Algebraic Topology via Differential Geometry (London Mathematical Society Lecture Note Series (No. 99))

the following diagram commutative:

9.4 Definition: (i) (ii) (iii)

For any K-vector space

E,

set

S°(E) = K SX(E) = E S(E) = ® S k (E). kdN

9.5 Theorem: The multiplication in

T(E)

induces on

S(E)

the structure

of a commutative graded algebra, known as the symmetric algebra of If a commutative algebra

C and a linear map

E.

: E •+ C are

given, there exists a unique algebra homomorphism Z, : S ( E ) •+ C

which makes the following diagram commutative:

S(E) 9.6 Theorem: Let

E,F be two vector s p a c e s . There i s a canonical isomorphism

of graded algebras S(E © F) = S(E) 8 s(F) . 9.7 Theorem: I f

E i s finite-dimensional,

{e , . . . , e } a b a s i s 1 n of E, then S (E) i s isomorphic with the polynomial algebra K['X , . . . , X ] , the ' i n c l u s i o n ' E c K[X , . . . , X ] being the map which sends e onto X. I n

(cf. 7.3) .

n = dim E,

i

i

27

10

DUALITY This section- applies to the finite-dimensional case only.

Recall t h a t , given an n-dimensional K-vector space basis { E , , . ... ., ,Ee n }},, the dual basis . I n E = L(E,K) defined by

i s the basis

{e*,...,e*} In

E with of

<e. , e .> = e . (e .) = S. . 1 j 1 j i:J

for a l l

i,

j e {l,...,n},

10.1 Theorem: Let

E, F

where

6

i s the Kronecker symbol (cf. 1 . 2 ) .

be two ( f i n i t e - d i m e n s i o n a l )

vector s p a c e s . The

map

X : E

B F -»• L(E,F)

: u 8 y \* (x |+ y)

i s an isomorphism. 4

*

Proof: The two spaces E (8 F and L(E,F) have t h e same dimension. Furthermore , i f {e , . . . ,e } i s a b a s i s of E and {ru , • • • ,n } a b a s i s of in

F,

L(E,F)

1

has a b a s i s whose elements are the maps

n.

0

Clearly

if

defined by

k = i

otherwise

<\>, . = X(e. a n . ) -

10.2 Corollary:

$. .

p

•

Under the same hypotheses,

*

*

*

E

HF

S (E H F)

E

BF

S JL(E,F ) = L ( E , L ( F , K ) ) S B ( E , F ; K )

Proof: a L(E B F,K)

= ( E 81 F ) '

•

28

1 0 . 3 Theorem

: Let

integer

there i s a natural

k,

E

be a ( f i n i t e - d i m e n s i o n a l )

v e c t o r s p a c e . For any

isomorphism

k * k * A (E ) 35 A (E)

induced by a b i l i n e a r map (where

Det

k * k A (E ) x A (E) •+ K :

(u

stands

for determinant) :

A...A U^/X

A. . .A x^) \+ Det

Proof: The map * k k ( E )) x E -> K : (u , . . . , u , x , . . . , x ) f+ D Deett 1 k 1 Tc

x.'s,

hence the b i l i n e a r map as

Now l e t

e

ix

form a b a s i s o f

A...A

(£,.,.,£ } 1 n £.

ik

,

1

e j

l

< . . .< i

I k

E.

A . . . A e. \

1

iff

i

= j r

onto

of

k distinct

* (e.

l
A . . . A E. )

i s th'e

0

for a l l

integers

r = l,...,k

r

otherwise.

l e t (i , . . . , i ) i

, j P

Det

respect

The elements

and

I K

sequences

and with

sends

onto

On t h e o t h e r hand

u. ' s

< n

k A (E) : s e e 8 . 1 3 . By d e f i n i t i o n

e l e m e n t o f t h e d u a l b a s i s which

3

asserted.

be a b a s i s of

1 < i

()

(
i s 2 k - l i n e a r and skew-symmetric with respect to the to the

1 < i < k

(<e. , e . > ) = ^"p J q

£

(j, , •- . /j , ) l k

be two

{l,...,n}:

q 0

unless

(jnf--.#j.) i s a permutation of ( i , . . . , i ) . S i n c e we r e s t r i c t 1 k 1 k o u r s e l v e s t o t h e c a s e when l < i <...
Det l
( < e . , e . >) p q

t h e map

= 1 .

29

A (E ) -+ A (E)

: u

A . . .A n

f* (x

A. . .A x

h" Det

() )

induced

l<j
11

M

* (e. A...A e. ) , X 1 l k

i.e.

MODULES If the vector space

module

onto

E

over a field

over a commutative ring

A

K

is replaced by a

(cf. 2.7), the 'universal1 construc-

tions of Sections 7, 8, 9 can still be performed: one obtains successively the tensor algebra the exterior powers the symmetric powers

If bases for

M

A (M)

T(M) A (M) S (M)

and exterior algebra

A (M) ,

and symmetric algebra S(M).

is a free module of finite type, Theorem 8.13 which gives remains valid.

30

II

DIFFERENTIAL FORMS ON AN OPEN SUBSET OF

We f i r s t r e c a l l u n s y s t e m a t i c a l l y

0

(and without proofs)

ELEMENTARY RESULTS OF DIFFERENTIAL CALCULUS We r e s t r i c t ourselves to the finite-dimensional, Throughout the section,

K-vector spaces, E

a few

n, p

E, F

real case.

will be two finite-dimensional

their respective dimensions;

(for the topology induced by the Euclidian norm) ;

U an open subset of f : U -»• F;

x e U.

0.1 First Order The map f i s said to be differentiable (necessarily unique) linear map

< > j e L(E,F)

at

x

iff there exists a

such that for a l l

h e E with

x+h £ U .-

f(x+h) = f(x)

where

lim

+ 4>(h) + ||h|| e(h)

e(h) = 0. h-K) The linear map

written

$

is called the derivative

of

f

at

x.

I t is

f' (x) :

f (x+h) = f (x) + f' (x) .h + | | h | | e(h) .

(Note the traditional

f' (x) .h

instead of f' (x) (h) .)

Differentiability at x

clearly implies continuity at x.

If

f itself is linear, it is differentiable and f'(x) = f for

If

g : E -*• F

all x.

is bilinear, it is differentiable and

31

g'(x,y).(h,k) = g(x,k) + g(h,y) .

(Generalises to multilinear maps).

If

f : U -*- F

is a constant map, it is differentiable and its

derivative is the zero map. Note that for

n=0 all maps have to be constant, hence

differentiable with zero derivatives (cf. definition formula where necessarily h=0).

If

E and F have bases

respectively,

f'(x)

can be written as a matrix: the Jacobian matrix.

elements are the partial

3f. r-^- (x)

{en/...,e } and {ri, #•-•#!) } In 1 p

,

derivatives

of

1 < i

f

at

x,

Its

written

1 <j
OX ,

The adjective

'partial 1 comes from the fact that th'e one-element matrix

3f

(

i (x)) 3x.

is the derivative at

0

of

7T. ° f ° T o o . : 3R -> ]R i x 3

where

o. : K -> E : 1 [t E.

i s the j

canonical injection,

IT, : F -> ]R : TI, I* 6 the i canonical projection and T : E + E : y k y+x ik , x a t r a n s l a t i o n . (All t h i s simply means: consider the i component of f and only allow

x t o move p a r a l l e l to the j Using the isomorphism

coordinate axis.)

L(E,F) = E B F of

I . 1 0 . 1 , one can

write 3f. 1=1

]=1

Considerable use i s made of the sufficient differentiability: hood of

i f a l l the p a r t i a l derivatives of

x and are continuous

at

x,

f

condition f

for

exist in a neighbour-

i s differentiable a t x.

32 Formal Properties If

f, g : U •+ F

differentiable at x

and

(f+g)'(x) = f'(x) + g'(x) .

If furthermore differentiable at x

are dif ferentiable at x, (f+g) is

F

has a multiplicative structure,

fg is

with derivative given by

(fg) ' (x) .h = f (x) (g1 (x) .h) + (f'(x).h)g(x) .

If such that

f : U •*• F, g : V •*• G

f(U) c v,

G

where

V

is an open subset of F

i s a finite-dimensional vector space,

differentiable at x, g

is differentiable at f (x) ,

then

f is

g « f is

differentiable at x and

(g°f) " (x) = g1 (f (x)) o f (X)

.

0.2 Second and Higher Orders Suppose

f

is differentiable at all points of U. There

results a map f' : U •* E

8 F : x|* f (x)

which may or may not be

differentiable. If it is differentiable at x e U, its derivative is the second derivative of f

at x, denoted by f"(x)

:

f " (x) e E* B (E* B F) = (E* B E * ) B F = (E»E) * B F = L (T2 (E) ,F) .

Iterating the process wherever possible, one gets the q derivative of f, a map f q ) : U -> L(Tq(E),F). Here too partial derivatives are of great utility. For instance the

k

32f component of f" (x) . (e .fie.) i s denoted by r (x) i 3 ox. dx\ 1 2 """ 9 f 3f k 8 k I t i s easy to check t h a t (x) = T (• ) (x) , 8x.3x. 3x. 3x. Of fundamental importance is

and so on.

etc.

Schwartz' Lemma If

32f -—r 3x.3x.

32f and - — r 3x.3x.

are defined in a neighbourhood of x y

33

and continuous at x, If

32f ^x 3 x

32f (x) = ^ 3 x

f, f' , f",..., f

said to be of class If

then

C

(x)

.

exist and are continuous in U, f is

in U.

q t h derivatives exist for all

q

(hence are continuous),

oo

f

is said to be of class C . To generalise Schwarz1 Lemma, notice that the projection

5 : Tq(E) -+ Sq(E)

of 1.9.2.

induces a monomorphism

5* : L(sq(E),F) -+ L(Tq(E) ,F) .

Generalised Schwartz' Lemma a (a) If f is of class c 1 , f ^ factors through

* E, , i.e. in

fact f(q)

: U + L(sq(E) ,F) .

Formal properties of higher derivatives like all kinds of details relevant to this section, including proofs, can be found in any standard textbook (see in particular [4]).

1

DIFFERENTIAL FORMS Notations remain the same as in Section 0 for the rest of the

Chapter, but F

is no longer arbitrary, chosen independently from E.

