2.1. The Functor K(X)

Back

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 definition

of ‘vector bundle’ which allows the fibers of a vector bundle p : E!X to be vector

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 over X, 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 as "n!X. Define two vector bundles E1 and E2 over X to be

stably isomorphic, written E1 _

s E2 , if E1

_"n _ E2

_"n for some n. In a similar vein

we set E1 _ E2 if E1

_"m _ E2

_"n for some m and n. It is easy to see that both _

s

and _ are equivalence relations. On equivalence classes of either sort the operation

of direct sum is well-defined, commutative, and associative. A zero element is the

class of "0 .

Proposition 2.1. If X is compact Hausdorff, then the set of _ equivalence classes of

vector bundles over X forms an abelian group with respect to _.

This group is called e K.X..

Proof: Only the existence of inverses needs to be shown, which we do by showing

that for each vector bundle _ : E!X there is a bundle E0!X such that E_E0 _ "m

for some m. If all the fibers of E have the same dimension, this is Proposition 1.9.

In the general case let Xi

. fx 2 Xjjdim_−1.x. . i g. These Xi ’s are disjoint open

sets in X, hence are finite in number by compactness. By adding to E a bundle which

over each Xi is a trivial bundle of suitable dimension we can produce a bundle whose

fibers all have the same dimension. tu

For the direct sum operation on _

s equivalence classes, only the zero element, the

class of "0 , can have an inverse since E_E0 _

s "0 implies E_E0_"n _ "n for some

n, which can only happen if E and E0 are 0 dimensional. However, even though

inverses do not exist, we do have the cancellation property that E1

_E2 _

s E1

_E3

implies E2 _

s E3 over a compact base space X, since we can add to both sides of

E1

_E2 _

s E1

_E3 a bundle E0

1 such that E1

_E0

1 _ "n for some n.

Just as the positive rational numbers are constructed from the positive integers

by forming quotients a=b with the equivalence relation a=b . c=d iff ad . bc , sowe

can form for compact X an abelian group K.X. consisting of formal differences E−E0

of vector bundles E and E0 over X, with the equivalence relation E1 − E0

1 . E2 − E0

2

iff E1

_E0

2 _

s E2

_E0

1 . Verifying transitivity of this relation involves the cancellation

property, which is why compactness of X is needed. With the obvious addition rule

.E1−E0

1...E2−E0

2. . .E1

_E2.−.E0

1

_E0

2., K.X. is then a group. The zero element is

the equivalence class of E−E for any E, and the inverse of E−E0 is E0 −E. Note that

every element of K.X. can be represented as a difference E−"n since if we start with

E − E0 we can add to both E and E0 a bundle E00 such that E0_E00 _ "n for some n.

There is a natural homomorphism K.X.!e K.X. sending E − "n to the _ class

of E. This is well-defined since if E − "n . E0 − "m in K.X., then E_"m _

s E0_"n ,

hence E _ E0 . The map K.X.!e K.X. is obviously surjective, and its kernel consists of

elements E−"n with E _ "0 , hence E _

s "m for some m, so the kernel consists of the

elements of the form "m −"n . This subgroup f"m −"ng of K.X. is isomorphic to Z.

30 Chapter 2 Complex K–Theory

In fact, restriction of vector bundles to a basepoint x0 2 X defines a homomorphism

K.X.!K.x0. _ Z which restricts to an isomorphism on the subgroup f"m − "ng.

Thus we have a splitting K.X. _ e K.X._Z, depending on the choice of x0 . The group

e 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

C.Sn. of isomorphism classes of k dimensional vector

bundles over Sn and _n−1U.k.. Under this bijection, adding a trivial line bundle

corresponds to including U.k. in U.k.1. by adjoining an .n.1.st row and column

consisting of zeros except for a single 1 on the diagonal. Let U .

S

k U.k. with the

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

C.Sn. _ _n−1U.k. induce a bijection e K.Sn. _ _n−1U .

Proposition 2.2. This bijection e K.Sn. _ _n−1U is a group isomorphism.

Proof: We need to see that the two group operations correspond. Represent two

elements of _n−1U by maps f ;g :Sn−1!U.k. taking the basepoint of Sn−1 to the

identity matrix. The sum in e K.Sn. then corresponds to the map f _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 _0U.2k. . 0, there

is a path _t

2 U.2k. from the identity to the matrix of the transformation which

interchanges the two factors of Ck_Ck . Then the matrix product .f _11._t.11_g._t

gives a homotopy from f _ g to fg _11.

