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2.2. Bott Periodicity

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The form of the Bott periodicity theorem we shall prove is the assertion that the

external product map _ : K.X.K.S2.!K.X_S2. is an isomorphism for all compact

Hausdorff spaces X. The present section will be devoted entirely to the proof of this

theorem. Nothing in the proof will be used elsewhere in the book except in the proof

of Bott periodicity for real K–theory in Chapter 4, so the reader who wishes to defer a

careful reading of the proof may skip ahead to x2.3 without any loss of continuity.

The main work in proving the theorem will be to prove the surjectivity of _.

Injectivity will then be proved by a closer examination of the surjectivity argument.

Clutching Functions

From the classification of vector bundles over spheres in x1.2 we know that vector

bundles over S2 correspond exactly to homotopy classes of maps S1!GLn.C., which

Bott Periodicity Section 2.2 39

we called clutching functions. To prove the Bott periodicity theorem we will generalize

this construction, creating vector bundles over X_S2 by gluing together two vector

bundles over X_D2 by means of a generalized clutching function.

We begin by describing this more general clutching construction. Given a vector

bundle p : E!X, let f : E_S1!E_S1 be an automorphism of the product vector

bundle p_11 :E_S1!X_S1 . Thus for each x 2 X and z 2 S1 , f specifies an

isomorphism f.x;z. :p−1.x.!p−1.x.. From E and f we construct a vector bundle

over X_S2 by taking two copies of E_D2 and identifying the subspaces E_S1 via

f . We write this bundle as .E; f ., and call f a clutching function for .E; f .. If

ft:E_S1!E_S1 is a homotopy of clutching functions, then .E; f0. _ .E; f1. since

from the homotopy ft we can construct a vector bundle over X_S2_I restricting

to .E; f0. and .E; f1. over X_S2_f0g and X_S2_f1g. From the definitions it is

evident that .E1; f1._.E2; f2. _ .E1

_E2; f1 _f2..

Here are some examples of bundles built using clutching functions:

1. .E; 11. is the external product E _1 . _.E; 1., or equivalently the pullback of E via

the projection X_S2!X.

2. Taking X to be a point, then we showed in Example 1.12 that .1; z. _ H where ‘1’

is the trivial line bundle over X, ‘z’ means scalar multiplication by z 2 S1 _ C, and

H is the canonical line bundle over S2 . CP1 . More generally we have .1; zn. _ Hn,

the n fold tensor product of H with itself. The formula .1; zn. _ Hn holds also for

negative n if we define H−1 . .1; z−1., which is justified by the fact that HH−1 _ 1.

3. .E; zn. _ E _ Hn . _.E;Hn. for n 2 Z.

4. Generalizing this, .E; znf. _ .E; f . bH

n where bH

n denotes the pullback of Hn

via the projection X_S2!S2 .

Every vector bundle E0!X_S2 is isomorphic to .E; f . for some E and f. To

see this, let the unit circle S1 _ C [ f1g . S2 decompose S2 into the two disks D0

and D1, and let E_ for _ . 0;1 be the restriction of E0 over X_D_ , with E the

restriction of E0 over X_f1g. The projection X_D_!X_f1g is homotopic to the

identity map of X_D_ , so the bundle E_ is isomorphic to the pullback of E by the

projection, and this pullback is E_D_ , so we have an isomorphism h_ : E_!E_D_ .

Then f . h0h−1

1 is a clutching function for E0 .

We may assume a clutching function f is normalized to be the identity over

X_f1g since we may normalize any isomorphism h_ : E_!E_D_ by composing it

over each X_fzg with the inverse of its restriction over X_f1g. Any two choices of

normalized h_ are homotopic through normalized h_ ’s since they differ by a map g_

from D_ to the automorphisms of E, with g_.1. . 11, and such a g_ is homotopic

to the constant map 11 by composing it with a deformation retraction of D_ to 1.