As a heuristic example we first take canonical basis

{ e ,. .. , e } , 1 n

So l e t entiable map. When

U be an open subset of F = IE,

as here,

appear l a t e r

(see Section 2) to write

differential

(not 'the derivative 1 ) of

to be R , with its

ZR

and

f : U •+• H

a differ-

i t i s customary for reasons which will df

instead of

f:

df : U •+ OR") * .

In p a r t i c u l a r , the i

E

and F = ]R.

projection

f'

and call i t the

34

TT

: U -* R : (x , . . . ,x ') (+ x.

1

U i s d i f f e r e n t i a b l e with the constant map n *

d(ir.

1

)

: U -»• (R )

i :

( x , . . . , x ) \+

* IT

= e

U as its differential. This is where a universal abuse of notation, hallowed by tradition (its origin being the confusion between the function and the value it takes at a point), and above all extremely convenient, creeps in: one writes dx. instead of d(ir ) thus turning the correct formula U df(x)

=

v 3f * I -1=- (X) e. 1 i=l 3x.

(cf. 0.1)

1

where

x = (x ....,x ) e U, into 1 n

df (x) = I

3 f

-+- (x) dx. (x) o X„

1=1

1-

,

i

hence the (in) famous

df = Now, even if

df

which l e a d s t o

1.1 -Definition: on

U

is

E

is not

]R ,

i t s t i l l is the case that

: U ->- E

the

A differential

a. d i f f e r e n t i a b l e

form

(resp. a d i f f e r e n t i a l

map ( r e s p . a map of c l a s s

form of c l a s s

C^*)

Cr )

(f> : U -»• E*

where

U

is

an open s u b s e t of t h e f i n i t e - d i m e n s i o n a l

v e c t o r space

E.

The r i g h t g e n e r a l i s a t i o n , which t h e r e s t of t h e book r e s t s i s given by t h e

upon,

35

1.2 Definition:

Let k

be a non-negative i n t e g e r . A k-diffexential

or d i f f e r e n t i a l form of degree

k,

form,

on U i s a d i f f e r e n t i a b l e map

k * <> ) : U -* A (E ) If form of degree

P <> j i s of c l a s s C , k and c l a s s C .

p

i t i s a C r - k - d i f f e r e n t i a l form, o r

1.3 Remarks: A O-form i s a r e a l - v a l u e d map. A 1-form i s a form i n the sense of 1 . 1 .

forms to be

For t h e sake of s i m p l i c i t y , from now on we s h a l l assume a l l 00 C . Adapting statements and proofs to the C -case, p e 3N, C .

Adi

is

virtually automatic. 0O

let

fi*(U)

The H-vector space of be © n (U) . kdN * A(E )

Since

]{

C -k-forms w i l l be denoted by

Q (U) ;

is endowed with a rich algebraic structure (see

1.8.7) it is reasonable to expect the

1.4 Theorem: For all

j, k e U,

there

exist bilinear maps

Sp (U) x fik(U) -+ fij+k(U)

which make (i) Q (U) k

(ii) n (U) * ( i i i ) 0 (U)

into a commutative ring, into an into an

£2°(U)-module, 0 Q (U)-algebra;

* (iv) furthermore, ft (U) module.

i s the exterior algebra of

SI (U)

Proof: Naturally the bilinear maps are defined by setting, for and

1 as an

0 SI (U)

<>| e fl (U)

<(> e fiNu) :


:

U->A

i+k

J

(E

*

)

:

x

Statements (i) to ( i i i ) are easily checked. Note that

§ A tp i s

36

of class

C

as the composite

x

k A

(E*) - AJ+k(E*)

where the second map, defined in 1.8.7, i s b i l i n e a r hence of class To check (iv) suppose without loss of generality t h a t with

{e , . . . , £ } i t s canonical b a s i s . If

e x i s t s a unique n-tuple <(>(x) =

j 1. (x) E . l i=l z A. ' s a r e , i . e . X. e f! (u) .

fl (U)

as an

and

x e U,

there

i f f a l l the

In the language of the i n t r o d u c t i o n , t h i s i s

n Q = £ X. dx. ,

saying t h a t

$ e fl (U)

( X, (x) , . . . , X (x)) e :R such that 1 n I t i s clear t h a t $ i s of class C°

C . E =K

which shows t h a t

{dx , . . . , d x } i s a b a s i s of

S2 (U)-module.

A similar argument shows that

{dx.

A.

. . A dx.

« (U)-module.

1 < i , <. . . < i,

< n} i s a b a s i s of

as an

D

1.5 Remark: Here too i t follows that for 1.6 Notation:

Si (If)

r > n, fir(U) = {0}

(cf.

To avoid complicated multiindices i t i s convenient to set

V and, for

I = (i , . . . , i , ) e J : 1 k n dx

= dx.

A.

. . A dx.

1

k

Then any

J e B (0)

* =

A

1

k

j

dx

i s uniquely written as

j

n where

X = X I

1.8.13).

.

e Si (U) .

! , . . . , !

The simplest way to construct a k-form i s shown in the

37

1.7 Example: Let f ,...,f 1

on

U. Then

df. A...A df e « k (U). 1 k

be

k

df ,...>df

k

real-valued maps of class

are 1-forms on (k+l) th

A

map

C

defined

0 and

X e fi°(U) still entails that

X df x A..-.-A df k e f2k(U) .

Now we have just seen that any k-form is a sum of forms of this type: writing i t as such often suffices (see below, e.g. proof of 3.1) without having to resort to explicit formulae involving the bases. 2

EXTERIOR DIFFERENTIAL

Writing

df

i n s t e a d of

f'

leads t o c a l l i n g

d

the map

d : ft (U) •*• ft (U) : f |+.df = f I t satisfies d(f+g) = df + dg

for a l l

f, g e «°(U) ,

d(Af) = X df

for a l l

X e M, f e ft°(U) ,

d(fg) = (df)g + f (dg)

for a l l

f, g e fl°(U) .

The temptation is great to try and extend this

d

to the whole of ft (u),

and that i s achieved in the 2.1 Theorem: There exists a unique map d : ft (U) -* ft (U) called the exterior

differential,

which satisfies the following conditions:

lc (i) for any (ii) d

is

k+1

k e W, d(ft (0) ) eft

(0) ,

E-linear,

(iii) for any a e ft^U), 8 e JJ4(U), d(a A g) = da A g + (-l) p a A dg , (iv) d ° d = 0 , (v) if

f e 0°(D) ,

df = f

.

38

Comments: The structure given to of an anticommutative differential

* SI (U)

by conditions (i) to (iv) i s that

graded algebra (DGA) .

The presence of a sign factor

(-1)

gct

in ( i i i )

should not

come as a surprise in an exterior algebra. To justify

(iv) a priori

one must be prepared to check (a

recommended exercise!) that i t f u l f i l s

a very reasonable condition, viz.

that the differential of a constant form of any degree be zero.

Proof:

I

Uniqueness Let

d

and

6

be two maps which s a t i s f y a l l the conditions.

(1) They are equal on for a l l

i = l,...,n , (2) Let

SI (U)

because of (v) . In p a r t i c u l a r ,

dx. = <5x. .

a =

n n [ a . dx. = £ a. Sx. e SI (0) x 1 i=l 1 i=l 1

where

a. e SI (U) . x

Because of ( i i ) , ( i i i ) and (iv) ,

da

=

n /,

=

I (da

d (a. dx.)

y da 1=1 X

,0

A dx± + (-1)

A

a

dtdxj)

dx.

Similarly Sa =

n £ {a. A 6x. 1

whence

da = 6a

because of (1).

(3) Take an integer up to degree a = £ g. A Y.

,

1

(k-1) . with

Write any

k > 2

and suppose t h a t

a e SI (U)

1 < deg 6. S k - 1 ,

d

and

6

coincide

as

1 < deg Y.

S

k-1

some f i n i t e set (use 1.7) . Then because of ( i i ) and ( i i i )

for a l l

i

in

39

da = 2, d ( B i i

A Y±)

I (dSi A Tj, + ( - l ) d e g 6 i Si A dYi) . i A similar expression i s obtained for

6a,

hence

da = 6a

on

O £2 (U) .

by induction

hypothesis.

II

Existence Condition

If

a =

I

a

dx

(v) d e f i n e s e Q (U) ,

d

where

a efJ°(U),

set

n da=

(cf.

Y , d a A d x = L -jk. I I I e J n

Y da. ^ . I ,. l < i , < . . . < i,
. . ,i

.Adx.A...Adx in a. k 1 k

'Uniqueness' ( 2 ) ) .

The d e f i n i t i o n i s extended t o

Si (U)

as usual for a d i r e c t sum; conditions

(i) to (iv) are checked on homogeneous forms: (i) i s obvious (ii)

Let a =

IeJ',

hence

a

d 1

\

'

B=

I v ^

n

0 a l , Bj e fi (U) . Since d i s ' o r d i n a r y ' k all

l v

d e r i v a t i o n on

d x

w i t h

i '

n

0 Q (U) ,

d ( a + 6 ) = do + dg I I I I

for

d(a+B) = da + dB .

Similarly

d(Aa) = A d a , X e ]R.

( i i i ) To a v o i d c o m p l i c a t e d c o m p u t a t i o n s , f i r s t remark t h a t , f o r 0 (J> e Q (U) and any k - t u p l e I = (i , . . . , i ) X K d(((> dx. A . . . A d x . ) = d<j> A d x . A . . . A d x . : 1 X 1 1 \ l k - if

I eJP, n

- if

I

t h i s i s the definition of

d;

contains the same integer twice, both sides are equal to zero;

- if all the integers in

I

are distinct but not in increasing order,

setting them right multiplies both sides by the same signature.

where

Now let a = I p a l dx± e J2P(U) e O "^ n a , BT e ii (U) . Since a

A 6

=

Y JP n

Y a B dx J J eJq J J n

A

and

B=

I q Sj dXj e flq(U) , J £J n

dx J

40

the opening remark ensures that

d(aAg)

y y d ( a B ) A dx A dx X J I £ J P J £ fi Z J n n L

I _ « dg

£ JP n

J £ Jq n

1 JJP JJjq n

6

T

than 6

A dx ) A (

£

n since

dB^ A dx

8x

All terms with

i=j

d(dcc) =

iA

dx

j •

the f i r s t

(-l)Pa A

is

= I dlit]

fi°(U)

A dx

8 x

i = ? I a^r^- ^

X

3 x a x

3

as

r^— (dx.Adx. + dx. A dx .) 3x.3x. ] I l j j x

(1.8.8 again!) a € $2 (U) ;

(cf.

d ( f dg) = d f A d g

1.8.8).

are zero. What remains can be rewritten using

= 0

d(d(f

dCda)

T , ^ . . „ l < i < 3 < n

(2) Take g e

J

Then

x

Schwarz' Lemma (0.2)

f,

dx

dx ) ;

(cf.

a e 0 (U) .