It remains to see that the matrix product fg represents the sum .f . . .g. in

_n−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 _n−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 from f . f0 to f1 and gt from g . g0 to g1 . Then ftgt is

a homotopy from fg to f1g1 . f . g. tu

This proposition generalizes easily to suspensions: For all compact X, e K.SX. is

isomorphic to hX;Ui, the group of basepoint-preserving homotopy classes of maps

X!U .

From the calculations of _iU in x1.2 we deduce that e 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 e K.Sn. _ e K.Sn.2. for all n _

0. More generally, there are isomorphisms e K.X. _ e K.S2X. for all compact X, 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 _: e K.X.!e K.S2X. defined

later in this section is an isomorphism. There is an equivalent form of Bott periodicity

involving K.X. rather than e K.X., an isomorphism _ : K.X.K.S2. _ ----! K.X_S2..

The map _ is easier to define than _, so this is what we will do next. Then we will

set up some formal machinery which in particular shows that the two versions of Bott

Periodicity are equivalent. The second version is the one which will be proved in x2.2.

Ring Structure

Besides the additive structure in K.X. there is also a natural multiplication coming

from tensor product of vector bundles. For elements of K.X. represented by

vector bundles E1 and E2 their product in K.X. will be represented by the bundle

E1

E2 , so for arbitrary elements of K.X. represented by differences of vector bundles,

their product in K.X. is defined by the formula

.E1 − E0

1..E2 − E0

2. . E1

E2 − E1

E0

2 − E0

1

E2 . E0

1

E0

2

It is routine to verify that this is well-defined and makes K.X. into a commutative ring

with identity "1 , the trivial line bundle, using the basic properties of tensor product

of vector bundles described in x1.1. We can simplify notation by writing the element

"n 2 K.X. just as n. This is consistent with familiar arithmetic rules. For example,

the product nE is the sum of n copies of E.

If we choose a basepoint x0 2 X, then the map K.X.!K.x0. obtained by restricting

vector bundles over x0 is a ring homomorphism. Its kernel, which can be

identified with e K.X., is an ideal, hence also a ring in its own right, though not necessarily

a ring with identity.

Example 2.3. Let us compute the ring structure in K.S2.. As an abelian group,

K.S2. is isomorphic to e K.S2._Z _ Z_Z, with additive basis f1;Hg where H is the

canonical line bundle over CP1 . S2 , by Proposition 2.2 and the calculations in x1.2.

We use the notation ‘H’ for the canonical line bundle over CP1 since its unit sphere

bundle is the Hopf bundle S3!S2 . To determine the ring structure in K.S2. we have

only to express the element H2 , represented by the tensor product HH, as a linear

combination of 1 and H. The claim is that the bundle .HH._1 is isomorphic to

H_H. This can be seen by looking at the clutching functions for these two bundles,

which are the maps S1!U.2. given by

z,

z2 0

0 1

!

and z,

z 0

0 z

!

With the notation used in the proof of Proposition 2.2, these are the clutching functions

fg_11 and f _g where both f and g are the function z,.z.. As we showed

there, the clutching functions fg_11 and f_g are always homotopic, so this gives the

desired isomorphism .HH._1 _ H_H. In K.S2. this is the formula H2.1 . 2H,

32 Chapter 2 Complex K–Theory

so H2 . 2H − 1. We can also write this as .H − 1.2 . 0, and then K.S2. can be

described as the quotient Z.H.=.H − 1.2 of the polynomial ring Z.H. by the ideal

generated by .H − 1.2 .

Note that if we regard e K.S2. as the kernel of K.S2.!K.x0., then it is generated

as an abelian group by H−1. Since we have the relation .H−1.2 . 0, this means that

the multiplication in e K.S2. is completely trivial: The product of any two elements

is zero. Readers familiar with cup product in ordinary cohomology will recognize

that the situation is exactly the same as in H_.S2; Z. and eH

_.S2; Z., with H − 1

behaving exactly like the generator of H2.S2; Z.. In the case of ordinary cohomology

the cup product of a generator of H2.S2; Z. with itself is automatically zero since

H4.S2; Z. . 0, whereas with K–theory a calculation is required.

