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these in §2.3, notably Adams theorem on the Hopf invariant with its corollary on the
nonexistence of division algebras over R in dimensions other than 1, 2, 4, and 8,
the dimensions of the real and complex numbers, quaternions, and Cayley octonions.
A further application to the J homomorphism is delayed until the next chapter when
we combine K theory with ordinary cohomology.
1. The Functor K(X)
Since we shall be dealing almost exclusively with complex vector bundles in this
chapter, let us take vector bundle to mean generally complex vector bundle unless
otherwise specified. Base spaces will always be assumed paracompact, in particular
Hausdorff, so that the results of Chapter 1 which presume paracompactness will be
available to us.
For the purposes of K theory it is convenient to take a slightly broader defini-
tion of vector bundle which allows the fibers of a vector bundle p:E X to be vec-
’!
tor spaces of different dimensions. We still assume local trivializations of the form
h:p-1(U) U×Cn, so the dimensions of fibers must be locally constant overX, but
’!
if X is disconnected the dimensions of fibers need not be globally constant.
The Functor K(X) Section 2.1 29
Consider vector bundles over a fixed base space X. The trivial n dimensional
vector bundle we write asµn X. Define two vector bundlesE1 andE2 overX to be
’!
stably isomorphic, written E1 H"sE2 , if E1 •"µn H"E2 •"µn for some n. In a similar vein
we set E1
and
of direct sum is well-defined, commutative, and associative. A zero element is the
class of µ0 .
Proposition 2.1. IfX is compact Hausdorff, then the set of
vector bundles over X forms an abelian group with respect to •" .
This group is called K(X).
Proof: Only the existence of inverses needs to be shown, which we do by showing
that for each vector bundle À:E X there is a bundle E X such that E•"E H"µm
’! ’!
for some m. If all the fibers of E have the same dimension, this is Proposition 1.9.
In the general case let Xi ={x"X| dimÀ-1(x)=i}. These Xi s are disjoint open
|
sets inX, hence are finite in number by compactness. By adding toE a bundle which
over eachXi is a trivial bundle of suitable dimension we can produce a bundle whose
fibers all have the same dimension.
For the direct sum operation on H"s equivalence classes, only the zero element, the
class of µ0 , can have an inverse since E•"E H"s µ0 implies E•"E •"µn H"µn for some
n, which can only happen if E and E are 0 dimensional. However, even though
inverses do not exist, we do have the cancellation property that E1 •"E2 H"s E1 •"E3
implies E2 H"s E3 over a compact base space X, since we can add to both sides of
E1 •"E2 H"sE1 •"E3 a bundle E1 such that E1 •"E1 H"µn for some n.
Just as the positive rational numbers are constructed from the positive integers
by forming quotientsa/bwith the equivalence relationa/b=c/diffad=bc, sowe
can form for compactX an abelian groupK(X)consisting of formal differencesE-E
of vector bundles E and E over X, with the equivalence relation E1 -E1 =E2 -E2
iff E1 •"E2 H"s E2 •"E1 . Verifying transitivity of this relation involves the cancellation
property, which is why compactness of X is needed. With the obvious addition rule
(E1 -E1)+(E2 -E2)=(E1 •"E2)-(E1 •"E2),K(X)is then a group. The zero element is
the equivalence class ofE-E for anyE, and the inverse ofE-E isE -E. Note that
every element ofK(X)can be represented as a differenceE-µn since if we start with
E-E we can add to both E and E a bundle E such that E •"E H"µn for some n.
There is a natural homomorphism K(X) K(X) sending E-µn to the
’!
of E. This is well-defined since if E-µn =E -µm in K(X), then E•"µm H"s E •"µn,
henceE
’!
elementsE-µn withE
elements of the formµm-µn. This subgroup {µm-µn} ofK(X)is isomorphic to Z.
30 Chapter 2 Complex K Theory
In fact, restriction of vector bundles to a basepointx0 "X defines a homomorphism
K(X) K(x0) H" Z which restricts to an isomorphism on the subgroup {µm -µn}.
’!
Thus we have a splittingK(X)H"K(X)•"Z, depending on the choice ofx0 . The group
K(X) is sometimes called reduced, to distinguish it from K(X).
Let us compute a few examples. The complex version of Proposition 1.10 gives a
bijection between the set Vectk(Sn)of isomorphism classes ofk dimensional vector
C
bundles over Sn and Àn-1U(k). Under this bijection, adding a trivial line bundle
corresponds to includingU(k)inU(k+1)by adjoining an(n+ 1)st row and column
consisting of zeros except for a single 1 on the diagonal. Let U = U(k) with the
k
weak topology: a subset of U is open iff it intersects each U(k) in an open set in
U(k). This implies that each compact subset of U is contained in some U(k), and it
follows that the bijections Vectk(Sn)H"Àn-1U(k)induce a bijectionK(Sn)H"Àn-1U.
C
Proposition 2.2. This bijection K(Sn)H"Àn-1U is a group isomorphism.
Proof: We need to see that the two group operations correspond. Represent two
elements of Àn-1U by maps f,g:Sn-1 U(k) taking the basepoint of Sn-1 to the
’!
identity matrix. The sum inK(Sn)then corresponds to the mapf•"g:Sn-1 U(2k)
’!
having the matrices f(x) in the upper left k×k block and the matrices g(x) in the
lower right k×k block, the other two blocks being zero. Since À0U(2k)= 0, there
is a path ±t " U(2k) from the identity to the matrix of the transformation which
interchanges the two factors of Ck×Ck. Then the matrix product(f•" 1 1 •"g)±t
1)±t(1
gives a homotopy from f•"g to fg•"1
1.
It remains to see that the matrix product fg represents the sum [f]+[g] in
Àn-1U(k). This is a general fact about H spaces which can be seen in the following
way. The standard definition of the sum in Àn-1U(k) is [f]+[g]=[f +g] where
the map f +g consists of a compressed version of f on one hemisphere of Sn-1
and a compressed version of g on the other. We can realize this map f +g as a
product f1g1 of maps Sn-1 U(k) each mapping one hemisphere to the identity.
’!
There are homotopies ft fromf =f0 to f1 andgt fromg=g0 tog1 . Thenftgt is
a homotopy from fg to f1g1 =f+g.
This proposition generalizes easily to suspensions: For all compact X, K(SX) is
isomorphic to X,U , the group of basepoint-preserving homotopy classes of maps
X U.
’!
From the calculations of ÀiU in §1.2 we deduce that K(Sn) is 0, Z, 0, Z for
n= 1, 2, 3, 4. This alternation of 0 s and Z s continues for all higher dimensional
spheres:
Bott Periodicity Theorem. There are isomorphisms K(Sn)H"K(Sn+2) for all ne"
0. More generally, there are isomorphismsK(X)H"K(S2X)for all compactX, where
S2X is the double suspension of X.
The Functor K(X) Section 2.1 31
The theorem actually says that a certain natural map ²:K(X) K(S2X) defined
’!
later in this section is an isomorphism. There is an equivalent form of Bott periodicity
H"
—"
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