<**=?£- * v

Now

dx

n = ( - 1 ) P dxT A dfi

(iv) (1) Take

with

A

n

da A g = (

da

dx X

Jd a i A

The second double sum i s none o t h e r £

A

J

.

write

it

a s a sum of forms of t y p e

f dg,

1.7).

+ f d(dg) = dfA dg

b e c a u s e of

(iii)

and ( 1 ) . Then

dg) ) = d ( d f A dg) = d(df)

A

dg - df

A

d(dg)

= 0 and

d(da) = O. (3)

Write any 1 < deg y.

Suppose k > 2 and ( i v ) e s t a b l i s h e d up t o d e g r e e k r a £ J2 (U) a s a = £ B. Ay. with 1 < deg B. £ k - 1 , 1 1 1 i < k - 1 for a l l i i n some f i n i t e s e t . Now i f

B £ fiP(U) , 1 < p < k - 1

d(BAY)

= dB Ay +

then (-DPBAdy;

and

(k-1) .

y e fiq(U) , 1 < q < k - 1 ,

41

proceed to d(dgAy) = d(dB) A dy + (-1) P + 1 dB A dy d6

because of ( i i i ) and induction hypothesis; and to d(6Ady) = dg A dy + (-D P gAd(dy) = dB

A

dy.

Finally d(d(BAy)) = (-1) P + 1 dB A dy + (-1) P dB A dy

and

d(da) = 0 .

•

To i l l u s t r a t e the situation, here are two examples, one c l a s s i c a l , the other somewhat more exotic. 2 2 R \{o} of R . 2 1 Denote the coordinates in K by (x,y) : the basis of fi (u)

2.2 Example: Let

U be the open subset

as an a (U)-module introduced in 1.4 will be denoted by Define

f, g c f! (U)

f (x,y) =

~Y 2 x +y

and

The differential form famous ('angle

1

g(x,y) = —^—j • x +y OJ = f dx + g dy c fl (U)

form) under the slightly incorrect notation

io = — — (-y dx + x dy) x +y 2 . 3 Theorem:

{dx,dy}.

by

du = O.

i s universally

42

Proof: Easy computation.

Q

This statement, which means that

-— = —2- , 3y

Schwarz1 Lemma, a necessary condition for

to

i s , because of

3x

to be the differential of a

O-form. However the condition i s not sufficient i t turns out

to has no antecedent under d.

the general usefulness of

to,

see 7.4 and 7.5.

For a generalisation to 2.4 Example: Let

n

(see Section 4) and as

For a proof of t h i s , and of

ft

OR \{o}) ,

see IV.6.5.

be a s t r i c t l y positive integer and

the vector 2 space of n x n-matrices with real coefficients, which i s isomorphic to HP . To keep to the matrix-oriented perspective, use pairs to index * the elements of the dual basis: e. . i s the linear form defined on M OR) n i} th th which sends any matrix

M into the entry in i t s i

Similarly, if ft (U)

n

row and j

U i s an open subset of

as an ft (U)-module will be denoted by

M OR)

M CR) ,

{dx..}.

the basis of

.

,

i ] l < i < n

dx. . (M) = e. . for all 13

column.

.

•

Thus

l < 3 < n

M eU.

ID

To go further in the same direction, one is led to introduce matrices of differential forms each coefficient of which is an element of * ft (U) .

F o ran example,

in ft (U) notation.

t h ematrix

will be denoted by

(dx. . ) . , . , . . . , i ] 1 < l < n, 1 < ] < n

with

dM in conformity with the usual abuse of * M (ft (U) ) ,

The result i s an It-algebra: the matrix algebra M (0) n

for short.

k

Note that i t is graded by the subspaces

in ft (U),

+

Mn (U) ,

derived from that

i s analogous to the multiplication of 'ordinary' matrices,

except that the, product of differential

forms obviously remains a n t i -

commutative. To simplify an a l l too cumbersome notation, Finally,

ij

or

k

M (U) = M (ft (U)) , k e N . n n Better s t i l l , the multiplication in *

a

coefficients

k e ft (U)

-

let

if

o = (a. . ) , . . , i j l S i S n ,

da

be t h e matrix

(da . ) ,

A will be omitted.

Mn((O) - i . e . e M

. _ . . l s j s n .

i) l < i < n ,

.

.

1<3
k+1 e M (U) .

n

We do not (for the time being?) study the situation systematically, but make two observations:

43

for

a s (f(tl) n

, J £ I^IU), n

i t s t i l l i s the case that

d(aB> = (da) B + (-l) P a dg; on the contrary, i t i s clearly not true in general that aa = 0 . if

a i s of odd degree. Now, l e t

U = GL(n^R) ,

whose elements are the invertible

n x n-matrices: i t is an open subset of continuous map. Denote by X

because the determinant i s a

X the map

0 •+• U

:

M OR)

: M \*- M

Thus

XdM £ M (U) . (Tradition would have us write M dM for t h i s rather n than XdM (cf. 2.2); we temporarily r e s i s t the temptation.) Set a = Tr ((XdM)P) eJ2P(U) where Tr i s the trace, i . e . n,p the sum of the diagonal coefficients. For instance, if

M

-i

=

i

xl^7

A

-\

) x/

\-y

and

/x n = 2, p = 1, M = I "2,1

=

i

y

V

z\ je M (R) = » ,

*'

xl=?l (t dx - z dy - y dz + x dt)

then to

fall into the notation of 2.2. In this case, a direct computation shows that j

and 2 xt~yz d<(> = - <J> B and dg = O,

,1

6 = t d x - z d y - y d z whence by 2 . 1 ( i i i )

= (d<)))B + *dB = 0 .

The general situation i s as follows: 2.5 Theorem: For any integers a =0 n,p do =0 n,p

if if

p p

n a 1, p > 1, i s even, i s odd.

+ xdt,

da_

2,1

then

= O: if

44 Proof: In an algebra of matrices with coefficients

in an anticommutative

graded ring,

Tr(AB) = (-l)Pq Tr (BA)

if the coefficients of

A

have degree 1

thus a.

p

and those of

B

degree

q:

1

= Tr ( (XdM) (XdM) "" ) n,p

and

a „ =0. n,2q Besides, if

M~ M = I or and

that

dx = -M d

I

is the identity matrix, it follows from

dX M + M~ dM = 0

dM M

(if the reader will forgive the notation),

. The previous remark, coupled with the fact that

Tr

commute with each other, shows that

do = -a „ n,2q-l n,2q

hence the second statement.

D

Remark: I t can be shown t h a t , for enough,

a n,p 3

odd,

has-no antecedent under

p | 1 mod 4,

and

n

large

d (see [1O]).

INVERSE IMAGE OF A DIFFERENTIAL FORM

Let let

p

U i toe an open subset of an n-dimensional vector space

V be an open subset of a p-dimensional vector space

F.

C -map <j) : U -*• V we wish to associate with i t a morphism of * H level. In degree

0

E;

Given a DGA's at

composition of maps yields a

<{>* : H°(V) •+ n ° ( U ) : f K f o $

which s a t i s f i e s

<> j (fg) = <J>* (f) <j>* (g) .

So the morphism we are a f t e r w i l l have t o reverse arrows variant

(contra-

case). To extend i t to higher degrees., f i r s t observe how i t behaves

under differentiation: if

x e U,

(f ° <> j) ' (x) = f ($(x) ) ° " (x)

45

which can be rewritten for yet unrevealed purposes as

%' (x) r or

( ' ( x ) » f ' ° <()) ( x )

.

(Recall that the transpose

A of X e L(E,F)

is such that,

for all h £ E, 5 £ F* = L(F,K) ,

here

for

A = <(.• (x) ,C = f ((x) ) .)

This suggests t h a t , given * 1 < > f (a) e H (U) would be through

oe e S3 (V) ,

(a) (x) = (% ' (x) ° a o $) (x)

f

x e

a reasonable

definition

u

at least that would ensure that

(df) .

Admittedly t h i s definition lacks formal elegance in so far as x

remains 'entangled inside the formula' . But any attempt at disentangling

it

(e.g. by using diagrams) only makes things worse. Besides, generalisation

to a l l degrees i s achieved without any r e a l difficulty

3.1 Theorem: Let U, v Given a map of class

(f>

of d e g r e e

0

as follows:

be open subsets of finite-dimensional vector spaces.

C <(>: U •+ V,

there exists a unique map

: S3 (V) ->• 0 (U)

such t h a t * < > j i s R-linear,

(i) (ii) (iii)

$ (a

A

B) = (a) A <(> (g) ,

a n y a , 6 e S3 (V)

$ ° d = d o <> f

(using the same letter

d

for both exterior differentials) ,

46

(iv) <(>*(f) = f o<J> i f

Furthermore, i f

T

dimensional vector space and * * (V)

(<(> ° ijl)

Finally,

= if,

if

f

e

fl°(V) .

i s an open subset of a t h i r d f i n i t e -

\ji ; T -*• U a C -map, * o (J,

U = V,

(Conditions (i) to ( i i i ) ensure t h a t

< > j

i s a DGA morphism.

Proof: I . Uniqueness, (v) and (vi) Any

*

<> f

Let L et

a = f dg1 A . .. ..A A ddg

which s a t i s f i e s

e ii (V)

with

( i i ) , ( i i i ) and ( i v )

f,

g ,..., g

£

ft°(V).

satisfies

<|)*(a) = <))*(f) <)>*(dg 1) A . . . A <((*(dgK ) * * * = lj) ( f ) d ° if) ( g ) A . . . A d o <(, ( g )

= (£«•)

d(g

° <)>)

A.