The rings K.X. and e K.X. can be regarded as functors of X. A map f :X!Y induces

a map f _ : K.Y.!K.X., sending E−E0 to f _.E.−f _.E0.. This is a ring homomorphism

since f _.E1

_E2. _ f _.E1._f _.E2. and f _.E1

E2. _ f _.E1.f _.E2..

The functor properties .fg._ . g_f _ and 11_ . 11 as well as the fact that f ' g

implies f _ . g_ all follow from the corresponding properties for pullbacks of vector

bundles. Similarly, we have induced maps f _ : e K.Y.!e K.X. with the same properties,

except that for f _ to be a ring homomorphism we must be in the category of basepointed

spaces and basepoint-preserving maps since our definition of multiplication

for eK

required basepoints.

An external product _ : K.X.K.Y.!K.X_Y. can be defined by _.ab. .

p_

1 .a.p_

2 .b. where p1 and p2 are the projections of X_Y onto X and Y . The tensor

product of rings is a ring, with multiplication defined by .ab..c d. . ac bd, and

_ is a ring homomorphism since _..ab..c d.. . _.ac bd. . p_

1 .ac.p_

2 .bd. .

p_

1 .a.p_

1 .c.p_

2 .b.p_

2 .d. . p_

1 .a.p_

2 .b.p_

1 .c.p_

2 .d. . _.ab._.c d..

Taking Y to be S2 we have an external product

_ : K.X.K.S2.!K.X_S2.

The form of Bott Periodicity which we prove in x2.2 asserts that this map is an isomorphism.

The external product in ordinary cohomology is called ‘cross product’ and written

a_b, but to use this symbol for the K–theory external product might lead to confusion

with Cartesian product of vector bundles, which is quite different from tensor product.

Instead we will sometimes use the notation a _ b as shorthand for _.ab..

Cohomological Properties

The reduced groups eK

satisfy a key exactness property:

Proposition 2.4. If X is compact Hausdorff and A _ X is a closed subspace, then the

inclusion and quotient maps A --!-i X q ----!X=A induce an exact sequence eK

.X=A. q_ ----!

Ke.X. i_ ----! eK

.A..

The Functor K(X) Section 2.1 33

Since A is a closed subspace of a compact Hausdorff space, it is also compact

Hausdorff. The quotient space X=A is compact Hausdorff as well, with the Hausdorff

property following from the fact that compact Hausdorff spaces are normal, hence a

point x 2 X − A and A have disjoint neighborhoods in X.

Proof: Recall that exactness means that the image of q_ equals the kernel of i_ .

The inclusion Im q_ _ Ker i_ is equivalent to i_q_ . 0. Since qi is equal to the

composition A!A=A>X=A and eK

.A=A. . 0, it follows that i_q_ . 0.

For the opposite inclusion Ker i_ _ Im q_ , suppose the restriction over A of a

vector bundle p : E!X is stably trivial. Adding a trivial bundle to E, we may assume

that E itself is trivial over A. Choosing a trivialization h:p−1.A.!A_Cn , let E=h be

the quotient space of E under the identifications h−1.x;v. _ h−1.y;v. for x;y 2 A.

There is then an induced projection E=h!X=A. To see that this is a vector bundle

we need to find a local trivialization over a neighborhood of the point A=A.

We claim that since E is trivial over A, it is trivial over some neighborhood of A.

In many cases this holds because there is a neighborhood which deformation retracts

onto A, so the restriction of E over this neighborhood is trivial since it is isomorphic

to the pullback of p−1.A. via the retraction. In the absence of such a deformation

retraction one can make the following more complicated argument. A trivialization

of E over A determines sections si :A!E which form a basis in each fiber over A.

Choose a cover of A by open sets Uj in X over each of which E is trivial. Via a local

trivialization, each section si can be regarded as a map from A \ Uj to a single fiber,

so by the Tietze extension theorem we obtain a section sij :Uj!E extending si. If

f'j;'g is a partition of unity subordinate to the cover fUj;X − Ag of X, the sum

P

j 'jsij gives an extension of si to a section defined on all of X. Since these sections

form a basis in each fiber over A, 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 _ q_.E=h.. In the commutative diagram at the right the

¡!

¡!

E¡¡¡¡¡!