Thus any two choices f0 and f1 of normalized clutching functions are joined by a

homotopy of normalized clutching functions ft .

40 Chapter 2 Complex K–Theory

The strategy of the proof will be to reduce from arbitrary clutching functions

to successively simpler clutching functions. The first step is to reduce to Laurent

polynomial clutching functions, which have the form `.x; z. .

jij_n ai.x.zi where

ai : E!E restricts to a linear transformation ai.x. in each fiber p−1.x.. We call

such an ai an endomorphism of E since the linear transformations ai.x. need not

be invertible, though their linear combination

i ai.x.zi is since clutching functions

are automorphisms.

Proposition 2.9. Every vector bundle .E; f . is isomorphic to .E; `. for some Laurent

polynomial clutching function `. Laurent polynomial clutching functions `0 and

`1 which are homotopic through clutching functions are homotopic by a Laurent

polynomial clutching function homotopy `t.x; z. .

i ai.x; t.zi .

Before beginning the proof we need a lemma. For a compact space X we wish

to approximate a continuous function f :X_S1!C by Laurent polynomial functions

jnj_N an.x.zn .

jnj_N an.x.ein_ , where each an is a continuous function X!C.

Motivated by Fourier series, we set

an.x. . 1

f.x;_.e−in_ d_

For positive real r let u.x; r; _. .

n2Z an.x.r jnjein_ . For fixed r < 1, this series

converges absolutely and uniformly as .x; _. ranges over X_S1 , by comparison with

the geometric series

n rn , since compactness of X_S1 implies that jf.x;_.j is

bounded and hence also jan.x.j . If we can show that u.x; r; _. approaches f.x;_.

uniformly in x and _ as r goes to 1, then sums of finitely many terms in the series

for u.x; r; _. with r near 1 will give the desired approximations to f by Laurent

polynomial functions.

Lemma 2.10. As r!1, u.x; r; _.!f.x;_. uniformly in x and _.

Proof: For r < 1 we have

u.x; r; _. .

n.−1

r jnjein._−t.f.x; t.dt

n.−1

r jnjein._−t.f.x; t.dt

where the order of summation and integration can be interchanged since the series

in the latter formula converges uniformly, by comparison with the geometric series

n rn . Define the Poisson kernel

P.r;'. . 1

n.−1

r jnjein' for 0 _ r < 1 and ' 2 R

Bott Periodicity Section 2.2 41

Then u.x; r; _. .

S1 P.r;_ −t.f.x; t.dt . By summing the two geometric series for

positive and negative n in the formula for P.r;'., one computes that

P.r;'. . 1

_ 1−r2

1−2r cos' . r 2

Three basic facts about P.r;'. which we shall need are:

(a) As a function of ', P.r;'. is even, of period 2_ , and monotone decreasing

on .0;_., since the same is true of cos' which appears in the denominator of

P.r;'. with a minus sign. In particular we have P.r;'. _ P.r;_. > 0 for all

r < 1.

(b)

S1 P.r;'.d' . 1 for each r < 1, as one sees by integrating the series for

P.r;'. term by term.

0 and the denominator approaches 2 − 2 cos' . 0.

Now to show uniform convergence of u.x; r; _. to f.x;_. we first observe that, using

(b), we have

u.x; r; _.−f.x;_.

___

P.r;_ −t.f.x; t.dt −

P.r;_ −t.f.x;_.dt

___

P.r;_ −t.

f.x; t.−f.x;_.

