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so by the Tietze extension theorem we obtain a section sij:Uj E extending si. If
’!
{Õj,Õ} is a partition of unity subordinate to the cover {Uj,X-A} of X, the sum
Õjsij gives an extension ofsi to a section defined on all ofX. Since these sections
j
form a basis in each fiber overA, they must form a basis in all nearby fibers. Namely,
over Uj the extended si s can be viewed as a square-matrix-valued function having
nonzero determinant at each point of A, hence at nearby points as well.
Thus we have a trivialization h of E over a neighborhood U of A. This induces
a trivialization of E/h over U/A, so E/h is a vector bundle. It remains only to verify
that E H" q"(E/h). In the commutative diagram at the right the
E E/h
--
--
-’!
p
quotient map E E/h is an isomorphism on fibers, so this map
’!
X X/A
and p give an isomorphism EH"q"(E/h). --
-q’!
-
-
There is an easy way to extend the exact sequenceK(X/A) K(X) K(A)to the
’! ’!
left, using the following diagram, where C and S denote cone and suspension:
( ) (( ) ) ( )
A X X*"CA X*"CA *"CX X*"CA *"CX *"C X*"CA
X/A SA SX
In the first row, each space is obtained from its predecessor by attaching a cone on the
subspace two steps back in the sequence. The vertical maps are the quotient maps
-
-
-
-
’!
’!
-
-
-
-
-
-
’!
’!
’!
34 Chapter 2 Complex K Theory
obtained by collapsing the most recently attached cone to a point. In many cases the
quotient map collapsing a contractible subspace to a point is a homotopy equivalence,
hence induces an isomorphism on K . This conclusion holds generally, in fact:
Lemma 2.5. If A is contractible, the quotient map q:X X/A induces a bijection
’!
q" : Vectn(X/A) Vectn(X) for all n.
’!
Proof: A vector bundle E X must be trivial over A since A is contractible. A
’!
trivialization h gives a vector bundle E/h X/A as in the proof of the previous
’!
proposition. We assert that the isomorphism class of E/h does not depend on h.
This can be seen as follows. Given two trivializations h0 and h1 , by writing h1 =
(h1h-1)h0 we see that h0 and h1 differ by an element of gx " GLn(C) over each
point x " A. The resulting map g:A GLn(C) is homotopic to a constant map
’!
x ±"GLn(C) since A is contractible. Writing now h1 =(h1h-1±-1)(±h0), we
see that by composingh0 with±in each fiber, which does not changeE/h0 , we may
assume that ± is the identity. Then the homotopy from g to the identity gives a
homotopy H from h0 to h1 . In the same way that we constructed E/h we construct
a vector bundle (E×I)/H (X/A)×I restricting to E/h0 over one end and to E/h1
’!
over the other end, hence E/h0 H"E/h1 .
Thus we have a well-defined map Vectn(X) Vectn(X/A), E E/h. This is an
’!
inverse to q" since q"(E/h)H"E as we noted in the preceding proposition, and for a
bundle E X/Awe have q"(E)/hH"E for the evident trivialization h of q"(E) over
’!
A
From this lemma and the preceding proposition it follows that we have a long
exact sequence of K groups
··· K(SX) K(SA) K(X/A) K(X) K(A)
’! ’! ’! ’! ’!
For example, if X = A("B then X/A = B and the sequence breaks up into split
short exact sequences, which implies that the map K(X) K(A)•"K(B) obtained by
’!
restriction to A and B is an isomorphism.
We can use this exact sequence to obtain a reduced version of the external prod-
—"
uct, a ring homomorphismK(X) K(Y) K(X'"Y)whereX'"Y =X×Y/X("Y and
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X("Y =X×{y0}*"{x0}×Y ‚"X×Y for chosen basepoints x0 "X and y0 "Y. The
spaceX'"Y is called the smash product ofX andY. To define the reduced product,
consider the long exact sequence for the pair (X×Y,X("Y):
( ))- (S X("Y - K - K
( ))- X'"Y - X×Y - K
(S X×Y - K- X("Y
K
’!