X¡¡¡¡¡! A q

p

X/

E/h

quotient map E!E=h is an isomorphism on fibers, so this map

and p give an isomorphism E _ q_.E=h.. tu

There is an easy way to extend the exact sequence eK

.X=A.!e K.X.!e K.A. to the

left, using the following diagram, where C and S denote cone and suspension:

A >X X CA

A SA SX

> [ >(X [ CA) [CX >((X [ CA) [CX ) [C(X [ CA)

X/

¡!

¡!

¡!

' ' '

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 eK

. 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

0 .h0 we see that h0 and h1 differ by an element of gx

2 GLn.C. over each

point x 2 A. The resulting map g :A!GLn.C. is homotopic to a constant map

x,_ 2 GLn.C. since A is contractible. Writing now h1 . .h1h−1

0 _−1.._h0., we

see that by composing h0 with _ in each fiber, which does not change E=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 _ 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. _ E as we noted in the preceding proposition, and for a

bundle E!X=A we have q_.E.=h _ E for the evident trivialization h of q_.E. over

A tu

From this lemma and the preceding proposition it follows that we have a long

exact sequence of eK

groups

___!e K.SX.!e K.SA.!e K.X=A.!e K.X.!e 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 e K.X.!e K.A._ e 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 product,

a ring homomorphism e K.X. e K.Y.!e K.X ^Y. where X ^Y . X_Y=X _Y and

X _ Y . X_fy0g [ fx0g_Y _ X_Y for chosen basepoints x0 2 X and y0 2 Y . The

space X ^Y is called the smash product of X and Y . To define the reduced product,

consider the long exact sequence for the pair .X_Y;X _Y.:

K(S(X Y ))¡¡!

»

¼

¼

©

£ K(S(X _Y ))¡¡! _

»

K(SX )

»

K(SY )

»

K(X )©

»

K(Y )

»

K(X Y )¡¡!

» K (X Y )¡¡!

»

£ 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

The Functor K(X) Section 2.1 35

obtained from S.X _ Y. by collapsing a line segment to a point. The last horizontal

map in the sequence is a split surjection, with splitting e K.X._ e K.Y.!e K.X_Y.,

.a; b.,p_

1 .a..p_

2 .b. where p1 and p2 are the projections of X_Y onto X and Y .

Similarly, the first map splits via .Sp1._ . .Sp2._ . So we get a splitting e K.X_Y. _

e K.X ^ Y._ e K.X._ e K.Y..

For a 2 e K.X. . Ker.K.X.!K.x0.. and b 2 e K.Y. . Ker.K.Y .!K.y0.. the

external product a _ b . p_

1 .a.p_

2 .b. 2 K.X_Y. has p_

1 .a. restricting to zero in

K.Y. and p_

2 .b. restricting to zero in K.X., so p_

1 .a.p_

2 .b. restricts to zero in both

K.X. and K.Y., hence in K.X _ Y.. In particular, a _ b lies in e K.X_Y., and from

the short exact sequence above, a _ b pulls back to a unique element of e K.X ^ Y..

This defines the reduced external product e K.X. e K.Y.!e 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.

K(X ) K(Y ) ¼ (K(X ) Y ) © © ©

»

K( )

»

K(X ) Y

»

K( )

»

¡!

¡!

==

==

==

­ ­ Z

K(X Y ) ¼ K(X Y © © ©

»

) K(X ) Y

»

K( )

»

£ ^ Z

Since Sn ^X is the n fold iterated reduced suspension ÖnX, which is a quotient

of the ordinary n fold suspension SnX obtained by collapsing an n disk in SnX to a

point, the quotient map SnX!Sn ^ X induces an isomorphism on eK

by Lemma 2.5.

Then the reduced external product gives rise to a homomorphism

_: e K.X.!e K.S2X.; _.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 eK

groups, the first row in the following diagram:

¼

K(S X) K( ( ))

» »

K(X) A

»

K( )

»

==

==

==

==

==

==

==

==

¡! ¡! X/ ¡! ¡! ¡! ¡! ¡!

-

2 K(S A) S A

» 2 K(SX ) K( )

» »

¡!

¼ ¡

¡!

