Appendix: Paracompactness

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A Hausdorff space X is paracompact if for each open cover fU_

g of X there

is a partition of unity f'_

g subordinate to the cover. This means that the '_ ’s are

maps X!I such that each '_ has support (the closure of the set where '_ . 0)

contained in some U_ , each x 2 X has a neighborhood in which only finitely many

'_ ’s are nonzero, and

P

_'_

. 1. An equivalent definition which is often given is

that X is Hausdorff and every open cover of X has a locally finite open refinement.

The first definition clearly implies the second by taking the cover f'−1

_ .0; 1.g. For the

converse, see [Dugundji] or [Lundell-Weingram]. It is the former definition which is

most useful in algebraic topology, and the fact that the two definitions are equivalent

is rarely if ever needed. So we shall use the first definition.

A paracompact space X is normal, for let A1 and A2 be disjoint closed sets in X,

and let f'_

g be a partition of unity subordinate to the cover fX −A1;X−A2g. Let 'i

be the sum of the '_ ’s which are nonzero at some point of Ai . Then 'i.Ai. . 1, and

Appendix: Paracompactness 25

'1 . '2 _ 1 since no '_ can be a summand of both '1 and '2 . Hence '−1

1 .1=2; 1.

and '−1

2 .1=2; 1. are disjoint open sets containing A1 and A2 , respectively.

Most of the spaces one meets in algebraic topology are paracompact, including:

(1) compact Hausdorff spaces

(2) unions of increasing sequences X1 _ X2 _ ___ of compact Hausdorff spaces Xi ,

with the weak or direct limit topology (a set is open iff it intersects each Xi in an

open set)

(3) CW complexes

(4) metric spaces

Note that (2) includes (3) for CW complexes with countably many cells, since such

a CW complex can be expressed as an increasing union of finite subcomplexes. Using

(1) and (2), it can be shown that many manifolds are paracompact, for example Rn .

The next three propositions verify that the spaces in (1), (2), and (3) are paracompact.

Proposition 1.16. A compact Hausdorff space X is paracompact.

Proof: Let fU_

g be an open cover of X. Since X is normal, each x 2 X has an open

neighborhood Vx with closure contained in some U_ . By Urysohn’s lemma there is a

map 'x :X!I with 'x.x. . 1 and 'x.X −Vx. . 0. The open cover f'−1

x .0; 1.g of

X contains a finite subcover, and we relabel the corresponding 'x ’s as '_ ’s. Then

P

_'_.x. > 0 for all x, and we obtain the desired partition of unity subordinate to

fU_

g by normalizing each '_ by dividing it by

P

_'_ . tu

Proposition 1.17. If X is the direct limit of an increasing sequence X1 _ X2 _ ___

of compact Hausdorff spaces Xi , then X is paracompact.

Proof: A preliminary observation is that X is normal. To show this, it suffices to find

a map f :X!I with f .A. . 0 and f.B. . 1 for any two disjoint closed sets A and B.

Such an f can be constructed inductively over the Xi ’s, using normality of the Xi ’s.

For the induction step one has f defined on the closed set Xi

[.A\Xi.1.[.B\Xi.1.

and one extends over Xi.1 by Tietze’s theorem.

To prove that X is paracompact, let an open cover fU_

g be given. Since Xi is

compact Hausdorff, there is a finite partition of unity f'ij

g on Xi subordinate to

fU_

\ Xi

g. Using normality of X, extend each 'ij to a map 'ij :X!I with support

in the same U_ . Let _i

.

P

j 'ij . This sum is 1 on Xi , so if we normalize each 'ij

by dividing it by maxf1=2;_i

g, we get new maps 'ij with _i

. 1 in a neighborhood

Vi of Xi . Let ij

. maxf0;'ij

−

P

k<i _k

g. Since 0 _ ij

_ 'ij , the collection f ij

g

is subordinate to fU_

g. In Vi all kj ’s with k > i are zero, so each point of X has a

neighborhood in which only finitely many ij ’s are nonzero. For each x 2 X there

is a ij with ij.x. > 0, since if 'ij.x. > 0 and i is minimal with respect to this

26 Chapter 1 Vector Bundles

condition, then ij.x. . 'ij.x.. Thus when we normalize the collection f ij

g by

dividing by

P

i;j ij we obtain a partition of unity on X subordinate to fU_

g. tu

Proposition 1.18. Every CW complex is paracompact.

