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London Mathematical Society Lecture Note Series. 99
Algebraic Topology via Differential Geometry M. Karoubi and C. Leruste U.FM. de Mattematiques, Universiti Paris VII
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CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney
CAMBRIDGE u n i v e r s i t y p r e s s
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521317146 © Cambridge University Press 1987 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1987 Reprinted 1989 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication Data ISBN 978-0-521-31714-6 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.
A Marie^Antoinette
M.K.
Contents
Introduction
ix
I.
Algebraic preliminaries
1
II.
Differential forms on an open subset of R n
III. Differentiable manifolds
30 70
IV. De Rham cohomology of differentiable manifolds
121
V.
161
Computing cohomology
VI. Poincare duality - Lefschetz' theorem
214
Appendix A. Stokes'theorem
319
Appendix B. Chern character and non-commutative De Rham cohomology
333
Bibliography
360
Index
361
INTRODUCTION
There are several ways of presenting Algebraic Topology: singular homology and cohomology, homotopy theory, K-theory, cobordism, etc. The one we have chosen is based on the De Rham cohomology of differentiable manifolds. The advantages of this presentation are many: an algebraic formalism reduced to its minimum (tensor calculus, exterior algebra); a pleasant and natural multiplicative structure for the cohomology; selfcontained proofs without reference to the literature. Another advantage lies in the kind of mathematical culture - varied and not too specialised which is necessary for the understanding of these notes: they therefore form a course accessible to students in their final year of undergraduate studies. On the other hand, this approach clearly has its limits in so far as the torsion phenomena of cohomology are not dealt with (though this can be achieved through the 'modern' presentation of De Rham cohomology due to Sullivan, Grothendieck and Miller: see the paper by H. Cartan [3]1 . Similarly, homotopy theory is virtually absent. So this course should be seen as a first step towards Algebraic Topology and the geometry of manifolds. For a deeper study, we refer the Reader to, e.g., the following: [6], [8], [12], [15], [16], [17], [13], [18]. With the exception of the two appendices, this book was first published, in French, by the Publications mathematiques de l'Universite Paris VII, in 1982. Stokes' theorem, being a classic related to the main topic and indeed hinted at in the text (VI.1.22 (ii) ) - was included in full as Appendix A. Moreover, the important developments in 'non-commutative' Differential Geometry which have occurred since 1982 led the first author to write Appendix B as an introduction to the theory of characteristic classes in this new setting. We hope it will make the Reader want to pursue the matter further.
Finally, we wish to express our warmest thanks to Professor I.M. James, not only for his constant support, but even for being kind enough to check and correct the t r a n s l a t i o n carefully - any incongruities l e f t are the second author's
responsibility:
Max Karoubi Christian Leruste Universite Paris VII December 1986.
N.B.
A number between square brackets
(e.g. [ 7 ] ) , i s a reference to the
Bibliography at the end of the book. Cross-references within the text are given - by two numbers in Arabic numerals (e.g. 3.17) i f they indicate another paragraph of the same chapter; - from chapter to chapter, by three numbers, with the chapter number, in front, in Roman numerals (e.g. IV.2.1).
I
ALGEBRAIC PRELIMINARIES
Algebra i s n o t s t u d i e d h e r e of t h i s
f o r i t s own s a k e : t h e only purpose
c h a p t e r i s t o i n t r o d u c e t h o s e concepts and r e s u l t s which w i l l be
needed i n t h e s e q u e l .
Some p r o o f s
a r e reduced t o s k e t c h e s
and t h e r e a d e r i s
invited to consult the relevant l i t e r a t u r e , which i s abundant and excellent (e.g.
[ 2 ] , [14]) .
Throughout the chapter, spaces are over maps
K.
f : E •+ F,
denoted by
E, F
field.
All vector
are two vector spaces, the set of a l l
linear
equipped with the canonical vector space s t r u c t u r e ,
is
L(E,F) .
1
BILINEAR MAPS
1.1 Definition: from
If
K i s a (commutative)
E x F
Let to
G.
the map y fr f (x,y) x \+ f(x,y)
E,
F,
G be t h r e e v e c t o r s p a c e s and
We say t h a t : F ->- G
: E ->• G i s a l s o
f
i s a bilinear
f
a function
map i f f f o r a l l
i s l i n e a r and f o r a l l
y e F
x e E
t h e map
linear.
The s e t of a l l such maps i s equipped n a t u r a l l y with a v e c t o r space s t r u c t u r e .
I t i s denoted by
1.2
E, F , G a r e f i n i t e - d i m e n s i o n a l ,
Theorem:
q = dim G,
the maps
If then
dim 8(E,F;G)
with
i]
6
is
r
s
lr
js
y,
'k
(l i s b i l i n e a r , then
( is bilinear,
^
factors through TT
and gives a unique linear map I(J which makes the following diagram commutative: EXF
,(EXP)
K(EXF)/R(E,F) One then takes 2.2 Remarks: 'I' \* ty ° j
E 8 F
t o be
K
X
/R(E,F)
and j = i • o. D
(i) One can rephrase a l l t h i s by saying t h a t the mapping
gives an isomorphism between ( i i ) The p a i r
(E 8 F, j)
L(E H F,G)
hence r e f e r r e d t o as ' t h e 1 t e n s o r product of Proof of (ii):
Suppose two p a i r s
and B(E,F;G) .
i s unique up t o canonical isomorphism, E and F.