. . A d(g

o <> |)

Use 1.7 and (i) to show t h a t two morphisms which s a t i s f y conditions (iv) are equal. Similarly, (f

G

t't)

(41 °
*

* * = << l °•

because they both send

a

(i)

to

onto

d

dig, ° <> ) ° if) A- • -A ( 9 v ° $ ° i|>) . As for (vi) i t i s

II.

trivial.

Existence 0 * On fl (V) , 1}) If

i s defined by (iv) .

k > 1, a e 0 (V)

and

x e U,

4>*(a) (x) = A k ( V ( x ) ) . a ( < J > ( x ) )

set

(cf.

1.8.14).

Then (i) i s obvious. ( i i ) A d i r e c t proof based on diagrams i s possible but so awkward t h a t we p r e f e r computations. Suppose without loss of g e n e r a l i t y t h a t {n,/...,Ti } of

SJ (V)

i t s canonical b a s i s . Denote by

as an

ii (V)-module.

F = Ir

{dy , . . . , d y }

with the usual b a s i s

47

By definition.

) (x) = Ak(t<|)'(x) ). (dy.

( d y . A . . . A d y .

I k

(ty(x))

«.••*

^

1 t * ( '(x).n.

=

t * A . . . A( ij>'(x).n. )

)

1 This e q u a l i t y h o l d s whether

(i , . . . , i

) £J

((x!

k •

k

o r n o t , so t h a t

* * * * (dyT A dy ) = <> f (dy ) A tb ( d y ) I J I J in a l l cases. Similarly, i f <> t ( a d y ) (x) = ( % ' ( x )

= a(<)>(x))

a £ £2 (V)

and whether

A . . . A % • ( x ) ) . (a(<)>(x))

dy

I £ J i

(<J>(x))

or n o t , A . . . A d y ± dj,(x))

(t*'(x).n* ) A...A ( V ( x ) . n * )

* * * t h a t i s : <> ) (ady ) = (() ( a ) < > j (dy ) . I a dy £ fir(V) , 6 = I g T dy e flS(V) I £J J £J p p Because o f ( i )

Now l e t a =

where

a , Q e Q (V) . I J **(ct) =

T <(>*(a dy ) X I I £Jr P

I,

I , 4>*(a )<|>*(dy ) X I I £ JJr r P as we have just seen. Similarly

4>(B)

=

y * ( g T ) * ( d y ) J J J £ JS P

Using t h e same remarks a g a i n t o g e t h e r w i t h ( i v ) and ( i ) , i t follows that

(a)

A $ ( g ) = ][

I

I ( a ) ((> ( 6 ) <(> ( d y T ) A ((>

J

J

J

J

(dy J

)

48

4> ( d Y l

I

J

I

J

dy

= ( a

A

A dy )

g)

A

(iii) Recall that, strictly speaking,

dy.

is the derivative of

- V ^y i • In the i n t r o d u c t i o n t o the p r e s e n t theorem, we have seen t h a t our d e f i n i t i o n ensures t h a t for all

( d f ) = &( ( f

Thus

*

f e n°(V) .

*

d ( 4 (dy ) ) = d ( 4 (dir.1 i

i

= d(d(<j>*(Tr^ = O .

Now l e t

a =

)" k

l e J

a

I

dy e 0 (V) I

where

a

I

€ fi (V) .

follows from ( i i ) t h a t

I

<J>*(cO =

I e J

and

d(<() ( a ) ) =

£

(aT)*(dy. ) A...A ()>*(dy. ) ; k

d(()> ( a ) ) if ( d y . )

i er P

x

A . . . A

$ (dy.)

x

i

\

as we have j u s t seen t h a t a l l the other terms are zero. Simultaneously,

da =

2. da A dy : X X I £J P

and, as above,

It

49

(da)

Hence

I . d(*(a) ) A a *(dy X X I eJk

d,

d°

When bases

using (ii) .

•

are given for

E

the inverse image of a differential

and

F,

explicit formulae giving

form can be obtained.

This i s what the r e s t of t h i s section i s devoted t o .

Suppose without loss of generality that with ())' (x)

{E , . . . , £

} and

E = 3R

and

F = Wr ;

{n , . . . , r i } t h e i r respective canonical bases. Then

i s represented by i t s Jacobian matrix

L

(x)

3<J> 3x

For a 1-form

a =

(cf. 0.1)

V 1 I a.dy. e ft (V) 1

with

1

0 a. e H (v) ,

it

1

follows immediately that

(a) (x)

t = V (x). (a(iKx)))

v =

? 1

1

-<x)

a (<(i(x) )

E*.

In other words:

(a) =

For a k-form

. dx.

with

k > 2,

(a. ° *) the computation i s a l i t t l e more

i n t r i c a t e but boils down to the same technique. F i r s t take

I = (i , . . . , i ) e J .

To develop

.

50

A ( " ( x ) ) .

) A . . . A (t())l(x).n* )

(n) = ( V ( x ) . n !

k n r=l

3

*i

r=l

observe t h a t the c o e f f i c i e n t

of

e

, J = (j1/...,j. ) e J # JK n

J

than the determinant

i s none o t h e r

3*, 3x.

L

-(x)

3x.

(x)

D (<J>) (x) = J

3*.

Hx)

3x

3x. 3

k -(x)

extracted from the transpose of the Jacobian matrix: the signatures of the permutations which occur in the development occur identically in the determinant. (We stick to the transpose in an attempt at remaining coherent.) I t follows that (x) =

Ik

Dj(
J eJ

J e .

Now l e t

a =

[ I

£

jk

a

I

dy

I

e flk(V) with

a

Then J> (a> =

I

v

* ( < O 4> ( a y )

P I£Jk P

dx a J e

.

I

e S2°(V) .

51

Which gives in a l l dimensions the

3.2 Explicit Formula If

a

a =

6

<|)(a) =

where

6 , e Q°(U)

dy

J e Jk

e JJ (V)

with

a

e fi°(V) ,

then

dx e

J

J

satisfies

8

=

J ,

(« T ° •) D*(<|>)

with

3x

3x.

3!

= D.

3x.

4

3x.

DB i?HAM COHOMOLOGY

Denote by

= d

d

the r e s t r i c t i o n of the e x t e r i o r V

V

differential

-4-1

(u)

k

f! (U)

I t i s the usual procedure with DGA' s to introduce the following * vector spaces in order to study $2 (U) : Z (U) = Ker B k (U) = Im d 0

k-1

, k € M, k * 0 ,

B (U) = {0} . If a k-form i s an element of

z k (u),

If a k-form i s an element of

B (0) , i t i s

We have seen (2.1(iv))

i t is closed. exact.

that an exact form is closed, but the

converse is false in general. Indeed our main i n t e r e s t is to find out to k k what extent Z (U) i s 'bigger' than B (u) , for which we introduce the

52 4.1 Definition: The k quotient

De Rham cohomology vector space of U

is the

H ^ U ) = Z~ (U) /B* (U) . k k If u e Z (U) , its class in H (U) is denoted by [u]. The De Rham cohomologg of U is the direct sum H*(U)

Hk(U)

9 kdN

.

4.2 Remark: If

U i s an open subset of

4.3 Remark: I f

U i s connected,

m", H (U) = {0}

a C -map

f:

U -+R

for

k > n.

satisfies

df = 0

iff

i t i s c o n s t a n t . This means t h a t the map

Z°(U) -*K : f \* f (x)

,

any

x e u

i s an isomorphism. I t follows t h a t

H°(U) = K .

More g e n e r a l l y , i f

U has

m connected components,

0 m H (U) =]R .

As e x p e c t e d we f i n d t h e comforting

4.4 Theorem: <>| : U -+ V

If

U i s an open s u b s e t of

n R ,

V

an open s u b s e t of

p ]R

and

a C -map, then t h e DGA morphism

<()* : fl*(V) -»• H*(U)

i n d u c e s a l i n e a r map of degree *

*



If

: H (V) -»• H (U) .

T

i s an open s u b s e t of

($ ° ty) = \j>

If

n = p

o <j)

<J>

3R

and

\j> : T •> U a C -map, then

: H (V) •+ H (T) .

and

D = V,

both a t

H

*

Remark: Writing

0

*

then

*

*

l e v e l and a t

U l e v e l c r e a t e s no r i s k of

53

confusion: the context makes i t clear which i s meant. * Proof: Statement ( i i i ) of Theorem 3.1 ensures t h a t *(Bk(V)) c Bk(U)

for a l l

k e W.

k

k

< } > (Z (V)) c z (U) and

The r e s t i s Standard s t u f f

from

elementary l i n e a r algebra. D 5

HOMOTOPY

We wish to study the situation created when a differential

form

i s allowed to vary differentiably depending on a parameter. From the point of view of cohomology the essential result i s Corollary 5.9.

5.1 Definition:

form depending on a parameter i s a C -map

: U x P •+• A k ( E * )

a

where

A k-differential

U is as before an open subset of a finite-dimensional vector space

E. If we set a (x) = a(x,t) , and

t |* a

is of class

then for all t e M, a

v e (I (U)

C.

We denote by fip (0)

the K-vector space of k-forms depending

on a parameter, and fiP*(U) = ffi fiPk(U) . kdN If notations are to be kept within manageable limits, some classical identifications are necessary. Lest ambiguities might ensue, we first examine in detail the * 5.2 S t r u c t u r e of fl ( U x i ) Traditionally 'inclusion'

1

E i s regarded as a subspace of

: E ->- E x M : x [*• (x,0)

E xn

under the

and p r o j e c t i o n

: E x B •* E : ( x , t ) W x . I f {e, » . . . , £ } i s a b a s i s of E, E X E i n h e r i t s the b a s i s 1 n {e , . . . , £ ,8} where 0 = (0,1) and e. ' = ' (e.,0) i.e. I i s in e f f e c t in i i E treated as an inclusion. * * Similarly, E appears as a subspace of (E XTR) with 'inclusion'

IT

54

t

TT£

and dual b a s e s

{e1,...,e

}

and {e , . . . , £

,8*}

respectively.