Given " > 0, there exists a _ > 0 such that jf.x; t.−f.x;_.j < " for jt −_j < _ and

all x, since f is uniformly continuous on the compact space X_S1 . Let I_ denote

the integral

P.r;_ −t. jf.x; t.−f.x;_.j dt over the interval jt −_j _ _ and let I0

denote this integral over the rest of S1 . Then we have

jt−_j__

P.r;_ −t. " dt _ "

P.r;_ −t.dt . "

By (a) the maximum value of P.r;_ −t. on jt − _j _ _ is P.r;_.. So

_P.r;_.

jf.x; t.−f.x;_.j dt

The integral here has a uniform bound for all x and _ since f is bounded. Thus by (c)

we can make I0

_ " by taking r close enough to 1. Therefore ju.x; r; _.−f.x;_.j _

.I0

_ 2". tu

Proof of Proposition 2.9: Choosing a Hermitian inner product on E, the endomorphisms

of E_S1 form a vector space End.E_S1. with a norm k_k . supjvj.1 j_.v.j .

The triangle inequality holds for this norm, so balls in End.E_S1. are convex. The

subspace Aut.E_S1. of automorphisms is open in the topology defined by this norm

since it is the preimage of .0;1. under the continuous map End.E_S1.!.0;1.,

_,inf.x;z.2X_S1 j det._.x; z..j . Thus to prove the first statement of the lemma it

will suffice to show that Laurent polynomials are dense in End.E_S1., since a sufficiently

close Laurent polynomial approximation ` to f will then be homotopic to

42 Chapter 2 Complex K–Theory

f via the linear homotopy t` . .1 − t.f through clutching functions. The second

statement follows similarly by approximating a homotopy from `0 to `1 , viewed as

an automorphism of E_S1_I , by a Laurent polynomial homotopy `0t

, then combining

this with linear homotopies from `0 to `0

0 and `1 to `0

1 to obtain a homotopy `t

from `0 to `1 .

To show that every f 2 End.E_S1. can be approximated by Laurent polynomial

endomorphisms, first choose open sets Ui covering X together with isomorphisms

hi :p−1.Ui.!Ui

_Cni . We may assume hi takes the chosen inner product in p−1.Ui.

to the standard inner product in Cni , by applying the Gram-Schmidt process to h−1

of the standard basis vectors. Let f'i

g be a partition of unity subordinate to fUi

and let Xi be the support of 'i , a compact set in Ui . Via hi , the linear maps f.x;z.

for x 2 Xi can be viewed as matrices. The entries of these matrices define functions

_S1!C. By the lemma we can find Laurent polynomial matrices `i.x; z. whose

entries uniformly approximate those of f.x;z. for x 2 Xi . It follows easily that `i

approximates f in the k _ k norm. From the Laurent polynomial approximations `i

over Xi we form the convex linear combination ` .

i'i`i , a Laurent polynomial

approximating f over all of X_S1 . tu

A Laurent polynomial clutching function can be written ` . z−mq for a polynomial

clutching function q, and then we have .E; `. _ .E; q. bH

−m. The next step is

to reduce polynomial clutching functions to linear clutching functions.

Proposition 2.11. If q is a polynomial clutching function of degree at most n, then

.E; q._.nE; 11. _ ..n . 1.E; Lnq. for a linear clutching function Lnq.

Proof: Let q.x; z. . an.x.zn . ___.a0.x.. Consider the matrices

BBBBBBBB@

1 −z 0 ___ 0 0

0 1 −z ___ 0 0

0 0 1 ___ 0 0

...

0 0 0 ___ 1 −z

an an−1 an−2 ___ a1 a0

CCCCCCCCA and

BBBBBBBB@

1 0 0 ___ 0 0

0 1 0 ___ 0 0

0 0 1 ___ 0 0

...

0 0 0 ___ 1 0

0 0 0 ___ 0 q

CCCCCCCCA

which define endomorphisms of .n . 1.E . We can pass from the first matrix to the

second by a sequence of elementary row and column operations in the following way.

In the first matrix, add z times the first column to the second column, then z times

the second column to the third, and so on. This produces all 0’s above the diagonal,

and the polynomial q in the lower right corner. Then for each i _ n, subtract the

appropriate multiple of the ith row from the last row.