Proof: Given an open cover fU_

g of a CW complex X, suppose inductively that we

have a partition of unity f'_

g on Xn subordinate to the cover fU_

\ Xng. For a

cell en.1

 with characteristic map Ø :Dn.1!X, f'_Ø

g is a partition of unity on

Sn . @Dn.1 . Since Sn is compact, only finitely many of these compositions '_Ø can

be nonzero, for fixed  . We extend these functions '_Ø over Dn.1 by the formula

_".r.'_Ø.x. using ‘spherical coordinates’ .r ;x. 2 I_Sn on Dn.1 , where _" : I!I

is 0 on .0;1−". and 1 on.1−"=2;1.. If " ." is chosen small enough, these extended

functions _"'_Ø will be subordinate to the cover fØ−1

 .U_.g. Let f j

g be a finite

partition of unity on Dn.1 subordinate to fØ−1

 .U_.g. Then f_"'_Ø; .1−_". j

g is

a partition of unity on Dn.1 subordinate to fØ−1

 .U_.g. This partition of unity extends

the partition of unity f'_Ø

g on Sn and induces an extension of f'_

g to a partition

of unity defined on Xn[en.1

 and subordinate to fU_

g. Doing this for all .n.1. cells

en.1

 gives a partition of unity on Xn.1 . The local finiteness condition continues to

hold since near a point in Xn only the extensions of the '_ ’s in the original partition

of unity on Xn are nonzero, while in a cell en.1

 the only other functions that can be

nonzero are the ones coming from j ’s. After we make such extensions for all n,

we obtain a partition of unity defined on all of X and subordinate to fU_

g. tu

Here is a technical fact about paracompact spaces that is occasionally useful:

Lemma 1.19. Given an open cover fU_

g of the paracompact space X, there is a

countable open cover fVk

g such that each Vk is a disjoint union of open sets each

contained in some U_ , and there is a partition of unity f'k

g with 'k supported in

Vk .

Proof: Let f'_

g be a partition of unity subordinate to fU_

g. For each finite set S of

functions '_ let VS be the subset of X where all the '_ ’s in S are strictly greater

than all the '_ ’s not in S . Since only finitely many '_ ’s are nonzero near any x 2 X,

VS is defined by finitely many inequalities among '_ ’s near x, so VS is open. Also,

VS is contained in some U_ , namely, any U_ containing the support of any '_

2 S ,

since '_

2 S implies '_ > 0 on VS . Let Vk be the union of all the open sets VS such

that S has k elements. This is clearly a disjoint union. The collection fVk

g is a cover

of X since if x 2 X then x 2 VS where S . f'_

j'_.x. > 0 g.

For the second statement, let f'

g be a partition of unity subordinate to the

cover fVk

g, and let 'k be the sum of those ' ’s supported in Vk but not in Vj for

j < k. tu

Exercises 27

Exercises

1. Show that a vector bundle E!X has k independent sections iff it has a trivial

k dimensional subbundle.

2. For a vector bundle E!X with a subbundle E0 _ E, construct a quotient vector

bundle E=E0!X.

3. Show that the orthogonal complement of a subbundle is independent of the choice

of inner product, up to isomorphism.

4. A vector bundle map is a commutative diagram

¡!

¡!

E ¡¡¡¡¡!E

B ¡¡¡¡¡!B f

f »

0

0

where the two vertical maps are vector bundle projections and e f is an isomorphism

on each fiber. Given such a bundle map, show that E0 is isomorphic to the pullback

bundle f _.E..

5. Show that the projection Vn.Rk.!Gn.Rk. is a fiber bundle with fiber O.n. by

showing that it is the orthonormal n frame bundle associated to the vector bundle

En.Rk.!Gn.Rk..

6. Show that the pair

􀀀

Gn.R1.;Gn.Rk.

_

is .k−n. connected, and deduce that Proposition

1.9 holds for finite-dimensional CW complexes. [The lowest-dimensional cell of

Gn.Rk.1. − Gn.Rk. is the e._. with _ . .1; 2; ___;n − 1; k . 1., and this cell has

dimension k . 1 − n.]

The idea of K–theory is to make the direct sum operation on real or complex vector

bundles over a fixed base space X into the addition operation in a group. There are

two slightly different ways of doing this, producing, in the case of complex vector

bundles, groups K.X. and e K.X. with K.X. _ e K.X._Z, and for real vector bundles,

groups KO.X. and gKO.X. with KO.X. _ gKO.X._Z. Complex K–theory turns out

to be somewhat simpler than real K–theory, so we concentrate on this case in the

present chapter.

Computing e K.X. even for simple spaces X requires some work. The case X . Sn

involves the Bott Periodicity Theorem, proved in x2.2. This is a deep theorem, so

it is not surprising that it has applications of real substance, and we give some of

these in x2.3, 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.

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