To follow on , i **

dx. w i l l denote b o t h t h e form 1 1 1 ) e Si (I) (I) x x ii || , w while o ff ccourse ( ( x , t ) \* e . ) e Si hile, o ourse,

and t h e form

(x \+ £ , ) £ SI (U) xx

d tt d

ii ss

( ( x , t ) \+ 6*) e nX(U x]R) . That way {dx, , . . . , d x } module and {dx , . . . , d x

i s a b a s i s of

, d t } i s a b a s i s of

Si (U)

SI (U x]R)

as an Si (U)-

as an

Si ( O x E ) -

module. Topping i t a l l up w i t h t h e ' i n c l u s i o n ' !)°(UXE)

:

A |* X o (IT

k one obtains inclusions

U

k

Si (U) c Si (U «1)

for a l l integers

k.

(The whole affair b o i l s down to considering a function which i s constant with respect to one of i t s variables as a function of one less variable'.) Finally i t i s clear that both d i f f e r e n t i a l s agree on Si (U) * * and therefore on SI (0) , because of which Si (U) ends up being a sub-DGA of Si (U xlR) . * SIP (U)

From that point of view, 1 SIP (U)

0 i s the

appears as a half-way house: * {dx.,,...,dx } only, and SIP (U) is

n (U xm) -module on

n

o its

Si (U xE) -exterior algebra. I t i s a subalgebra of

Si (UxE) ,

but not a

sub-DGA since

d(np°(u))
o

f £ 8P (0) = Si (U xEl .

Then

df = I •£*- dx. + | i dt 1=1

and

-— * 0 at

i

in general.

Now set Ef = ^7 3 ;r— dx x 2 . i=l

i

and, for a = Ie

J , aI dxI

^ -jk J

n

with

55

a

1

e fi (U x]R) ,

(so

set

5a =

T . 5a lef

Z

dx

A

X

n 5 is the ' x-component' of the differential). Then for a l l k

k eU

k+1

5(fip (u)) c np

x

(u) .

There follows a useful 5.3 Lemma: If

k 5 1,

any form

id = a + B

A

a e ftPk(U) and

with

to e !) (U x]R)

can be w r i t t e n uniquely as

dt

B e np k ~ 1 (U)

.

Furthermore doo = X + y A d t X = 5a e «P k+1 (U)

with

and

u = 58 + (-1) k f2- e HPk(U) . 0 t

k Proof: An element of the usual b a s i s of fi (U XJO

O as an fl (U *!*) -module

i s e i t h e r of the form dx.

A...Adx.

, 1 < i

A...A dx.

Adt

<

< i


or of the form dx.

,

l < j < . . . < j < n .

The unique development £ k al dx I J I eJ J eJ n a . dx, + 7 n, , B_ dx_ A dt with a , B_ e S (U XR) , thus provides a unique a = I

unique

J

I , a

l£jk fK

T

6 =

Y

BT dx

n

dx I

and a I

as a s s e r t e d .

J eJ Furthermore and

dB = 5B +

writing ^ - for

/.

du = da + dB A d t

- 5 — d t A dx

2

k

n

"g^~ dx .

where

da = 5ct +

hence t h e second r e s u l t ,

0

I . -r— dt A dx n

56

We need one more technical gadget before homotopy can be defined: the integration of differential forms depending on a parameter, (a more general notion of integral for differential forms will be introduced later: see VI.1.18. Our purpose here is much less ambitious.)

5.4 Definition: Let k be an integer. Suppose a e QP (U), a, b e E . r 0 a = 2, v a d x with a e 0 (UxR), and a (x) = a ( x , t ) . I I I t J e J k n , a

dt

i s d e f i n e d t o b e t h e form

a

Y I eJ

0

dx

Write

e £2 (U)

n where

6

e Q (U)

8

is the map

: U -+R: x \* \ a (x,t) dt. 'a

JRemari: In the present context, having to write

dt inside the integral is

most unwelcome (cf. e.g. 5.3). It cannot be helped for historical reasons.

There is a 'differentiation under the integral sign' theorem in the following guise: k 5.5 Lemma: If a e UP (U) ,

d( I

a a

t-

k+1 then in S2 (U) :

dt) =

(£a)

dt .

I t

'a

Proof: With the notations of 5 . 4 ,

fb it

1'

does not depend on

t.

o

dt =

a a

V -rk.

B. dx X X

where

I eJ

Hence

fb d(

v a

•> a

t

dt) =

) I ef

dg n

I eeJJ

A dx

X

kk

n

n 3 6r n i=l 33xxii

T

X

X

(Of course many terms in t h i s double sum are zero; never mindt) On the other hand,

6

X

0 e ft (U)

57 n 3a

5a =

I I eJ n

g^- dx i

A

dx x .

I €J n

(Same remark.) So the lemma follows from the 'elementary1 theorem

36

(b 3a

— (x) = j — (X,t) dt .

•

And now the concept which gives i t s t i t l e to the section can enter the stage. Actually i t does so in two definitions which at first might seem somewhat unrelated:

5.6 Definition:

A homotopy operator for

U i s an K-linear map of degree

(-1)

K : S (U xM) -*• Q (U)

such t h a t

d ° K + K ° d = J., - J where 1 0

d

reoresents both

differentials

and J

: U ->• U »E : x |+ ( x , i ) , i = O, 1

i s t h e usual i n c l u s i o n .

5.7 Definition:

Let

U, V be open subsets of two finite-dimensional

vector spaces. Two C -maps

fQ

, f1

: U •* V

are said to be Adifferentiably)

homotopic iff there exists a C -map

f : DxE+V

such that, for a l l

x e U,

f (x) = f(x,O)

If s o , we write

f-

f

.

and

f (x) = f ( x , l )

58

As it turns out, the two notions are very close to each other as will appear in the proofs of the two fundamental results of this section:

5.8 Theorem: A homotopy operator can always be constructed.

5.9 Corollary: If

f

= f , then

* * * * fQ = fj, : H (V) •+ H (U) . Proof of 5.9: The hypothesis is precisely that

f.

=fj.

,i

Assume t h a t 5 . 8 h o l d s : * f at

fl

= O,

then

* - f

1 .

* = (J,-JJ

* °f

K°d)°f

level. If

a £ n (V)

(f*-f*)(a)

with

= dB + 0 * * k k (f - f ) ( Z (V)) c B (U) f

- f

= 0

da = 0 ,

=d°K°f*(a) = d(K o f

So

* =(d°K +

at

+K°d°f*(cx)

(a) ) + K ° f with

8 e

(da)

by 3 . 1 . ( i i i )

fik~1(U).

and, p a s s i n g t o De Rham cohomology, H

level.

k * Proof of 5.8: (i) L e t w = a + 8 A d t e f l (U xK) : see 5 . 3 . To compute J . (aj) o b s e r v e t h a t f o r a l l x e U, J'. (x) = I ( 5 . 2 ) ; t h e r e f o r e t h e m a t r i x of 1

% ! (x)

is

Id*

E

59

From

tjlie ) = O

follows that

J.(dt) = O, hence

J. (a)) = J. (a + 6 A dt) = J. (a) .

= Idm*

From

E

follows t h a t

V ~"

J.(a)(x)

" ^

= a(x,i)

for a l l

x

e

U.

Finally,

(J -J ) (ui) : x^<x(x,l) - a(x,O) .

(ii) For such an

u,

set

K(u) = (-1)

k-1

f1

g

dt.

•"0

(N.B.: i f the sign bothers you, write

to = a + dt A y

and adapt the proof

accordingly.)

Then

d(K(id)) = (-I) 1 *" 1 I

(SBK dt

after 5.5.

On the other hand,

dm = X + y A dt

(notations of 5.3)

and k f K(do)) = (-1)

J

ji O

fc

dt

as

/•I = (-1)

.-I (IS),

o by definition of

k+l du e SI (U xK)

t

dt +

J

o

(^) dt dt t

ji-

l t follows that rl (d°K + K»d) (to) =

Now, by d e f i n i t i o n ,

(f^-) dt . J o *t t ( ^ ) . (x) = f^-

(x,t) ,

whence

(d ° K + K o d) (OJ) : x [* I f £ (x,t) dt = a ( x , l ) - a(x,0) J O 3t

.

Q

60

6

COHOMOLOGY OF K This our f i r s t

computation exhibits right away homotopy as an

essential tool: a l l cases are reduced by i t

We begin with the partial

to the case

n = 0.

result:

0 0 H (E ) = E . k 0 For any i n t e g e r k > 0 , H OR ) = { o } . 6 . 1 Lemma :

(By the way, K

i s , of c o u r s e , k

Proof: For k

H OR°)

k a 1,

{o}.)

0

k O

A OR ) = { o } , whence

ft

QR ) = { o } , whence

= {O}. Any function

defined on K

i s differentiable

(O.I):

therefore

ft OR ) = FOR ,K)

and t h a t i s c a n o n i c a l l y isomorphic with M (as v e c t o r 0 0 0 0 spaces) . As t h e d i f f e r e n t i a l can only be z e r o , Z OR ) = ft OR ) , whence

H°CR°) = Z ° a R 0 ) / B 0 ( R ° )

=R/{0} = K (cf. 4 . 3 ) . •

To extend t h i s t o the g e n e r a l c a s e , we use t h e canonical injection is

i

: K

clear t h a t

->• OR

: O |->- O and p r o j e c t i o n

p : ]R -+ K

: x \* O.

It

p °i = IdO . iR

6.2 Lemma: The composite

q = i ° p : I

-+M : x [* O

is homotopic to the identity

f : Enx E -*• TRn

Proof: Define c l a s s C°°.

Moreover, ^(x)

I(L n .

= f(x,l)

= x,

f(x,t)

f (x) = f(x,O) i.e.

f1 = Id^n .

6 . 3 Theorem: For any i n t e g e r

n.

if k

by

k = O

n

H CR ) = {0}

if

k > 1

=0, D

= tx . i.e.

This map i s notoriously of f

= q;

and

61

Proof: I f n = O, use 6 . 1 . I f n S l , Lemma 6.2 shows t h a t i * : H*CRn) •* H CR°) and * * 0 * n p : H (E ) + H (E ) are isomorphisms, inverse t o each o t h e r : indeed (cf. 4.4 and 5.9) * * * * p o i = ( i o p) = ( I d ^ ) = Id and

i

°p

= ( p ° i ) * = (-1%p'> =

Id

H*

6.4 Remark: The same i s true more generally of any open subset such that

Id

X

X •*• X : x f* x

(x

Such an X i s said t o be 7

X of K

i s homotopic to a constant map e X) .

contractible.