(E B F , j ) and
c o n d i t i o n s . One gets two commutative diagrams
(E K ' F , j ' )
s a t i s f y the
E xF
E B F where t h e p a r t of 2 . 1 ' s (E 8 F , j ) ,
(G,)
i s p l a y e d s u c c e s s i v e l y by
(E B ' F , j ' )
and
The diagram
E xF
E B F o b t a i n e d by composition i s t h e r e f o r e commutative and \b' ° tb = I d E IS F by t h e u n i q u e n e s s p r o p e r t y . S i m i l a r l y , I t i s traditional to write
i|) » I | I ' = I d x Hy
for
, . Q j(x,y) .
The
computational properties of tensor products are then summed up in the 2. 3 Proposition (i)
For X e K , x e E , y £ F : x 8 0
(ii)
for
x , x ' e E , y € F : (x+x 1 )
for
x £ E , y , y ' £ F : x ® (y+y 1 ) = x 8 y
Any z £ E 8 F
If
E,
+ x » y ' . sum z = ][ x . B y ,
where
This decomposition i s n o t unique, G a n d any
$ £_B(E,F;G) , ty i s d e f i n e d b y
x. e E, y. £ F . F
a r e f i n i t e - d i m e n s i o n a l with r e s p e c t i v e dimensions
respective bases dimension
Proof:
® y == x S y + x 1 8 y
c a n b e w r i t t e n a s a finite
F o r any v e c t o r s p a c e
where (iv)
hence
= 0 H y =0
x. £ E , y . £ F . (iii)
A(x K y) = (Ax) K> y = x B(Xy) ;
np
{e , . . . , £
and b a s i s
}
a n d {n , . . . , n
(i) , ( i i ) , ( i i i ) are l e f t
np.
Now, i f
D
Ea F
G has dimension
dim 8(E,F;G) = npq (cf. 1.2) : so must be independent.
then
p and
E K F has
1 < i < n, 1 < j < p } .
t o the p a t i e n t
(iv) Because of (i) and ( i i ) , B g e n e r a t e s fore n o t exceed
•-V-} ,
B = { e. 8 n .
n,
dim E SI F
reader. whose dimension can t h e r e q,
must be
dim L(E IS F,G) = np
and t h e
e. H r\.
2.4 Notation
{Abuse of):
Because of
"the l i n e a r map form to the s l i g h t l y
(ii)
we s h a l l
abbreviate
IJJ: E » F + G which sends an element of the
x K y to z" incorrect
"ip: E B F ->• G :
xHyh>-z".
The special case when one of the vector spaces involved is the ground field keeps recurring in the sequel. We give a full description of it as a first example. The very structure of any vector space
E
includes a bilinear
map
y: K x E -s- E : U,x) h- Ax.
2.5
Theorem
: The p a i r
tensor product
Proof: the
Let
F
commutative,
the conditions
which d e f i n e t h e
K B E.
be a v e c t o r
space and
the only p o s s i b i l i t y
JI(1,X)
i)j°yU,x)
(i)
fulfills
f.
KxE + F
a bilinear
map. To make
diagram
since
2.6
(E,u)
= x
= IJJUX)
by d e f i n i t i o n . = *(l,Xx)
i s , for a l l
x e E,
to set
ii(x)
= (l,x)
Then
= A(l,x) = =
<j>(A,x)
w h i c h shows t h a t
=
Ai[)(x) w h i c h shows t h a t
<j) = iji ° p , I|I
is linear.
D
Corollary: There e x i s t s
(ii)
If
E
the b a s i s
is
a canonical
finite-dimensional
I t i s clear that
v: K B E - > - E
with b a s i s
{1 8 E , . . . , 1 « E } ( c f . 1 n
r e g a r d t h i s isomorphism as an
Proof:
isomorphism
2.3(iv))
identification.
t h e diagram
: 1 B x [* Ax.
B = { E , , . . . , E }, v sends n 1 onto B. I t i s u s u a l t o
Kx E
KKE
i s c o m m u t a t i v e . Use 2 . 2 ( i i ) .
2.7 Tensor product
Q
of Abelian
groups
A similar construction can be performed with
7L instead of
K
and E-modules ( i . e . Abelian groups) instead of K-vector spaces. The r e s u l t i s the tensor product of the two Abelian groups. This e n t i r e section
(except 2.3(iv)
and 2 . 6 ( i i ) '.) and most -
though not a l l - of what follows remain v a l i d . We shall not look into the matter in any d e t a i l since we hardly need more than the bare
3
COMMUTATIVITY.
definition.
ASSOCIATIVITY
The formal p r o p e r t i e s gathered in t h i s section w i l l not come as a surprise.
3.1 Theorem: Let
E,
F
be two vector spaces. The map
i s an i s o m o r p h i s m .
Proof: The 'switch map' T
T: E xF + F * E : (x,y) h- (y,x)
induce the l i n e a r maps
< > j and
T
$' x
ExF
-F x E
E H F-
-F IS E-
and i t s inverse
in the following diagram:
-1
•ExF
r with
<j>(x 0 y) = y H x
•E IS F
by d e f i n i t i o n .
