K"(P(E)) 0
’! ’! ’!
’!
and KerÁ must be generated as a K"(X) module by some polynomial of the form
Ln +an-1Ln-1 + ··· +a01 with coefficients ai " K"(X), namely the polynomial
(-1)i»i(E)Ln-i in Proposition 2.25, regarded now as an element of K(P(E•"1)).
i
The class U " K(T(E)) mapping to (-1)i»i(E)Ln-i is the desired Thom class
i
since when we restrict over a point of X the preceding considerations still apply, so
the kernel ofK(CPn) K(CPn-1)is generated by the restriction of (-1)i»i(E)Ln-i
’!
i
to a fiber.
[More applications will be added later: the Gysin Sequence, the Künneth formula,
and calculations of the K theory of various spaces including Grassmann manifolds,
flag manifolds, the group U(n), real projective space, and lens spaces.]
-
-
-
-
’!
’!
Characteristic classes are cohomology classes in H"(B;R) associated to vector
bundles E B by some general rule which applies to all base spaces B. The four
’!
classical types of characteristic classes are:
1. Stiefel-Whitney classes wi(E)"Hi(B; Z2) for a real vector bundle E.
2. Chern classes ci(E)"H2i(B; Z) for a complex vector bundle E.
3. Pontryagin classes pi(E)"H4i(B; Z) for a real vector bundle E.
4. The Euler classe(E)"Hn(B; Z)whenE is an orientedn dimensional real vector
bundle.
The Stiefel-Whitney and Chern classes are formally quite similar. Pontryagin classes
can be regarded as a refinement of Stiefel-Whitney classes when one takes Z rather
than Z2 coefficients, and the Euler class is a further refinement in the orientable case.
Stiefel-Whitney and Chern classes lend themselves well to axiomatization since
in most applications it is the formal properties encoded in the axioms which one uses [ Pobierz całość w formacie PDF ]