COHCMOLOGY OF JR 2 \{0>

Even though we shall eventually describe more powerful methods (V.2.4) to determine t h i s cohomology - and also that of M \{o},

any

n

-

directly, the computations of t h i s section do not lack interest in spite, or because, of their 'do-it-yourself Let

approach.

U = K 2 \{0} .

Remarks 4.2 and 4.3 show that

Hn(U) = {0}

For the rest, the shape of

irresistibly suggests that 'polar

for

n > 3 and that

H°(U) =K. U

coordinates' would be the desirable thing to introduce. Here is how it"is done rigorously: Let

V =B

(hence fin(V) = {0} for n > 3) and

x»

: V -> U : (r,8) |+ (r cos 6, r sin 9) . 00

I t i s clear that (j> ° T = <> | if

<> ( i s of c l a s s

C ,

s u r j e c t i v e and t h a t

T i s the t r a n s l a t i o n T : V •*• V : ( r , 9 ) |+ (r,e+2ir) . * Denote by

0

* t h e subalgebra (and even sub-DGA) of fi (V)

62

i n v a r i a n t under the action of

T : the previous remark shows t h a t

<j>*(ft*(U)) c 0* . Description of

0 :

O 1 7.1 Lemma: I f f e H (V) ,a = u dr + v d8 e n (V) , then f e 0° i f f f o T = f , a £ 0

iff

u ° x = u and

ai £ 0

iff

woT = w .

2 uj = w dr A de e ft (V) ,

v°T = v ,

Proof: Cbvious. D As a matter of f a c t . 7.2 Theorem: The homomorphisra

<> f

provides an isomorphism between Proof: 6

is mono: On fl (U) ,

induced by the "change of c o o r d i n a t e s ' * * U (U) and 6

t h i s i s a consequence of

ij> being s u r j e c t i v e , s i n c e

* (f) = f On ft (U) : l e t valued C -maps defined on

a = a dx + b dy where U.

Then

a

and b

$ (a) = u dr + v d9

are r e a l -

with

u(r,8) = a o <j>(r,9) cos 8 + b o $(r,6) sin 6 (S) v(r,9) = r (-a ° i)>(r,8) sin 9 + b ° <|>(r,9) cos 8) . cos 8

sin 9

r * O, i t

Since the determinant

follows from u = v = 0 t h a t is surjective. 2 On ft (U) : l e t u * defined on U. Then <> j (OJ) = here too w = 0 implies c =

a°<j>=b°(j) = O,

whence

= c dx A dy where w dr A d8 with 0.

c

a =b = 0

as

$

°° i s a r e a l - v a l u e d C -map

w(r,8) = r c ° $ ( r , 8 )

and

63

<(>

is

epi:

Of course

i)> does not have an inverse. Failing that, prove

without difficulty, i f not with pleasure, the 7.3 Technical Lemma: (i) The functions Arctan (y/x) \

:

:

if x > 0

(x,y)

Arctan (y/x) + TT i f x < 0 Arccotan (x/y)

if y >0

y : K x M* -*• M : (x,y) Arccotan (x/y) + IT i f y < O agree on the f i r s t t h r e e (open) quadrants and d i f f e r by 2IT on t h e f o u r t h . ( i i ) Wherever i t / t h e y i s / a r e defined, U°4>(r,8)

i s / a r e equal t o Now take

If

x * O, s e t

if

y * 0,

of c l a s s

X »
f £ 9 , (x,y) e U. .y) = f ( / c 2 + y 2 , A ( x , y ) ) ;

s e t g (x,y) = f ( / x +y , y(x,y)) . C l e a r l y C

and/or

6 mod. 2TT .

g

and g

are

and, because of 7 . 3 ( i ) , they agree on t h e i n t e r s e c t i o n of

t h e i r domains, which are open: -there r e s u l t s a map g e SI (U) which agrees with

g

and/or Take

Obtain

g

everywhere on U. By 7 . 3 ( i i ) ,

a e 9 :

g°$ = f .

a f t e r 7 . 1 , a = u dr + v d6 with

h , k e f t (U) t h e r e s p e c t i v e antecedents of

u, v € 0 .

u, v as above. A

d i r e c t computation involving System (S) shows t h a t , i f a(x,y) = h(x,y)

- k(x,y) / 2 2 /x +y

2

2

x +y

and

b(x,y) = k(x,y)

the form

2 2 x +y

+ h(x,y)

6 = a dx + b dy e SI (U) s a t i s f i e s Take

u €9 ,

/ 2 2 / x +y < > j (B) = a .

i . e . tu = w d r A d6 with

be t h e antecedent of w obtained as above. S e t

we0 .

Let z e Q (U)

64

c(x,y) = — — — z(x,y) /2 2 +y 2 a = c dx A dy e B (U) :

and

then

* < > | (a) = to . D

This isomorphism enables us to describe the cohomology of U by determining the algebraic structure of 0 . To detect 0 , consider the linear form A : ^(V) -*1R defined by A (a) =J

v(l,6) d8 0

if a = u dr + v d8. polar coordinates'.)

('Integration along the unit circle expressed in

7.4 Theorem: The restriction Ker I = d0°). 1

Proof: Obviously I

I of A to 0 n z (V) i s epi and

1

f

d9 e 0 n Z (V) . Now I(d6) =

is epi because i t is linear.

_

2iT

If

fe6°,

d8

= f(l,2ir)

Jo

d6 = 2TV * 0, so that

df = | i dr + ff- d6 and or

39

3f

~

o3e

(1,8)

- f(l,O)

a= u d r + v d 9 e 0

Conversely, take da = ( | ^ - |^-) d r A d8 = 0

3r

= 0.

and t h a t

38

v(l,9)

such that d8 = 0 .

JQ

For

f to satisfy

df = a,

it is necessary that

r

f that

f(r,8) =

u(s,8) ds + X(6) with

X a C -map. If so

9f - — = u, dr

i.e.

65

f> 6) ds

+

§(

v(r,6) -

-J;

J u s t choose

A (8) =

'o

f(r,8) = which satisfies

v(l,t)

dt

and you g e t

u(s,8) ds +

df = a

v(l,t) dt

and rr

f(r,8+2ir) - f(r,9) = because

u

and

v

f8+2ir (u(s, 6+2TT)-u(s,9) ) ds +

v(l,t) dt = O

are periodic. D

1 ( * Set as usual for a e ft (U) : -.a = Ad|> (a)) • The map

7.5 Corollary:

' S

J : Q (U) -*-!* : a |-»- J a S induces an isomorphism between

Proof: The isomorphism B 1 (U)

* $

H (U)

and R.

of 7.2 c a r r i e s

to d(0°) and J

1 Z (U) over t o

1 1 Z (V) n 0 ,

to I. Z1(U)

Theorem 7.4 now says that J

is epi and that its kernel Z (U)

is

B (U) , hence the asserted isomorphism since m sim

H (U) = Z (U)/B (U) .

(N.B. What form should correspond to d8 under old

X 2

2

(-y dx + x dy)

of 2.21)

x +y The situation is even simpler in dimension 2:

7.6 Theorem:

dtG1) = Q2 .

Proof: Let w = w drAde

with

w e 0 . Set for instance

$

but dear

66

v(r,8) =

frw(s,6)

ds

and

a = v d6:

•>1 then

v(r,e+2ir) = v ( r , 6 ) ,

7.7 Corollary:

hence

The cohomology of

1 a e 0 ,

9v and da = -— d r A d6 = ou. D or

U i s zero i n dimension 2 .

* 2 2 2 1 Proof: The image by of fi (U) i s 0 , t h a t of B (U) i s d(0 ) . 2 2 2 2 2 Since B (U) c z (U) c Q (u) , i t follows from 7.6 t h a t B (U) = 0 (U) and H2(U) = { 0 } . [] 7.8 Recapitulation:

H°OR2\{0}) =H 1 CR 2 \{0}) = M HnCR2\{0}) = {0} for n > 2.

8

DIFFERENTIAL FORMS WITH COMPACT SUPPORTS

In preparation for future developments (Chapters V and VI) we give special attention to differential forms with compact supports: these provide a notion of cohomology with compact supports which bears a certain analogy with 'ordinary' cohomology even though i t s structure is less rich. As before space,

k

U i s an open subset of a finite-dimensional vector

an integer and

a e SI (U) .

Recall that the support of (topological) closure of

a,

denoted by

{x e U a(x) * 0}:

Supp a,

i s the

thus a form with compact

support i s a form which i s zero outside a compact subset of

U.

Clearly the k-forms with compact supports constitute a subspace k (U k r space SI of the vector space SI (U) , denoted by ^ c

set n*(u) =

As i t turns out, 8.1 Lemma: The subspace

Q (U) i s a sub-DGA and even an i d e a l of

Proof: The only n o n - t r i v i a l p o i n t i s t h a t

d(n (U)) c o (u) .

the support "can only s h r i n k ' under t h e action of

d,

take

SI (u) .

To see t h a t k a e fl (U) ,

K = Supp a, V = U \ K, 1 : V - > 0 the i n c l u s i o n . By d e f i n i t i o n of

I

( e . g . use 3.2 and t h e remark t h a t the d e r i v a t i v e

i s t h e i d e n t i t y a t a l l p o i n t s of

V) ,

67

and

1 (a) = a = O; ; * I * = 1 (da) = d(i (a)) = 0

(da)

which shows that

Supp (da) c K . Q

This i s reasonable ground for the

8.2 Definitions: k > 1,

For any

k e H,

k k k Z (U) = fl (U) n Z (U) .

For any

k e D,

k

B (U) = d t ^ ^ f U ) ) . (Beware! This i s not simply

n B

^(U'

(U) : the support of the

antecedent by d must be compact too!) If k = 0, B c < u ) = {°} • F o r any k e n , Hk(U) = Z k (u)/B k (U). Finally, H*(U) = ffi H*(U) defines the C keN De Rham cohomology with compact supports. 8.3 Important Remark: A C -map

$ : U -> V between open subsets of f i n i t e -

dimensional vector spaces being given, i t i s out of the question to hope for * * * * homomorphisms between B (V) and £2 (U) , or between H (V) and H (U) , in C

C

C

L-

the manner of 3.1 and 4.4. Indeed there i s no reason why the support of the inverse image of a form with compact support should be compact. A very simple example i s provided by the case <> ( : B -* R

U = V = ]R; k = O; f e fl OR) , f * 0;

a constant map such that

f ° <> j * 0:

then

Supp (> (f) — 1R . If a similar theory i s nevertheless requested, r e s t r i c t i v e conditions will have t o be imposed on

$: see IV.7.6.