The second matrix is a clutching function for .nE; 11._.E; q.. The first matrix

has the same determinant as the second, hence is also invertible and is therefore an

automorphism of .n.1.E for each z 2 S1 , determining a clutching function which we

denote by Lnq. Since Lnq has the form A.x.z.B.x., it is a linear clutching function.

Bott Periodicity Section 2.2 43

The two displayed matrices define homotopic clutching functions since the elementary

row and column operations can be achieved by continuous one-parameter families of

such operations. For example the first operation can be achieved by adding tz times

the first column to the second, with t ranging from 0 to 1. Since homotopic clutching

functions produce isomorphic bundles, we obtain an isomorphism .E; q._.nE; 11. _

..n . 1.E; Lnq.. tu

Linear Clutching Functions

For linear clutching functions a.x.z . b.x. we have the following key fact:

Proposition 2.12. Given a bundle .E; a.x.z.b.x.., there is a splitting E _ E._E−

with .E; a.x.z . b.x.. _ .E.; 11._.E−; z..

Proof: The first step is to reduce to the case that a.x. is the identity for all x.

Consider the expression:

._. .1. tz.

a.x.

z . t

1 . tz

. b.x.

a.x. . tb.x.

z . ta.x. . b.x.

When t . 0 this equals a.x.z . b.x.. For 0 _ t < 1, ._. defines an invertible

linear transformation since the left-hand side is obtained from a.x.z . b.x. by first

applying the substitution z,.z . t.=.1 . tz. which takes S1 to itself (because if

jzj . 1 then j.z.t.=.1.tz.j . jz.z.t.=.1.tz.j . j.1.tz.=.1.tz.j . jw=wj . 1),

and then multiplying by the nonzero scalar 1.tz. Therefore ._. defines a homotopy

of clutching functions as t goes from 0 to t0 < 1. In the right-hand side of ._. the

term a.x. . tb.x. is invertible for t . 1 since it is the restriction of a.x.z . b.x.

to z . 1. Therefore a.x. . tb.x. is invertible for t . t0 near 1, as the continuous

function t,infx2X

det.a.x. . tb.x..

is nonzero for t . 1, hence also for t near

1. Now we use the simple fact that .E; fg. _ .E; f . for any isomorphism g : E!E.

This allows us to replace the clutching function on the right-hand side of ._. by the

clutching function z . .t0a.x. . b.x...a.x. . t0b.x..−1 , reducing to the case of

clutching functions of the form z . b.x..

Since z . b.x. is invertible for all x, b.x. has no eigenvalues on the unit circle

S1 .

Lemma 2.13. Let b : E!E be an endomorphism having no eigenvalues on the unit

circle S1 . Then there are unique subbundles E. and E− of E such that:

(a) E . E._E− .

(b) b.E_. _ E_ .

E. all lie outside S1 and the eigenvalues of bjj

E− all lie

inside S1 .

Proof: Consider first the algebraic situation of a linear transformation T :V!V with

characteristic polynomial q.t.. Assuming q.t. has no roots on S1 , we may factor q.t.

44 Chapter 2 Complex K–Theory

in C.t. as q..t.q−.t. where q..t. has all its roots outside S1 and q−.t. has all its

roots inside S1 . Let V_ be the kernel of q_.T . :V!V . Since q. and q− are relatively

prime in C.t., there are polynomials r and s with rq..sq− . 1. From q..T .q−.T . .

q.T. . 0, we have Im q−.T . _ Ker q..T ., and the opposite inclusion follows from

r.T.q..T . . q−.T .s.T . . 11. Thus Ker q..T . . Im q−.T ., and similarly Ker q−.T . .

Im q..T .. From q..T .r.T . . q−.T .s.T . . 11 we see that Im q..T . . Im q−.T . . V ,

and from r.T.q..T . . s.T.q−.T . . 11 we deduce that Ker q..T . \ Ker q−.T . . 0.

Hence V . V._V− . We have T.V_. _ V_ since q_.T ..v. . 0 implies q_.T ..T.v.. .