By t h e u n i q u e n e s s p r o p e r t y
| and
$'
a r e i n v e r s e t o each
other.
3.2 Theorem: Let
E, F, G be three vector spaces. The map
(E H F) 8 G •* E B (F S G) : (x B y) IS z t* x H (y IS z)
i s an isomorphism.
Proof: I t i s an immediate corollary of t h e following construction
(which
does not lack i n t r i n s i c i n t e r e s t ) : Let
E, F, G, H be four vector spaces. A map
i s s a i d t o be trilinear
i f f for a l l
x e E, y e F, z e G,
: E x F x G + H the maps
E -> H : 5 h- 4.(5,y,z) F •+ H : n b- (x,ri,z) G -*- H : S H- $(x,y,C)
are l i n e a r (cf. 1.1). The aim, by analogy with 2 . 1 , i s now t o construct a vector space, t o be denoted by
E 8 F SG,
and a t r i l i n e a r map
such t h a t , given a vector space
u : E X F X G + E B F KG
H and a t r i l i n e a r map
-. E X F xG ->• H,
there e x i s t s a unique l i n e a r map to: E 8 F 8 G •» H which makes the following diagram commutative:
E x F x G-
E 8 F IS G "
3.3 Theorem: This construction i s always possible and the s o l u t i o n i s unique up t o canonical isomorphism.
Proof: (i) Uniqueness as in 2 . 2 ( i i ) . ( i i ) Left-handed proof of e x i s t e n c e : F i r s t show t h a t there e x i s t s a unique b i l i n e a r map A: (E 8 F) x G -*• H which makes the following diagram commutative:
E xF x G j X I d
•
•• H
Q
(E H F) x G
This i s true because, due t o
< > j being t r i l i n e a r ,
for a l l
z e G the map
: E x F -*• H : (x,y) \* ^(,x,y,z)
i s b i l i n e a r and therefore induces a unique
F •+ H which makes the following diagram commutative
l i n e a r map if (use 2.1) :
E H F
clearly the one possible candidate for
A i s defined by
A(x & y,z) = i> | (x 53 y) = H which makes the following diagram commutative:
(E H F) x G
V (E S3 F) ® G
J u s t check t h a t
un = j
( (E & F) B G, u ) (iii)
°(j
x
Id )
is
trilinear
as above, b u t s t a r t with
the s o l u t i o n i s now (E B (F H G) , u
Because of
morphic under
(i) , a
(E Si F) S G
as
and the p a i r
seen t o be a s o l u t i o n .
Right-handed proof of e x i s t e n c e :
instead: (iv)
is
1
and
where
E (a (F a G)
E x (F KG)
u (x,y,z) = x
are c a n o n i c a l l y
(y C9 z ) • iso-
follows:
ExFx G
(E
i.e.
a
F)
a
G
E 8 (F a G)
a((x S y) (Hz) = x (2 (y H z) .
3.4 Remark: Theorem 3.2 and Construction 3.3 can e a s i l y be extended t o v a r i a b l e s , n > 4.
In a l l cases, canonical isomorphisms are t r e a t e d as
i d e n t i f i c a t i o n s , brackets are dispensed with, or placed according to convenience or fancy.
n
4
TENSOR PRODUCT OF LINEAR MAPS Let
f
E, E1 , F, F '
: E ->• E' , g : F -»• F
f xg
4 . 1 Theorem:
1
be four v e c t o r spaces
and
two l i n e a r maps. Consider t h e map
: ExF-»E' XF' :
There e x i s t s
(x,y)
^
(f(x),g(y)).
a unique l i n e a r map, denoted by
f 53 g,
which
makes the following diagram commutative: f xg E' x F1
ExF j
j E H F
-E
1
1
8 F1
f H g In o t h e r words, t h i s map i s all
defined by
Proof: J u s t check t h a t
j
1
° (f xg)
is bilinear,
4.2 Remark: This can be extended t o f.
(f (3 g) (x H y) = f (x)
& g(y)
for
x e E, y e F .
£ L(E.,E!),
i = l,...,n,
n
which i s
l i n e a r maps,
trivial. Q
n > 3:
given
the l i n e a r map
f, « . . . 18 f :E B...8E •» E1 B . . . g Ein 1 n 1 n 1 is
defined by
(f, S . . . a f ) (x, 1 n l (cf.
H . . . B x ) = f, (x.) nil
B ... B f
(x ) nn
3.4).
The computational p r o p e r t i e s are summed up for
n = 2
4. 3 Proposi tion: (i) f
Let
E, E 1 , E " , F , F 1 , F "
: E' •+• E " , g : F ->- F ' , g
(f 1 ° f)
1
be s i x v e c t o r s p a c e s : F
1
-* F "
B ( g 1 o g) = ( f
four
B g1)
o
and
l i n e a r maps.
( f a g ) .
f
: E -»• E' Then
in
the
10 (ii) For any vector spaces
E, F,
I i , SI IoL = Id rj
(iii)
f : E •+ E1 , g : F -+ F '
If
isomorphism and (f B g)
= (f
(iv) L e t E, E1 , F , F 1 g, g 1 : F -»- F 1
r
H
F
a r e i s o m o r p h i s m s , then ) 81 (g
Bg = f ® g + f '
f 8! g
i s an
).
be four v e c t o r s p a c e s ,
four l i n e a r maps and X e K.
