From calculus to cohomology: de Rham cohomology and characteristic classes

=

): HP(V)

f*, where f: U

-+

-t

HP(U)

V is a smooth map homotopic to ¢.

Theorem 6.8 For p E l and open sets U, V, W in Euclidean spaces we have

(i) If t/>o, ¢1 : U

-+

V are homotopic continuous maps, then
(ii) If ¢>: U -+ V and 'If;: V

-+

'If;*: HP (W ) -t HP(U). (iii) If the continuous map ¢: U

-t

HP(u).

W are continuous, then ('If; o
-+

V is a homotopy equivalence, then

¢*: HP(V) -t HP (U) is an isomorphism.

=


o

6.

42

HOMOTOPY

f: U V with and 7/J, follows from the formula Proof. Choose a smooth map

¢1

---+

�

Dl('l/; o ¢) = DP(¢) o DP('!f;). In the general case, choose smooth maps f: U ---+ V and g: V ---+ and 7/J � g. Lemma 6.3 shows that 7/J o � g o f, and we get

W

with := f

(7/J 0 ¢)* = (g j)* = j* g* = * 'l/;*. 0

If 7/J: V

---+

0

0

U is a homotopy inverse to ¢, i.e. '1/J o := idu and

then it follows from (ii) that '!/;*: HP(U)

---+

o 7/J � idv,

HP(V) is inverse to *.

0

This result shows that HP(U) depends only on the homotopy type of U. In particular we have: Corollary 6.9 (Topological invariance) A homeomorphism h: U ---+ V benveen open sets in Euclidean spaces induces isomorphisms h*: HP(V) ---+ HP(U) for all p.

The corollary follows from Theorem 6.8.(iii), as h- 1 : V homotopy inverse to h.

Proof.

Corollary 6.10 If U � Rn is an open contractible set, then p > 0 and H0 ( U) = R.

---+

U is a 0

HP(U) = 0 when

Proof. Let F: U x [0, 1] -+ U be a homotopy from fo = idu to a constant map h with value xo E U. For x E U, F(x , t) defines a continuous curve in U,

which connects x to xo. Hence U is connected and H0(U) = R by Lemma 3.9. If p > 0 then Dl(ft): Dl(U) - DP(U) is the zero map. Hence by Theorem 6.8.(i) we get that ids,(u) =

and thus HP(U)

= 0.

fo = fi = o

0

In the proposition below, Rn is identified with the subspace Rn x {0} of Rn+l and R · 1 denotes the !-dimensional subspace consisting of constant functions.

6.

HOMOTOPY

Proposition 6.11 For an arbitrary closed subset

isomorphisms

43

A of R with A

i=

Rn we have

HP+ l (Rn+l - A) � HP(Rn - A) for p � 1 1 +l H (Rn - A) � H0(Rn - A)/R 1 H0(Rn+ l - A) � R. ·

Rn+ 1 = Rn x R, U1 = Rn x ( O,oo) U (Rn - A) x (-1, oo) u2 = Rn X ( -oo, 0) u (Rn - A) X (-oo, 1).

Proof. Define open subsets of

Then ul u u2 = Rn+l - A and ul n u2 = (Rn - A) X ( - 1 , 1). Let ¢: ul -+ ul be given by adding 1 to the (n + 1)-st coordinate. For x E U1 , U1 contains the line segments from x to ¢(x) and from ¢(x) to a fixed point in Rn x (O,oo). As in Example 6.5 we get homotopies from idu1 to ¢ and from ¢ to a constant map. It follows that ul is contractible. Analogously u2 is contractible, and HP(Uv) is described in Corollary 6.10. Let pr be the projection of U1 n U2 = (Rn - A) x ( -1, 1) on Rn - A. Define i : Rn - A -+ U1 n U2 by i(y) = (y, 0). We have pr o i = idR"-A and i o pr � idu1nu2• From Theorem 6.8.(iii) we conclude that is an isomorphism for every

p. Theorem 5.2 gives isomorphisms

8* : HP(Ul

n

U2 )

-+

HP+l (Rn+l - A)

for p � 1 . By composition with pr* one obtains the first part of Proposition 6.1 1 . Consider the exact sequence

0 -+ H0 (Rn+l - A) J..: H0(U1) E9 H0(U2) !; H0(U1 n U2 ) � H1(Rn+ l - A) -+ 0. An element of H0(UI) ffi H0(U2) is given by a pair of constant functions on U1 and U2 with values a1 and a2. Their image under J* is by Theorem 5.2 the constant function on ul n u2 with the value a l - a2. This shows that Ker 8* = Im J*

=R·1,

and we obtain the isomorphisms

H1 (Rn+l - A) � H0(U1 n U2)/R-1 � H0(Rn - A)/R · 1 .