T.q_.T ..v.. . 0. All eigenvalues of Tjj

V_ are roots of q_ since q_.T . . 0 on V_.

Thus conditions (a)–(c) hold for V. and V− .

To see the uniqueness of V. and V− satisfying (a)–(c), let q0

_ be the characteristic

polynomial of Tjj

V_, so q . q0

.q0

−. All the linear factors of q0

_ must be factors of

q_ by condition (c), so the factorizations q . q0

.q0

− and q . q.q− must coincide up

to scalar factors. Since q0

_.T . is identically zero on V_ , so must be q_.T ., hence

V_ _ Ker q_.T .. Since V . V._V− and V . Ker q..T ._ Ker q−.T ., we must have

V_ . Ker q_.T .. This establishes the uniqueness of V_ .

As T varies continuously through linear transformations without eigenvalues on

S1 , its characteristic polynomial q.t. varies continuously through polynomials without

roots in S1 . In this situation we assert that the factors q_ of q vary continuously

with q, assuming that q, q. , and q− are normalized to be monic polynomials. To

see this we shall use the fact that for any circle C in C disjoint from the roots of q,

the number of roots of q inside C , counted with multiplicity, equals the degree of

the map :C!S1 , .z. . q.z.=jq.z.j . To prove this fact it suffices to consider the

case of a small circle C about a root z . a of multiplicity m, so q.t. . p.t..t −a.m

with p.a. . 0. The homotopy

s.z. . p.sa . .1 − s.z..z − a.m

jp.sa . .1 − s.z..z − a.m j

gives a reduction to the case .t − a.m, where it is clear that the degree is m.

Thus for a small circle C about a root z . a of q of multiplicity m, small perturbations

of q produce polynomials q0 which also have m roots a1; ___; am inside

C , so the factor .z − a.m of q becomes a factor .z − a1. ___ .z − am. of the nearby

q0 . Since the ai ’s are near a, these factors of q and q0 are close, and so q0

_ is close

to q_ .

Next we observe that as T varies continuously through transformations without

eigenvalues in S1 , the splitting V . V._V− also varies continuously. To see this,

recall that V. . Im q−.T . and V− . Im q..T .. Choose a basis v1; ___ ; vn for V such

that q−.T ..v1.; ___ ; q−.T ..vk. is a basis for V. and q..T ..vk.1.; ___ ; q..T ..vn. is

a basis for V− . For nearby T these vectors vary continuously, hence remain independent.

Thus the splitting V . Im q−.T ._ Im q..T . continues to hold for nearby T ,

and so the splitting V . V._V− varies continuously with T .

Bott Periodicity Section 2.2 45

It follows that the union E_ of the subspaces V_ in all the fibers V of E is a

subbundle, and so the proof of the lemma is complete. tu

To finish the proof of Proposition 2.12, note that the lemma gives a splitting

.E; z.b.x.. _ .E.; z.b..x.._.E−; z.b−.x.. where b. and b− are the restrictions

of b. Since b..x. has all its eigenvalues outside S1 , the formula tz . b..x. for

0 _ t _ 1 defines a homotopy of clutching functions from z.b..x. to b..x.. Hence

.E.; z.b..x.. _ .E.; b..x.. _ .E.; 11.. Similarly, z.tb−.x. defines a homotopy of

clutching functions from z . b−.x. to z, so .E−; z .b−.x.. _ .E−; z.. tu

For future reference we note that the splitting .E;az . b. _ .E.; 11._.E−; z.

constructed in the proof of Proposition 2.12 preserves direct sums, in the sense that

the splitting for a sum .E1

_E2; .a1z.b1._.a2z.b2.. has .E1

_E2._ . .E1.__.E2._ .

This is because the first step of reducing to the case a . 11 clearly respects sums, and

the uniqueness of the _ splitting in Lemma 2.13 guarantees that it preserves sums.

Conclusion of the Proof

The preceding propositions imply that in K.X_S2. we have

.E; f . . .E; z−mq.