K(SA) X/A

»

K (X ) K ( )

» »

K (X) A

»

K ( )

» ¡! ¡! ¡! ¡! ¡! ¡ ! ¡! X 2 -K(

»2

A) - 1 A K - 1(X)

»

K - 1(

»

, A) K ( )

»

0 X 0 0

K (X) A

»

K ( )

» 0 ¡! 0

,A

¯ ¯

If we set eK

−n.X. . e K.SnX. and eK

−n.X;A. . e 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 isomorphisms _ commutes

since external tensor product with H−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 of eK

n to positive n via periodicity,

and then the six-term exact sequence can be written:

K (X ) A

»

K ( )

»

K( )

»

0 X,A 0 0

A K (X )

»

K ( )

»

K( )

»

1 1 1 X,A

¡¡¡¡!

¡¡!

¡¡!

¡¡¡¡! ¡¡¡¡!

¡¡¡¡!

A product eK

i.X.eK

j.Y .!eK

i.j.X ^ Y. is obtained from the external product

e K.X. e K.Y.!e K.X^Y. by replacing X and Y by SiX and SjY . If we define eK

_.X. .

eK

0.X._eK

1.X., then this gives a product eK

_.X.eK

_.Y .!eK

_.X ^ Y.. The relative

form of this is a product eK

_.X;A.eK

_.Y ; B.!eK

_.X_Y;X_B[A_Y., coming from

the products e K.Öi.X=A.. e K.Öj.Y =B.. -! e K.Öi.j.X=A ^ Y=B.. using the natural

identification .X_Y.=.X_B [ A_Y. . X=A ^ Y=B.

If we compose the external product eK

_.X.eK

_.X.!eK

_.X ^ X. with the map

eK

_.X ^ X.!eK

_.X. induced by the diagonal map X!X ^ X, x,.x;x., then we

obtain a multiplication on eK

_.X. making it into a ring, and it is not hard to check that

this extends the previously defined ring structure on eK

0.X.. The general relative form

of this product on eK

_.X. is a product eK

_.X;A.eK

_.X; B.!eK

_.X;A [ B. which is

induced by the relativized diagonal map X=.A [ B.!X=A ^ Y=B.

Example 2.6. Suppose that X . A [ B with both A and B contractible, as happens

for example if X is a suspension and A and B are its two cones. Then the product

eK

_.X.eK

_.X.!eK

_.X. is identically zero since it is equivalent to the composition

eK

_.X;A.eK

_.X; B.!eK

_.X;A[B.!eK

_.X. and eK

_.X;A[B. . 0 since X . A[B.

As a particular case we see that the product in eK

_.Sn. _ Z is trivial for n > 0. (For

n . 0 the multiplication in eK

_.S0. _ Z is just the usual multiplication of integers

since RmRn _ Rmn.)

Whereas multiplication in e K.X. is commutative, in eK

_.X. this is only true up to

sign:

Proposition 2.7. __ . .−1.ij__ for _ 2 eK

i.X. and _ 2 eK

j.X..

Proof: The product is the composition

e K.Si ^ X. e K.Sj ^ X. -!e K.Si ^ Sj ^ X ^ X. -!e K.Si ^ Sj ^ X.

where the first map is external product and the second is induced by the diagonal

map on the X factors. Replacing the product __ by the product __ amounts to

switching the two factors in the first term e K.Si^X. e K.Sj^X., and this corresponds

to switching the Si and Sj factors in the third term e K.Si ^ Sj ^ X.. Viewing Si ^ Sj

as the smash product of i . j copies of S1 , then switching Si and Sj in Si ^ Sj is

a product of ij transpositions of adjacent factors. Transposing the two factors of

S1 ^S1 is equivalent to reflection of S2 across an equator. Thus it suffices to see that

The Functor K(X) Section 2.1 37

switching the two ends of a suspension SY induces multiplication by −1 in eK.SY.. If

we view e K.SY. as hY;Ui, then switching ends of SY corresponds to the map U!U

sending a matrix to its inverse. We noted in the proof of Proposition 2.2 that the group

operation in K.SY. is the same as the operation induced by the product in U , so the

result follows. tu

Proposition 2.8. The exact sequence at the right is an

exact sequence of eK

_.X. modules, with the maps homomorphisms

of eK

_.X. modules. A

K (X )

»

K ( )

»

K( )

»

X,A ¡¡¡¡!

¡!

¡!