K SX SY K X Y
The second of the two vertical isomorphisms here was noted earlier, and the first
vertical isomorphism arises in similar fashion using Lemma 2.5 since SX("SY is
H"
H"
The Functor K(X) Section 2.1 35
obtained from S(X("Y) by collapsing a line segment to a point. The last horizon-
tal map in the sequence is a split surjection, with splitting K(X)•"K(Y) K(X×Y),
’!
" "
(a,b) p1(a)+p2(b)wherep1 andp2 are the projections ofX×Y ontoX andY.
Similarly, the first map splits via (Sp1)" +(Sp2)" . So we get a splitting K(X×Y)H"
K(X'"Y)•"K(X)•"K(Y).
For a " K(X) = Ker(K(X) K(x0)) and b " K(Y) = Ker(K(Y) K(y0)) the
’! ’!
" " "
external product a"b = p1(a)p2(b) " K(X×Y) has p1(a) restricting to zero in
" " "
K(Y)andp2(b)restricting to zero inK(X), sop1(a)p2(b)restricts to zero in both
K(X) and K(Y), hence in K(X("Y). In particular, a"b lies in K(X×Y), and from
the short exact sequence above, a"b pulls back to a unique element of K(X'"Y).
—"
This defines the reduced external productK(X) K(Y) K(X'"Y). It is essentially a
’!
restriction of the unreduced external product, as shown in the diagram below, so the
reduced external product is also a ring homomorphism, and we shall use the same
notation a"b for both reduced and unreduced external product, leaving the reader
to determine from context which is meant.
( ) —" ( ) H" ) —" )) X Y Z
X Y ( X Y
K K K K K K
( ) H"
X×Y X'"Y X Y Z
K K K K
SinceSn'"X is then fold iterated reduced suspension £nX, which is a quotient
of the ordinaryn fold suspensionSnX obtained by collapsing ann disk inSnX to a
point, the quotient map SnX Sn'"X induces an isomorphism on K by Lemma 2.5.
’!
Then the reduced external product gives rise to a homomorphism
²:K(X) K(S2X), ²(a)=(H- 1)"a
’!
where H is the canonical line bundle over S2 = CP1 . The version of Bott Periodicity
for reduced K theory states that this is an isomorphism. This is equivalent to the
unreduced version by the preceding diagram.
As we saw earlier, a pair(X,A)of compact Hausdorff spaces gives rise to an exact
sequence of K groups, the first row in the following diagram:
(
K S2X - K S2A -K S X/A K SX -K SA - K X/A - K X - K A
’!
2 2 1 1 1
( ) ( )- ( ) ( ) ( ) ( )-K0A
( )-K A - K X,A K X -K A - K0 X,A -K0 X
( )
K X
’!
² H" ² H"
( ) ( )
K0 X - K0A
’!
If we set K -n(X) = K(SnX) and K -n(X,A) = K(Sn(X/A)), this sequence can be
written as in the second row. Negative indices are chosen here so that the coboundary
maps in this sequence increase dimension, as in ordinary cohomology. The lower
left corner of the diagram containing the Bott periodicity isomorphisms²commutes
since external tensor product withH-1 commutes with maps between spaces. So the
- - ’!
- - ’!
==
==
==
==
==
==
==
==
==
==
==
’!
’!
-
-
-
-
36 Chapter 2 Complex K Theory
long exact sequence in the second row can be rolled up into a six-term periodic exact
sequence. It is reasonable to extend the definition ofK n to positivenvia periodicity,
and then the six-term exact sequence can be written:
( ) ( ) ( )
K0X,A -- K0 -- K0
-’! -’!
- X - A
( ) ( ) ( )
K1 AX
K1K1X,A
—"
A product K i(X) K j(Y) K i+j(X'"Y) is obtained from the external product
’!
—"
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