. .E; q.bH

−m

. ..n . 1.E; Lnq.bH

−m − .nE; 11.bH

−m

. ...n . 1.E..; 11.bH

−m . ...n . 1.E.−; z.bH

−m −.nE; 11.bH

−m

. ..n . 1.E..H−m . ..n . 1.E.−H1−m − nEH−m

This last expression is in the image of _ : K.X.K.S2.!K.X_S2.. Since every vector

bundle over X_S2 has the form .E; f ., it follows that _ is surjective.

To show _ is injective we shall construct _ : K.X_S2.!K.X.K.S2. such that

__ . 11. The idea will be to define _..E; f .. as some linear combination of terms

EHk and ..n . 1.E._Hk which is independent of all choices.

To investigate the dependence of the terms in the formula for .E; f . displayed

above on m and n we first derive the following two formulas, where deg q _ n:

(1) ..n . 2.E; Ln.1q. _ ..n . 1.E; Lnq._.E; 11.

(2) ..n . 2.E; Ln.1.zq.. _ ..n . 1.E; Lnq._.E; z.

The matrix representations of Ln.1q and Ln.1.zq. are:

BBBBBB@

1 −z 0 ___ 0

0 1 −z ___ 0

... ...

...

0 0 0 1 −z

0 an an−1 ___ a0

CCCCCCA

and

BBBBBB@

1 −z 0 ___ 0 0

0 1 −z ___ 0 0

...

0 0 0 ___ 1 −z

an an−1 an−2 ___ a0 0

CCCCCCA

46 Chapter 2 Complex K–Theory

In the first matrix we can add z times the first column to the second column to

eliminate the −z in the first row, and then the first row and column give the summand

.E; 11. while the rest of the matrix gives ..n . 1.E; Lnq.. This proves (1). Similarly,

in the second matrix we add z−1 times the last column to the next-to-last column to

make the −z in the last column have all zeros in its row and column, which gives the

splitting in (2) since .E;−z. _ .E; z., the clutching function −z being the composition

of the clutching function z with the automorphism −1 of E.

In view of the appearance of the correction terms .E; 11. and .E; z. in (1) and (2),

it will be useful to know the ‘_’ splittings for these two bundles:

(3) For .E; 11. the summand E− is 0 and E. . E.

(4) For .E; z. the summand E. is 0 and E− . E.

Statement (4) is obvious from the definitions since the clutching function z is already

in the form z . b.x. with b.x. . 0, so 0 is the only eigenvalue of b.x. and hence

E. . 0. To obtain (3) we first apply the procedure at the beginning of the proof of

Proposition 2.12 which replaces a clutching function a.x.z . b.x. by the clutching

function z . .t0a.x. . b.x...a.x. . t0b.x..−1 with 0 < t0 < 1. Specializing to the

case a.x.z . b.x. . 11 this yields z . t−1

0 11. Since t−1

0 11 has only the one eigenvalue

t−1

0 > 1, we have E− . 0.

Formulas (1) and (3) give ..n . 2.E.− _ ..n . 1.E.− , using the fact that the

_ splitting preserves direct sums. So the ‘minus’ summand is independent of n.

Suppose we define

_..E; z−mq.. . ..n . 1.E.−.H − 1. . EH−m 2 K.X.K.S2.

for n _ deg q. We claim that this is well-defined. We have just noted that ‘minus’

summands are independent of n, so _..E; z−mq.. does not depend on n. To see

that it is independent of m we must see that it is unchanged when z−mq is replaced

by z−m−1.zq.. By (2) and (4) we have the first of the following equalities:

_..E; z−m−1.zq... . ..n . 1.E.−.H − 1. . E.H − 1. . EH−m−1

. ..n . 1.E.−.H − 1. . E.H−m − H−m−1. . EH−m−1

. ..n . 1.E.−.H − 1. . EH−m

. _..E; z−mq..