¤ ¤

¤

The eK

_.X. module structure on eK

_.A. is defined by ___ . i_._._ where i is the

inclusion A>X and the product on the right side of the equation is multiplication

in the ring eK

_.A.. To define the module structure on eK

_.X;A., observe that the

diagonal map X!X ^ X induces a well-defined quotient map X=A!X ^ X=A, and

this leads to a product eK

_.X.eK

_.X;A.!eK

_.X;A..

Proof: To see that the maps in the exact sequence are module homomorphisms we

look at the diagram

K(S SA) K( ( ))

» »

j j X/ K(S X )

» j K(S A)

» ¡¡¡¡¡! S A ¡¡¡¡¡! ¡¡¡¡¡! j

K( S SA) K( ( ))

» »

SiX j j X/ K( S X )

» j K( S A)

» ¡¡¡! S A ¡¡¡! ¡¡¡! j

K(S SA) K( ( ))

» »

i j X/ K(S X )

» j K(S A)

» i +j ¡¡¡¡¡! S + A ¡¡¡¡¡! i + ¡¡¡¡¡! i +j

^ S Xi ^ S Xi ^ S Xi ^

¡¡!

¡¡!

¡¡!

¡¡!

¡¡!

¡¡!

¡¡!

¡¡!

where the vertical maps between the first two rows are external product with a fixed

element of e K.SiX. and the vertical maps between the second and third rows are

induced by diagonal maps. What we must show is that the diagram commutes. For the

upper two rows this follows from naturality of external product since the horizontal

maps are induced by maps between spaces. The lower two rows are induced from

suspensions of maps between spaces,

A S X^ X^X/A X^X X^A ¡¡¡! ¡¡¡! ¡¡¡!

A S X/A X A ¡¡¡¡! ¡¡¡¡! ¡¡¡¡!

¡¡!

¡¡!

¡¡!

¡¡!

so it suffices to show this diagram commutes up to homotopy. This is obvious for the

middle and right squares. The left square can be rewritten

A S X^ X^ X ¡¡¡!

A S X CA ¡¡¡¡!

¡¡!

¡¡!

[

( [CA)

where the horizontal maps collapse the copy of X in X[CA to a point, the left vertical

map sends .a; s. 2 SA to .a; a; s. 2 X ^ SA, and the right vertical map sends x 2 X

38 Chapter 2 Complex K–Theory

to .x;x. 2 X [ CA and .a; s. 2 CA to .a; a; s. 2 X ^ CA. Commutativity is then

obvious. tu

It is often convenient to have an unreduced version of the groups eK

n.X., and this

can easily be done by the simple device of defining Kn.X. to be eK

n.X.. where X. is

X with a disjoint basepoint labeled ‘+’ adjoined. For n . 0 this is consistent with the

relation between K and eK

since K0.X. . eK

0.X.. . e K.X.. . Ker.K.X..!K.... .

K.X.. For n . 1 this definition yields K1.X. . eK

1.X. since S.X.. ' SX _ S1 and

e K.SX_S1. _ e K.SX._ e K.S1. _ e K.SX. since e K.S1. . 0. For a pair .X;A. with A . ;

one defines Kn.X;A. . eK

n.X;A., and then the six-term long exact sequence is valid

also for unreduced groups. When A . ; this remains valid if we interpret X=; as

X. .

Since X.^Y. . .X_Y.., the external product eK

_.X.eK

_.Y .!eK

_.X^Y. gives

a product K_.X.K_.Y .!K_.X_Y.. Taking X . Y and composing with the map

K_.X_X.!K_.X. induced by the diagonal map X!X_X, x,.x;x., we get a

product K_.X.K_.X.!K_.X. which makes K_.X. into a ring.

There is a relative product Ki.X;A.Kj.Y ; B.!Ki.j.X_Y;X_B [ A_Y. defined

as the external product e K.Öi.X=A.. e K.Öj.Y =B.. -!e K.Öi.j.X=A ^ Y=B.., using

the natural identification .X_Y.=.X_B [ A_Y. . X=A^Y=B. This works when

A . ; since we interpret X=; as X. , and similarly if Y . ;. Via the diagonal map

we obtain also a product Ki.X;A.Kj.X; B.!Ki.j.X;A [ B..

With these definitions the preceding two propositions are valid also for unreduced

K–groups.

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