However there e x i s t s a reasonably i n t e r e s t i n g case when a morphism between cohomologies with compact supports does appear, but then in the covariant direction: Let E

and

a

U be an open subset of a finite-dimensional vector space

a k-form with compact support

i s an open subset of

E which contains

K c u,

i.e.

a e Si (U) .

U and

I : U -> V the inclusion, define a k-form l*(a) I

= a

and

l*(a) e Q (V) ^
by s e t t i n g

If

V

68

This definition works because the condition

Supp a c u

guarantees that

00

l,(oc)

i s of class

C .

Moreover, of

V too, hence

Suppd^la)) = Supp a = K which i s a compact subset

l*(ci) e fi (V) .

There follows the 8.4 Theorem: Any inclusion

I : U c V between open subsets of the same

finite-dimensional vector space induces a monomorphisni of DGA's : J1C(U) -> QC(V) and a l i n e a r map of degree : H (U) C

I

Proof: I f

0

•+ H ( V ) ^—

f e ti° (U) ,

.

i t i s well known t h a t

l*(df) = d U t ( f ) ) .

Extend

t h i s property t o higher dimensions with the h e l p of b a s e s . The r e s t , a t J2

level, is t r i v i a l .

As a consequence, hence the map l t

at

H

l^(^(U))

level.

c

z£
and 1*
c B

c(V) '

D

To end w i t h , and by way of elementary examples, we compute H (H°)

and H*CR) .

8.5 Theorem:

H°CR°) - R .

For any integer k > O, Proof: Since M

k 0 H (R ) = {o}.

(in contrast to all other JR 's -) is compact,

* o * o HC(R ) = H CR ) . D 8.6 Theorem: H CR) — R. For any integer

k * 1,

Proof: We know (1.5) that H^CR) = {0}

if

H CR) = {0}. Si OR),

hence fl CR) ,

are zero for

n S 2.

Thus

n > 2.

The only constant map defined on H

that has a compact support

69 is the zero map: thus

Z OR) = {o}, and H OR) too.

Because of the supports being compact, the linear map

: H^OR) • + » : f dx |+ |

f(x)

dx

(notations more uncomfortable than ever!) is well defined. I t clearly i s epi

(e.g. recall the existence of non-zero, positive, maps with compact

supports). Now, for

f e fl OR) ,

x f(t) dt is a primitive of

o t h e r words,

f whose support is compact iff

1

B OR) = Ker I . C 1 Then Hc OR) R since

f(t) dt = 0: in J —oo

1 1 Zo OR) = flcOR)

8.7 Remarks: Chapter V w i l l provide us with means of computing any i n t e g e r

n

* n H OR )

for

(see V.4.19) .

The fact that

H*OR0) * H* OR) C

C-

confirms - if need be - that the

cohomology with compact supports is not a 'true' cohomology.

70

III

DIFFERENTIABLE MANIFOLDS

The previous chapter was dedicated to open subsets of JRn: differential

forms were defined on them and their De Rham cohomology was

introduced. The next chapter will extend these notions to a more general family of mathematical objects: differentiable

manifolds.

They are topological spaces which locally look very much like H

(Definition 1.1), with a guarantee that the various local images can be

put together satisfactorily

(Definition 5.2). On such a manifold

forms can be defined by a 'patchwork' method

differential

(see Chapter IV) and a notion

of De Rham cohomology i s obtained in this enlarged setting.

The present chapter gives the necessary definitions together with a l i s t of examples which are intended to show that the notion covers a reasonably wide range of classical cases.

As for these examples, we have aimed at their being as descriptive as possible: we preferred ad hoc proofs, be they lengthy, to references to Big Theories which would have taken us away from our main subject without receiving proper treatment in the process (e.g. the very name of 'Lie group1 does not appear) .

1

TOPOLOGICAL MANIFOLDS

1.1 Definition: We say that

Let

M be a topological space,

n

a non-negative integer.

M is a topological manifold of dimensdon n (or n-manifold)

iff (i) the topology of (ii) the topology of a countable family

F

M i s Hausdorff M has a countable basis,

of open subsets of

M i s the union of a subfamily of

F:

i . e . there exists

M such that any open subset of

M i s then said to be

first-countable.

71

(iii) for any x e M, there exist an open neighbourhood x

in M, an open subset

a triple

(U,<j>,A)

A

of K

and a homeomorphism

is called a chart of the manifold

M

U of

<(> : U -> A: sue at x.

1.2 Remarks (1) The basic i n t e r e s t of Condition (ii) i s to allow the essential construction of Chapter IV Section 5 to be performed. (2) I t i s not to be believed that (i) i s a consequence of ( i i i ) : see [18]. (3) I t i s not a consequence of (ii) either (even though Condition (ii) i s occasionally referred to as (4) The dimension later

n

M being 'separable')

.

i s uniquely determined, as will be shown

(V.5.3). (5) The value

n = 0

i s not excluded: then

]R

= {0},

and a

O-manifold i s just a countable discrete topological space.

Our f i r s t three results are immediate consequences of the definition:

1.3 Theorem: Let open subset

E be a finite-dimensional vector space,

M of

n = dim E.

An

E i s an n-manifold.

In p a r t i c u l a r ,

Proof: F i r s t assume

3R

E =K .

i s an n-manifold.

Then

(i) i s well-known; (ii) e.g. take contained in

F

to be the family consisting of those open b a l l s

M whose radii are rational and whose centres have rational

coordinates; (iii)

for any

If

x e M,

E * 3R ,

and any basis of

E;

]R ,

i s carried over to

E.

i s an n-manifold.

U = A = M and

ij> = Id

choose one of the equivalent norms ddefined on E <

these t define respectively the topology of

homeomorphism with

1.4 Theorem: Let

take

E

and a

1through which the manifold structure j u s t established D

M be an n-manifold,

N an open subset of

M.

Then N

72

Proof: (i)

classical;

(ii)

obvious with family

(iii)

for any

(uriN, |

{ v n N | v e F};

x e N, l e t . •(DON)]

(U,<(>,A)

be a c h a r t of

i s a chart of

N at

M at

x:

then

an n ' - m a n i f o l d ,

then

x. D

Iu nN 1.5 Theorem: I f i s an

M1

M*M'

(n+n')-manifold.

Proof: Take c h a r t of at

M i s an n—manifold and

(x,x')

M'

at

(x.x1).

e M«M', (U,,A)

x' :

a chart of

1

then

M at 1

(UxU , <J> x <{>', A X A )

x,

(U'jiji'.A 1 )

i s a c h a r t of

a

«x«'

Q

On s e v e r a l occasions (cf. V.4.4) we s h a l l need the notion of submanifold.

We give the d e f i n i t i o n h e r e , b u t w i l l n o t e x p l o i t i t too

s y s t e m a t i c a l l y in the examples t h a t

1.6 Definition:

Let

M be an n-manifold,

non-negative i n t e g e r , dimension x

p

such t h a t

A nir

p < n.

i f f for any tf>(x) = 0

are homeomorphi sms)

follow.

We say t h a t

x e N,

N a subspace of

M and

N i s a submanifold

there e x i s t s a chart

(U,<j>,A)

of

p

a

M of of

M at

(which i s no r e s t r i c t i o n since t r a n s l a t i o n s i n K

and t h a t

(where as always

$1

3R

i s a homeomorphism between

i s considered as a subspace of

TR

UnN and under the

'inclusion' ID

n _

The integer

X

1'""XP n-p

The submanifold

2

i

*

x1,...,xp, , . . . ,

i s the codimension of N i s then a p-manifold

N

(in

M) .

in i t s own r i g h t .

FIRST EXAMPLES

2.1 Example: The circle

S

i s a topological manifold of dimension 1.

Proof: (i) and ( i i ) as a subspace of (iii)

Identify

The map e : 3R -*• S surjective.

W? with

: 8 f* exp(2in6)

m .

<E and

S1

with

{z e
i s known t o be continuous, open and

73

Fig. 3 . 1 . A submanifold and a chart with n = 2, p = 1 (see Definition 1.6). For any

x e S ,

choose

1

U = S \{x}, A = ] 9 - l / 2 , 6+1/2 [

2.2 Example: For any n e U ,

and

6 e ]R such that l

.

< > f = (e

the sphere

S S

x = exp(2iir6) ,

(See also 2.2.)

set

•

is a topological manifold of

dimension n.

Proof: (i) and (ii) as a subspace of

For (iii)

]R

since

there are two classical methods (but see also Section

3): (1) Make use of stereographic projections. Let

be the "North Pole" of S n and E its "equatorial

v = (O, ...,0,1)

plane' (of equation

x

parallel to E intersects point distinct from

= O) . A straight line drawn through E

at a (unique) point and S

v non

at a (unique)

V: see Figure 6.4. This procedure defines two maps:

74

: ,x

and n+1

n+1

S"\{V} :

1

( 2 x . , . . . , 2 x ,K2+...+x2-l) 1 '-IX 1 n

I t i s not too d i f f i c u l t to check that

<> (

and

n

inverse to each other: the t r i p l e

iji

.

are homeomorphisms

n

(s \{v},<(> ,R )

i s a chart of

s".