To obtain the second equality we use the calculation of the ring K.S2. in Example 2.3,

where we derived the relation .H − 1.2 . 0 which implies H.H − 1. . H − 1 and

hence H − 1 . H−m − H−m−1 for all m. The third and fourth equalities are evident.

Another choice which might perhaps affect the value of _..E; z−mq.. is the constant

t0 < 1 in the proof of Proposition 2.12. This could be any number sufficiently

close to 1, so varying t0 gives a homotopy of the endomorphism b in Lemma 2.13.

Bott Periodicity Section 2.2 47

This has no effect on the _ splitting since we can apply Lemma 2.13 to the endomorphism

of E_I given by the homotopy. Hence the choice of t0 does not affect

_..E; z−mq...

It remains see that _..E; z−mq.. depends only on the bundle .E; z−mq., not on

the clutching function z−mq for this bundle. We showed that every bundle over X_S2

has the form .E; f . for a normalized clutching function f which was unique up to homotopy,

and in Proposition 2.10 we showed that Laurent polynomial approximations

to homotopic f ’s are Laurent-polynomial-homotopic. If we apply Propositions 2.11

and 2.12 over X_I with a Laurent polynomial homotopy as clutching function, we

conclude that the two bundles ..n . 1.E.− over X_f0g and X_f1g are isomorphic.

This finishes the verification that _..E; z−mq.. is well-defined.

It is easy to check through the definitions to see that _ takes sums to sums since

Ln.q1 _ q2. . Lnq1 _ Lnq2 and, as previously noted, the _ splitting in Proposition

2.12 preserves sums. So _ extends to a homomorphism K.X_S2.!K.X.K.S2..

The last thing to verify is that __ . 11. The group K.S2. is generated by 1 and H,

so in view of the relation H .H−1 . 2, which follows from .H − 1.2 . 0, we see that

K.S2. is also generated by 1 and H−1 . Thus it suffices to show __ . 11 on elements

EH−m for m _ 0. We have __.EH−m. . _..E; z−m.. . E−.H −1..EH−m .

EH−m since E− . 0, the polynomial q being 11 so that (3) applies.

This completes the proof of Bott Periodicity. tu

Elementary Applications

With the calculation eK

_.Sn. _ Z completed, it would be possible to derive many

of the same applications that follow from the corresponding calculation for ordinary

homology or cohomology, as in [AT]. For example:

— There is no retraction of Dn onto its boundary Sn−1 , since this would mean that

the identity map of eK

_.Sn−1. factored as eK

_.Sn−1.!eK

_.Dn.!eK

_.Sn−1., but

the middle group is trivial.

— The Brouwer fixed point theorem, that for every map f :Dn!Dn there is a point

x 2 Dn with f.x. . x. For if not then it is easy to construct a retraction of Dn

onto Sn−1 .

— The notion of degree for maps f : Sn!Sn , namely the integer d.f . such that the

induced homomorphism f _ : eK

_.Sn.!eK

_.Sn. is multiplication by d.f .. Reasoning

as in Proposition 2.2, one sees that d is a homomorphism _n.Sn.!Z.

In particular a reflection has degree −1 and hence the antipodal map of Sn ,

which is the composition of n.1 reflections, has degree .−1.n.1 since d.fg. .

d.f .d.g.. Consequences of this include the fact that an even-dimensional sphere

has no nonvanishing vector fields.

However there are some things homology can do that K–theory cannot do in such an

elementary way, since eK

_.Sn. can distinguish even-dimensional spheres from odd48

Chapter 2 Complex K–Theory

dimensional spheres but it cannot distinguish between different even dimensions or

different odd dimensions. This, together with the fact that we have so far only defined

K–theory for compact spaces, prevents us from obtaining some of the other classical

applications of homology such as Brouwer’s theorems on invariance of dimension

and invariance of domain, or the Jordan curve theorem and its higher-dimensional

analogs.

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