(f+f)
. E
f,
: E •+ E1 ,
f
Then
Kg
(Af) S3 g = A(f B g) f B (g+g 1 ) = f B g + f B g ' f B (Xg) = X(f Proof-
Left t o t h e r e a d e r ;
5
B
g) .
( i i i ) i s a consequence of ( i ) and ( i i ) . D
TENSOR PRODUCT WITH A DIRECT SUM ( Recall t h a t , i f
indexed by a s e t
I,
(E. ) .
the direct
'DISTRIBUTIVITY')
i s a family of vector spaces
S :
((x. ) .
, y) |+ ( x i B y ) ,
and makes t h e following diagram commutative f o r a l l
j
e
I :
11
E B Fa j
id F
B
E. K F
Conversely i t i s precisely the universal property of there exists a unique linear map diagram commutative for a l l
y : S -> E 0 F
S
that
which makes the following
j € I :
•E IS F
a . 8 Id E. B F 3 There results for a l l
j e I
a commutative diagram
j E. B F-
hence
S ° y = Id
•E. 8 F 3
s by the uniqueness property for d i r e c t sums,
A direct computation shows that
Y ° 6((x.
i
= Y(
iel
iel
a.(x.)»y=(x.).
By.
iel
(Use previous diagrams. All sums are f i n i t e . )
5.2 Corollary:
For any family of v e c t o r spaces
(F.) . J j £ J
space
E,
EB(ffl F.) — ® (E B F . ) . 3 jeJ -1 jeJ
For any p a i r of
families
( © E. ) a ( ® F.) = e (E. B F.) . D iel 1 jeJ : (i, j)el x J 1
and any v e c t o r
12
Proof: First assertion via 3.1; second assertion in two steps.
5.3
Corollary:
Let
E,
F
be two vector
spaces
and
{ e . } .
D
,{ n . } .
i i £I respective bases { E , a TI } , .
l Proof:
( f i n i t e o r n o t : compare w i t h 2 . 3 ( i v ) ) .
.,
i s a b a s i s of
E SI F .
3 (i,]) eI x j By d e f i n i t i o n
of a basis,
6S K e . a n d F = ® Kn . . X 3 iel jeJ KE . & Kn . S K ( E . S n . ) : t h i s i s 2 . 3 ( i v ) w i t h
Apply 5 . 2 knowing t h a t n = p = 1.
E =
[]
6 6.1
j j£ J
Then
EXACT SEQUENCES
Definitions:
(i) Let
G, G1 , G" be three Abelian groups, a : G' ->• G, g : G •+ G" two G1 -»• G -»• G"
group homomorphisms. The sequence
i s said to be exact i f f
Im a = Ker g. (ii) Let
X be a subset of 7L ( f i n i t e or i n f i n i t e
at e i t h e r or both ends)
Let
(G ) be a family of Abelian groups and, for a l l n £ X such that n n £X (n+1) £ X, l e t v : G ->• G be a group homomorphism. n n n+1 The sequence . . . +G
—Z±*0
n-1n
n
- — ^ —G * ... n+1
is said to be exact iff all 3-term sequences contained in i t are exact in the sense of (i) . (iii) In particular, a 5-term exact sequence of the type
0 •+ G1 " G * G" •+ 0
(i.e.
a mono, g epi
and
Im a = Ker g)
i s known as a short
exact
sequence
(s.e.s) .
6.2 Remark: The above definitions are applicable to vector
spaces
and
linear
maps in l i e u of Abelian groups and group homomorphisms.
6.3 Examples: (i)
The sequence
O ->• G
the c a n o n i c a l i n j e c t i o n
->• G ffi G
-*• G
and p r o j e c t i o n ,
is
-*• 0 ,
where t h e arrows
a s.e.s.
represent
13
(ii) The sequence
O -*- 7L/2 Z -*-ffi/4 Z5 * 2 / 2 ZZ ->• O , where
multiplication by 2 and g by
Id
,
f
i s induced by
is a s.e.s.
However 7Z/4 TZ and ZS/2 EfflZ/2 zZ are not isomorphic with each other, hence the 6.4 Definition: A s.e.s.
0->-A->-B-*C-»-0
i s said t o be split
i f f there
exists an isomorphism $
B •+ AffiC which makes the following diagram
commutative
• AffiC
where
a
i s the canonical injection,
ir
the canonical p r o j e c t i o n .
A
6.5 Lemma: The following propositions are equivalent: (i) The s . e . s .
0
(ii) The epimorphism
>A -2-* B -=-» C
>O i s s p l i t
g has a right inverse (or retraction) &
(iii) The monomorphism
a has a left inverse
e.
Proof: ( i i ) =* ( i i i ) : for a l l b e B, b - 6 ° 6 (b) e Ker g = Im a: there e x i s t s a unique
a e A such t h a t
( i i i ) =* ( i i ) : i f and b e B with 6(o) (i)
ct(a) = b - 6 » B ( b ) .
g(b) = c, b - a ° e(b)
=b-a°e(b). (ii) and (iii) : projection
injection
a
Set e(b) = a.
b e Ker g = Im a, a ° e (b) = b .
is a right inverse of
Therefore, i f
c e C
depends on c only. Set is a left inverse of
TT .