The stereographic projection from the South Pole a = (0, . . . , 0 , - 1 )

gives another chart

(S \{a},4> JR ) a

and

S

i s an n

manifold because it is the union of S \{v} and S \{a}. (2) Cover For

i = l,...,m+l,

S

with hemispheres. n { (x , . . . , s

) e JRn + 1 x. > 0}

set H T =

and

= s

n {

) e K n + 1 | x i < 0}

x1,... , x

these are open subsets of S Denote by D

the (open) unit ball of ]R : the orthogonal

projection parallel to the i

f

i

•

coordinate axis induces

x

"i

x

n+l'

n+l}

and

which are homeomorphisms with respective inverses

:.,

V

x. , . . . ,x )

n

l

n

and

X

/

n'

"71

- i ! ' " A " I^j ' xi

V •

(See F i g u r e 3 . 1 . ) The c h a r t s

( H + , i)i + ,D n )

t o p o l o g i c a l manifold s t r u c t u r e since

(H.,(ji.,D n )

and S

=

also give

n+1 u (H. u H . ) . Q L 1 i=l

s"

its

75

2.3 Remark: The sphere

S

is in fact a submanifold of E

. Check that

the following construction gives charts which satisfy the conditions of 1.6: for x e S n , let P be the hyperplane orthogonal to x: it is x n isomorphic to E (as normed vector spaces), so that the projection parallel to

Ex

induces a continuous map

Take a real number

TT : S ->• E .

a, O < a < 1;

denote by

U the i n t e r -

section of the open 'annulus' contained between the spheres of radii and 1+a

with the open half-space which contains

x;

1-a

define then

v

-I ; Fig.

En (x)=O

*

3.2

Our next examples are the ubiquitous projective

spaces (real

or complex). Originally introduced to simplify and unify statements of elementary geometry (see [ I 1 ] Chapter 4), they play a considerable part in Algebraic Topology, as do their generalisations: the Grassman manifolds (see 4.15 sqq.).

At f i r s t from the set-theoretical point of view:

2.4 Definitions: The n

Let th

be a non-negative integer. real projective

space, denoted by 3RP ,

i s the set of

a l l straight lines through the origin (1-dimensional subspaces) in The n

complex projective

space, denoted by


E

i s the set

of a l l vector subspaces of (complex) dimension 1 in (L Note that the definition holds for are reduced to one point.

n = 0,

when IP

and
76 We shall then endow these sets with natural structures of topological manifolds, in the real and complex cases successively. In either case two methods are available, which we shall prove to be equivalent.

Two partial results which occur several times in the proof are elevated to the dignity of independent lemmas and proved first:

2.5 Lemma: Let X be a topological space, X.

Its graph* {(x,y) e X |xRy}

R

an equivalence relation on

is denoted by GrR, the quotient equipped

with its canonical topology by X/R, and the projection (continuous by definition) by p : X •+• X/R. 2 Then: if GrR is closed in X

and p

is open,

X/R is

Hausdorff.

Proof Let

£,n

be two d i s t i n c t points in

antecedents in Then

(x,y) i GrR V

(V x V ) n GrR

I t follows that

= 0.

z e V

of

and, because

open neighbourhoods

would e x i s t

X/R;

let

x, y

be respective

X: p(x) = £, p(y) = n-

and

x

and

t e V

V

of

GrR y

i s closed, there e x i s t

in

X such that

p(V ) np(V ) = 0

such that

(otherwise there

p(x) = p(t) ,

whence

(z,t) e (v xv ) n GrR ) . Of course Finally, open.

5 e p(V ) x p(V )

continuously on

p(V )

are open in

X be a topological space,

X/R

because

p

is

Proof: Let iff

p :

X •*• X/G

V be an open subset of -1

p (p(v)) But

G a topological group operating

X.

The projection

p

i s open in (p(V)) =

which i s homeomorphic t o on

n e p(V ) . y

D

2.6 Lenma: Let

X/G

and

and

V,

X.

i s open.

By definition,

p(V)

i s open in

X.

u g.V geG

where

g.V

hence open, since

X. D

And now to the protective spaces!

denotes

{g.x|x e V}

G operates continuously

77

Real Case Two relations can be defined: either on R n + \{o} : xRy iff there exist a real number

X *O

such that

y = Ax, or on S : xSy iff y = ±x. Clearly these are equivalence relations. The classes consist of straight lines (with the origin deleted) in the first case, of pairs of antipodal points in the second. Clearly too the two quotients are in one-to-one correspondence with each other and with

RP

(via: line through the origin

••--••line with origin deleted *-*• antipodal points of intersection with sphere) . Better still:

2.7 Theorem: The above one-to-one correspondence between R

\{o}/R

S /S and

is a homeomorphism with respect to the quotient topologies. The quotient spaces are compact.

Proof: Denote by < ( > the one-to-one correspondence. It appears in the following commutative diagram

i—> * n + \

sn

q L

\{o}/R

where

i

i s the canonical injection and If

U i s an open subset of

p, q R

n+1

\{O}/R,

is open in S , so c f > (U) is open in S /S:

Multiplication by scalars in ]R * n+.l multiplicative group JR = R\{0} multiplicative group

{+1,-1}

they define the relations

R

on M

the canonical projections.

thus

< j > is continuous.

induces an operation of the

\{o] and an operation of the

on S . Both operations are continuous and and S

respectively: thus

open by 2.6. The map

y : (s") 2 •+ R : (x,y) [•• || x+y||.|| x-y||

p

and q are

78

is obviously continuous. The graph of closed in

S

is

GrS = y

({o}),

which is

(S ) . The map

6 : CE n + 1 \{0}) 2 x.

is obviously continuous. The graph of closed in

OR

R

is

( ({o}) , which is

\{O}) .

Lemma 2.5 now proves that M Furthermore,

GrR = S

y.

S /S

is compact because

Finally,

$

\{o}/R S

and

S /S

axe Hausdorff.

is compact.

is a continuous bijection from a compact space to

a Hausdorff space, hence a homeomorphism between compact spaces. Q

Thanks to this theorem, the topology of IRP

can be defined

through either process depending on which is more convenient at the time. In so doing the homeomorphism

()> is regarded as an identifica-

tion, i.e. the previous diagram is reduced to L

The inverse of

\{0}

would give the commutative diagram p

3RP

where

p : x

IM'

The (provisional) culmination of the matter is contained in the

79

2.8 Theorem: For any n £ W , K P

is a compact topological manifold of

dimension n.

Proof: Condition (i) of 1.1, and compactness, were proved in 2.7. (ii) Let {X } Since

p

be a countable basis of the topology of S '.

is open (2.7) , p(X ) is an open subset of K P If

U

is an open subset of E P , there exists a sequence of n

-1

integers

for all r.

(r.) such that

p

(U) = u x j£N

hence, because

p is

j

surjective. = p ( u x ) = u p ( x )

U =

r

J3N

Therefore

{p(X )}

j

r

j<3J

.

j

i s a basis of the topology of r<3R

(iii) Clearly

L.

For i e { l , . . . , n + l }

i s an open s u b s e t of B

let

e 1R Rn

L. = { (x , . . . ,x

\ { o } . L e t U. = q ( L . ) ,

q

a s above.

Observe t h a t

Uj

since

(x , ...,x

= q"1(q(Li))

= L±

. ) R(y. , ... ,y

,) and y, *• 0

imply

x. * 0.

So U, is

an open subset of Now define

±

n+1,

: L.

X

X

and

: 3R

then |>.

(that

: U. -s- M

such t h a t

$ . = <j>. ° q

$. does factor is trivial) and

jj, = q °
•*• U .

:

'

i

X

X

' ' ' " ' X

i

X

i

80

(N.B.: i f

n = 2,

tfi

i s the 'passage to homogeneous coordinates' of

c l a s s i c a l projective geometry.) Checking that

$.

and

IJJ.

are one-to-one, inverse to each

other, i s easy. As for

q

L

,

since the operation of So the continuity of and

IJJ

i t is continuous by definition, and open by 2.6

i It*

$.

on ]R and

¥. ,

\{o}

r e s t r i c t s to an operation on

which i s obvious, implies t h a t of

L. . $.

.. Finally

(U.,cj).,]R )

i s a chart of

HP

It only remains to remark that E follows that ]RP =

u U, because

q

\{O} =

n+1 u L., whence it

is surjective. Q

1

Complex Case Here the two r e l a t i o n s a r e : n+1 either

on

y = Xx,

<E or on

morphic with such t h a t

\{o}: S

R

y = Xx

xR'y ,

*

i f f there e x i s t s

X e (E =
which i s the unit sphere of

(as Euclidian spaces): (taking

S

1

t o be

xS'y

<E

iff

,

such t h a t itself

there e x i s t s

isoXe S

{z e
The classes respectively consist of ([-vector spaces of dimension 1 with the origin deleted, and of c i r c l e s . Analogously to the r e a l case:

2.9 Theorem: The i n j e c t i o n

between compact spaces.

i'

: S

-• I

\{o}

induces a homeomorphism

81

Proof: copied from 2.7, with the groups K and S

and {+1,-1}

respectively; the map <5 replaced by

6'

definition but for complex variables; the map y

replaced by <E

which has the same formal by S' (S

The topology of

fflP

2n+1.2-

D

i s then defined by either method and t h i s

time, 2.10 Theorem: for any n e U, (EP dimension

i s a compact topological manifold of

2n.

Proof: copied from 2.8 with fl: instead of K. because

<E

i s isomorphic to ]R

3

THE IMPLICIT

The dimension i s

(as Euclidian spaces) .

2n

D

FUNCTION THEOREM

The second method used i n t h e case of spheres ( 2 . 2 ( i i i ) (2)) can be s y s t e m a t i c a l l y g e n e r a l i s e d and provides a very p l e a s a n t type of proof which we s h a l l often r e s o r t t o i n the sequel. The standard s i t u a t i o n i s as follows: two i n t e g e r s OT n M ; p C -maps f , . . . , f

n, p

such t h a t

: W •* K;

1 < p < n;

a point

W an open subset of

(a , . . . , a ) e W such t h a t

f (a , . . . , a ) = . . . = f (a , . . . , a ) = 0 1 1 n p 1 n and t h a t t h e Jacobian matrix 3f. (

1 < j
p

columns on the right is non-zero, and adapt the notations

accordingly: set q = n-p; call the first q coordinates principal; if n q (x ,...,x ) e R , write y. for x , x = (x,...,x) e K , y = (y,, . . . ,y J e E , and (x, , . . . ,x ) = (x,y) e E *]R = TR . Similarly 1 p In

and in particular,

a

, = b., a = (an,...,a ) e R ,

82

b = ( ^ . . . . . b ) e Vp, ( a 1 , . . . , a n ) = (a,b) e E 9 x» P = The hypotheses can then be rephrased as: . . . = f (a,b) = 0 P 3f.

(a,b)

* O

< j