Set
e = IT ° | and
a
and A
& =
(i)
(ii) or (iii) if either 5 or e are given, construct the other as above ( i f both are given and do not t a l l y , drop one of them -) . Set (b) = ( e ( b ) , 8 ( b ) ) , < > j (a,c) = a (a) + S(c)
and check t h a t the requirements
are met. D 6.6 Corollary:
Any s . e . s . of vector
spaces i s s p l i t .
Proof-. Let { c . } . T be a basis of C. For a l l i e I , choose b . e B i iel i (c.) = b : this defines a retraction. Q such that g (b ) Set i i i
14 As could be e x p e c t e d , we s h a l l now i n v e s t i g a t e t h e b e h a v i o u r of e x a c t sequences under t e n s o r p r o d u c t .
6.7 Theorem:
Let
O •+ G1 -»- G -> G" -* O be a s . e . s . of Abelian groups and H
an Abelian g r o u p . Then t h e sequence d
H
G" • H
*G BH
> G" K H
is exact.
Proof: o
1 ) That 2°)
BKId
i s epi i s
clear.
(i) Im (aBId ) c Ker (8 B i d )
because
o (aBIdjj) = (8 ° a) H 1 ^ = 0 B Id f i = 0 . ( i i ) Conversely: because of 1 ° ) , G" B H = G B H / Ker (gKId ) ; thanks to 2 ) (i) , BHId factors through Im (aHId ) i n t o a map H H 6: G B H / Im (a 8 Id ) ->• G" 8 H :
it suffices to show that x
ee G G be be such such that
6 is an isomorphism. Now, let x" e G", x and
g(x) = 8(x ) = x":
then
x - x
e Ker g = Im a and,
for all y e H,
x B y - x B y =
Hence a w e l l - d e f i n e d
(x-x ) 81 y e Im
(and o b v i o u s l y b i l i n e a r ) map
X : G" x H -s- G B H / Im (a 8 I d )
: (x",y) f* C l a s s of (x a y) .
H One e a s i l y checks t h a t t h e induced map p: G" B H + G B H / I n (a B I d )
i s t h e i n v e r s e of
6 . 8 Important
afild
Remark:
s.e.s. 6.3(ii),
In g e n e r a l
H = 2Z / 22Z
6.0
i s n o t mono: f o r i n s t a n c e t a k e t h e
and f i n d o u t what happens t o
1 8 1 € 71/2 7L B 2Z /2Z, .
The only 'favourable' particular
case i s that of s p l i t s . e . s ' s , hence in
15
6.9 Theorem: A s . e . s . of vector any vector space
F
0 •+• E1 + E I E" + 0
spaces
a s.e.s.
oBId E1 8 F
> E HF
giSId —=-> E" B F ——•—~> 0 .
Proof: As all s.e.s.'s of vector spaces are split, £
induces for
a has a left inverse
(6.5) . Then
(e m Idp) o (a B Idp) = (e - a) S Id p = Id E , B Idp = ldE,
which shows t h a t
a H IcL
g p
i s mono.
rv
ft
6.10 Corollary. Let E* -*• E -»• E" be an exact sequence of vector spaces. Then for any vector space F the sequence ixiaid E'
gs-HId
EIXF
> E H F
> E" H F
i s exact.
Proof: By canonical decomposition of homomorphisms one gets the s.e.s
> E1
0
> Ker a
O
» Ker g
0
> Im g
a
where
following
. 's
a. ° a
= a,
2
"l 6
*E
» E"
g , ° g. = 8,
0
> Ker a ® F
O
> Ker 8 B F
1
> Im a
>0
> Im g
> O
* Coker g
>O
Coker g = E " / I m 0 . * E1 8 F a
2 ISIcL,
Hence t h e s . e . s . ' s
«,8lIdF > Im a H F
>O
* Im 8 8 F
> 0
8 8Id »E HF
2
Im g B F where
(a
8 Id ) o (a
—
=-* E" B F
.B Id ) = a 8 Idp
and
» Coker g (S F
16
(6
81 Id ) o (A
g Id ) = g » I d
,
which shows t h a t
Im (a B Id ) = Im (a 2 81 Id ) = Ker (g
» Id p )
= Ker (g ® Id )
6.11 Remark: The equivalent of 6.10 for Abelian groups i s false (cf.
in
•
general
6.8) . The b e s t
'improvement' available for 6.7 c o n s i s t s in weakening ex 6 G' -+ G •*• G" -* 0 being e x a c t , with the same conclusion.
the hypothesis to
7
TENSOR ALGEBRA The question t h i s section deals with i s : how to endow a linear
situation with multiplicative resources?
Recall that a graded algebra over
K i s an algebra (with unit)
A whi ch - as a vector space i s the direct sum of a countable family of K-vector spaces
(A )
: A = ffl A kW
€
-
as a r i n g s a t i s f i e s
for a l l
7.1 Definitions:
n,
If
peW,
the
condition
A . A c A n p n+p
E i s a K-vector space, s e t
( i ) T°(E) = K (ii)
T X (E) = E
k Ok ( i i i ) f o r any i n t e g e r k > 2 , T (E) = E = E B...H E (iv) T(E) = e T ( E ) . keHN 7.2 Theorem: that, for
T h i s T(E) can b e g i v e n t h e s t r u c t u r e o f a g r a d e d K - a l g e b r a such a =
y
a
n
•«
and b =
.b =
Tb , a
n
P
neW
e T n (E) ,
the map
n
and
e T P (E) ,
pen
Y c L q qdN
where
p
c = q
a 8b L7 n p n+p=q
are s t r i c t l y p o s i t i v e ,
(x, S. . .H x ) B (y
1
b P
I t i s known as the tensor algebra
Proof: If
(k f a c t o r s )
n
l
a . . . B y ) •+ x
pi
of
E.
T (E) 8 r 8. . .8 x
(E) — T
S y
n
l
* (E)
8. . .8 y
p
under
17
which we agreed to regard as an identification (cf. 3.2, 3.4) . If
n = 0
or p = 0
K H TP(E)
the isomorphism is that of 2.6:
: X H X " Ax
or i t s analogue Tn(E)
B K •+ T n ( E )
: x B> A \-+ Ax
which we regard also as i d e n t i f i c a t i o n s . r a Bb e T"' Tn+P (E) (E) when n p a e T (E) and b £ Tr (E) . Since only a f i n i t e number of a ' s and n p n b 's are non-zero, only a finite number of c ' s are non-zero: a product
So in a l l cases we consider that
i s thus defined on task.
Checking i t s formal properties i s a routine
D
7.3 Remark: If (E)
T(E).
E has dimension 1 with basis
i s 1-dimensional generated by
convention that
X = 1 e K) :
By definition
{X},
X f : E -»- A
a linear map.
There exists a unique homomorphism of algebras A : T(E) -* A which makes the following diagram commutative:
T(E) Proof: (i) Uniqueness: as a vector space
T (E)
is generated by the elements
x, where x. £ E, i = 1, ,k, so that two algebra k homomorphi sms defined on T(E) are equal iff they coincide on T (E) = E. of the form x,
Now the diagram requests that A = , which implies that two solutions are necessarily equal.
18
( i i ) Existence: for
k > 1,
the map
i s k - l i n e a r , hence induces a l i n e a r map A : Tk(E) •* A : X B.. . .H x, [• Set
1
: K + S : a [*• a . l ,
A=
J
A
and check the formal p r o p e r t i e s . Q
7.5 Remark: As in a l l s i m i l a r cases already seen or about to be seen, and for the same reasons (cf. 2.2 ( i i ) ) , any other s o l u t i o n to t h i s u n i v e r s a l problem i s n a t u r a l l y isomorphic with
T(E) .
In the s p i r i t of Section 4 we turn t o maps. Recall t h a t a homomorphism of graded algebras of degree ( : A ->• B such t h a t for a l l
0
i s an algebra homomorphism
n e W • T k (F) : x k times
8
». . . B jt
|+- f (x ) B. . . B f (x ) „
EXTERIOR POWERS. EXTERIOR ALGEBRA
Amongst k - l i n e a r maps, some play a p a r t i c u l a r l y important role in t h i s book (- and elsewhere) : those which 'behave well 1 under permutation of t h e i r v a r i a b l e s . What we mean by ' w e l l ' i s t h i s :
19
8.1
Definition:
spaces)
A k - l i n e a r map y : E •* F
i s s a i d t o b e skew-symmetric
e x i s t two i n d i c e s
i
iff
(where
y(x , . . . , 5 t )
E, F = 0
are vector whenever
and j , l < i < k , l < j < k , i * j ,
such
there
that
x. = x . . A classical combinatorial computation proves the
8.2 Proposition: U,...,k},
If y
is skew-symmetric and a
then
0(1)'"""' a (k) where
e
is a permutation of
a
1'**"' K
is the signature of a.
We then proceed to construct an algebra which can handle this new situation.
8.3 Definitions: Let E (i) Say that
be a vector space.
x B...E x
exist indices
i
e T (E)
is an element with repetitions iff there
and j , 1 < i < k,
1 < j < k, i * j,
such that
x. = x .. (ii) The quotient of T (E) by the subspace generated by the elements with th k repetitions is called the k exterior power of E and denoted by A (E). The image of x
K...B x
T (E) •+ A (E) is denoted by x
under the canonical epimorphism
A...A x .
8.4 Remark: By definition, in A (E) elements with repetitions are zero: x
A...A x
= 0 whenever there exist indices
1 < j < k, i * j ,
such that
i
and j, 1 < i < k,
x. = x..
As could be expected,
A (E)
i s the solution of a universal
problem:
8.5 Theorem: Let
E, F
symmetric k - l i n e a r map.
be two vector spaces and
y ; E -»• F
There e x i s t s a unique linear
which makes the following diagram commutative:
map
a skew-
6 : A (E) •+ F
2O
Ak(E) k k u : E -> T (E) i s the k - l i n e a r map defined in Section 3 (see 3.3 k k and 3.4) and T T : T (E) -*• A (E) i s the canonical epimorphism.
where
Proof: (i)
If
6
i s a solution,
r e l a t i v e to IT
T (E).
& ° TT
Thus
5 °n
i s a solution of the universal problem i s uniquely determined:
so i s
S
since
is epi.
(ii)
Let
a) : T (E) ->• F
Because of
y
be the map deduced from
being skew-symmetric,
y
io(Ker TT) = O,
as in Section 3. hence
10
factors
into
6. • 8.6 Definition:
If
(i)
A°(E) = K
(ii)
A1(E) = E
(iii)
A(E) =
8.7 Theorem:
E
i s a vector space,
e Ak(E) . kdN This
A(E)
whose m u l t i p l i c a t i o n
can be given t h e s t r u c t u r e of a graded K-algebra
i s defined using b i l i n e a r maps
which send
(x, A...A X n , y, A . . . A y ) 1 " 1 p I t i s known as the exterior algebra of
Proof:
Let
set
6 n,p
the isomorphism
: T (E) x T^ (E) ->• T T (E) K Tr(E) — T
P
onto
A (E) xA (E) ->- A
xn A . . . A x A y 1 n l
(E)
A...A y . p
E.
(E)
* (E)
be t h e b i l i n e a r map which induces of 7 . 2 . Consider the commutative
diagram n
Since
p
T n (E)
x TP(E)
An(E)
x AP(E)
IT » i o — n+p n+p
n+p
^ £ —
T n + P (E)
i s skew-symmetric with r e s p e c t to i t s f i r s t
n
and
21
its
last
p
arguments
(indeed with r e s p e c t to all
its
arguments!) ,
-n i n t o the d e s i r e d b i l i n e a r map. IT o8 f a c t o r s through 7 r x IT it n+p n ,p n p simply remains t o check the formal p r o p e r t i e s .
8.8 Remark: The algebra then i n
A
P
(E):
A (E)
xAy = ( - l )
i s anticommutative: np
y A x.
if
Moreover,
x e A (E) ,
xAx = 0
if
x
It
y e A" (E) , i s of odd
degree The way
A(E)
i s c o n s t r u c t e d makes i t
following u n i v e r s a l problem (cf.
8.9 Theorem: Let such t h a t ,
A
for a l l
the s o l u t i o n of the
7.4):
be an a l g e b r a (with u n i t ) 2 x e E, <j>(x) =0.
and ij> : E -»• A
There e x i s t s a unique homomorphism of a l g e b r a s
a l i n e a r map
8 : A(E) -> A
which makes the following diagram commutative:
ME)'
(Note t h a t the obvious map
E -+• A(E)
i s mono) .
Proof: (i) by
For reasons 6(x
(ii)
To check
suffices that
similar
to 7.4(i),
t h e only possible
candidate
is
defined
A. . . A x ^ ) = (x ) . . .<J>(x. ) = 0 Now,
for
definition A vanishes
does work, on
whenever t h e r e
(y)(x)
) . . .(x)cj>(y)
x., ) • . .ef>(x. ) • •. < ( > ( x . ) • • -if>(x. ) = ±(x
start
Ker(T(E)
words
with x. = x . . 1 2 2 -1 2 = (x+y) - (x) - § (y) 0, hence
* j
. .
i s deleted.)
When we undertake t o compute the e x t e r i o r sum, we s h a l l r e a l i s e
A of 7.4. I t in other
Q
a l g e b r a of a d i r e c t
t h a t we need an a l g e b r a s t r u c t u r e on the t e n s o r product
22
of two a l g e b r a s .
8.10 Theorem:
If
In t h e n o n - g r a d e d c a s e ,
A
and
B
a r e two a l g e b r a s ,
b,
b
Proof:
Start
(where
a , a 1 £ A,
from t h e composite
Id A X T x I d B
A
i s an a l g e b r a w i t h
e B) .
PA»PB
AxB xA xB
where
A» B
(a 53 b) ( a ' K b ' ) = a a ' & b b 1
a m u l t i p l i c a t i o n d e f i n e d by 1
t h i s i s done n a t u r a l l y :
x
(resp.
i s t h e ' s w i t c h map1 and B) .
, A xB -—-
A xA xB xB
\i
(resp.
\i )
AH B
i s the multiplication in
D
Of c o u r s e t h i s can be e x t e n d e d unchanged t o t h e graded c a s e , but the signature
c o n d i t i o n i n 8.2 warns u s t h a t c a u t i o n s h o u l d be e x e r c i s e d .
Recall t h a t i f bigraded
A
(5.2) . I t i s (simply)
[A S B] = q 8.11 Theorem: I f
A
and
or
(i) (ii)
are graded,
A8 B
i s naturally
grading
ffi A IS B n p n+p=q
and
B
a r e two g r a d e d a l g e b r a s ,
into a g y raded alg y e b r a i n two ways: y either
B
g r a d e d by t h e ' t o t a l '
for
(a H b ) ( a ' S b ' ) = aa
ABB
can be made
a e A , b e B , a' e A , b ' e B , n' p' q r' 1
& bb'
(a 81 b) (a' B b ' ) = ( - l )
Pq
a a ' s bb 1 .
To d i s t i n g u i s h between t h e two c a s e s , we keep t h e o l d n o t a t i o n A
A
B
for the f i r s t
and write
A 81 B
for the second.
Proof: (For (ii) , instead of the ordinary switch map
T,
use the
'switch-
map-with-sign-condition':
x'(b,a)
= (-l)Pq
(a,b)
if
Checking the details i s straightforward.
a € A , b e B . q p Q
Thanks to t h i s , computing the exterior algebra of a direct sum
becomes easy:
set
23
8.12 Theorem: Let
E, F
be two vector spaces. There e x i s t s a canonical
isomorphism of graded algebras A(E a F) = A(E)
In p a r t i c u l a r ,
A
A(F) .
for any integer
Ak(E 8 F) =
K,
9 A1(E) B AJ(F) i+j=k
as vector spaces.
Proof: ( i ) The map | : E ffl F -> A(E) SI A (F) 4>(x,y) 2 = (x m l ) = x
2
: (x,y) [ > x B l * l B y
i s l i n e a r and
+ (x a l ) ( i 8 y) + ( l B y) (x a l ) + ( l a y ) 2
B l +
x B y - x B y + l B y
2
= 0.
Hence a u n i q u e homomorphism o f a l g e b r a s
9 : A(E ® F) •> A(E) B A(F)
after 8.9. (ii)
C o n v e r s e l y , t h e c a n o n i c a l monomorphisnis
e : E -• E © F ,
n : F -• E ® F
i n d u c e a l g e b r a homomorphisms
e : A(E) -* A(E © F)
which s e n d
,
n : A(F) •+ A(E ® F)
respectively x, A . . . A x I n
e A (E)
onto
(x_ ,0)
A . . . A (x , 0 ) e A (E ® F) I n
yn A . . . A y 1 p
6
AP(F)
onto
(0,y.) A . . . A ( 0 , y ) 1 p
and
e A
P
( E 9 F ) .
24
Set check t h a t
: A n (E) a A P (F) -* An
*
the
i|i
define
P
( E ffi F)
: a B b \+ £ (a)An (b) ;
an a l g e b r a homomorphism
1J1 : A(E) » A(F) ->• A(E 9 F) ;
check t h a t
9
and
ifi
are inverse t o each o t h e r .
There follows next
,...,E 1
}
E
be a f i n i t e - d i m e n s i o n a l
a b a s i s of
the
— kl (n-k) I AE.
is
A...A£,
such t h a t Ak(E)
k > n,
1 £ i
(p f a c t o r s ) :
2
£.
conditions,
of the = (£)
for
S...S
c
where
are equal t o
k < n
Furthermore
o f degree
and
(cf.
= K 9 Ke.
Ak(E)
a.
= O
h a s a b a s i s made
or
1: t h i s proves = {0}
for
8.12(i)) , the image of a
for a l l other i n d i c e s .
£ D
with
1.
1,
£.
=1,
and
that
k > n. e.
A...AE 1
isomorphism i s p r e c i s e l y
1
ct
£ a. ' s
1
p £ 2,
[A(Ke ) K---B A(KE ) ]
a up w i t h t h e p r o d u c t s
for
{e. H. . . S £'. }
A(Ke. ) = A°(K£. ) ffi A 1 (Ke.)
and
1 of degree 0,
In t h e s e
k
the singleton
t h i s b e i n g an e l e m e n t w i t h r e p e t i t i o n s
A P (K£. ) = {0}
dim Ak(E)
< i
algebras. Now
exactly
has dimension
(iterated),
A(E) = A(KeJ 1
basis
k 1 i k £ n, A (E)
an i n t e g e r ,
Because of 8.12
as g r a d e d
n = dim E;
and a b a s i s whose elements are the products
If
Proof:
k
vector space;
E.
n
If
£.
a r e s u l t which w i l l be used e x t e n s i v e l y in
chapter:
8.13 Theorem: Let {E
D
under the k
a S... H e
with
a.
= . . . = a.
= 1, a.
= 0
25
Finally, as with tensor algebras, a linear map induces a homomorphism between e x t e r i o r algebras: Let any integer
E,F be two vector spaces,
k > 1, A k (f)
f : E -> F a linear map. For
define
: Ak(E) -* Ak(F)
by s e t t i n g A k (f){x 1 A...A x^) = f(X;L) A...A f f x ^
.
8.14 Theorem: These maps are linear and their direct sum defines the unique homomorphism of algebras which makes the following diagram commutative:
A(E) A(f) Proof: Straightforward: here 9
f(x)
= f(x) A f(x) = 0.
SYMMETRIC POWERS. SYMMETRIC ALGEBRA
Having dealt with skew-symmetric maps, one could expect that amongst k-linear maps plain symmetric maps would behave 'even b e t t e r ' . As i t turns out, from our point of view they only behave 'almost as w e l l ' , in spite of which we dedicate t h i s section to them. However proofs w i l l be omitted as they can be adapted automatically from the previous section. 9.1 Definition: (x , . . . x . ) e E
x(x
A k - l i n e a r map x
and any permutation
a(l)
9.2 Definition: The k of
: E
^(k)'
= X(X
"* F o
1
1S
of
symmetric iff,
for a l l
{l,...,k},
V
symmetric power of a vector space
E
is the quotient
T (E) by the subspace generated by all the elements of the form
x, 63. . .43 x, - x ,,, 8. ..8 x with x 1 k ad) a(k) I It is denoted by
S (E) .
e E, o
a permutation of
{l,...,k}.
26 9.3 Theorem: Let
F be a vector space and
k-linear map. There e x i s t s a unique linear
X:
E
"*" F
a
symmetric
map
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 ]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) :
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 , 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. = 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, B and dg = O,
,1
6 = t d x - z d y - y d z whence by 2 . 1 ( i i i )
= (d 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 ' ° *(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).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(*(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 ( 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( (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
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 » 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) ^ 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