We also have that dim(Im (!*)) = dim(Ker (J*)) = 1 so H0(Rn+l - A ) � R.D ,

44

6.

Addendum 6.12

HOMOTOPY

In the situation of Proposition 6. 1 I we have a diffeomorphism

R: Rn+ l - A

�

Rn+ l - A

defined by R(x1 , . . . , Xn , Xn+l) = (x1, . . . , Xn , - Xn+I). The induced linear map

is multiplication by -1 for p � 0. Proof.

In the notation of the proof above we have commutative diagrams, which the horizontal diffeomorphisms are restrictions of R:

m

In the proof of Proposition 6 . 1 I we saw that

is surjective. Therefore it is sufficient to show that R* o 8*([w]) = -8*([w]) for an arbitrary closed p-form w on U1 n U2. Using Theorem 5 . 1 we can find Wv E 0P(Uv), v = 0, 1, with w = jj(w1 ) -j2(w2)· The definition of 8* (see Definition 4.5) shows that 8*([w]) = [T] where T E QP+ l (Rn+ 1 - A) is determined by i� (T ) = d;,;.;v for v = 1, 2. Furthermore we get

-R{jw = R0 jz(w2) - R{j ji(w1) = ji(Riw2) - J2( R2w 1 ) ii(R*T) = Rj(i2T) = Rj(dw2) = d(Riw2) i2(R*T) = R;(ijT) = R2(dwl ) = d(R2wi). o

o

These equations and the definition of 8* give 8*(-[R0w]) (2)

8* R(i([wJ) = -RQ o

For the projection pr: U1 n U2 the composition

�

o

= [R*T]. Hence

8* ([w]).

Rn - A we have that pro Ro = pr and therefore

is identical with pr* . Since pr* is an isomorphism, RQ is forced to be the identity map on HP(U1 n U2), and the left-hand side in (2) is 8*[wj. This completes the proof. 0

6.

HOMOTOPY

45

Theorem 6.13 For n � 2 we have the isomorphisms

HP (Rn

_

{ O}) �

{ 0R

zfp =

�,

n

-1

otherwzse.

Proof. The case n = 2 was shown in Example 5.4. The general case follows from induction on n, via Proposition 6. 1 1 . 0

An invertible real n a diffeomorphism

x

n matrix A defines a linear isomorphism Rn - Rn, and

For each n � 2, the induced map fA.: Hn-1 (Rn - {0}) Hn-1(Rn - {0}) operates by multiplication by det A/Jdet AI E { ±1 }.

Lemma 6.14

Proof. Let B be obtained from A by replacing the T-th row by the sum of the T-th row and c times the s-th row, where T =f- s and c E R,

B = (I + cEr,s)A,

where I is the identity matrix and Er,s is the matrix with entry 1 in its T-th row and s-th column and zeros elsewhere. A homotopy between fA and fB is defined by the matrices (I + tcEr,s) A,

0 � t � l.

From Theorem 6.8 it follows that fA. = f'B· Furthermore detA = detB. By a sequence of elementary operations of this kind, A can be changed to diag ( l, . . , l, d), where d = det A. Hence it suffices to prove the assertion for diagonal matrices. The matrices

.

(

. JdJtd dtag l, . . . , l, ldl

),

yield a homotopy, which reduces the problem to the two cases A = diag(l, . . . , l, ±l), so fA is either the identity or the map R from Adden­ dum 6.12. This proves the assertion. 0 From topological invariance (see Corollary 6.9) and the calculation in Theorem 6.13, supplemented with if p = 0 if p =!= 0 we get

6.

46

Proposition 6.15 If n i=

HOMOTOPY

m then Rn and Rm are not homeomorphic.

Proof. A possible homeomorphism Rn

Rm may be assumed to map 0 to 0, and would induce a homeomorphism between Rn - {0} and Rm - {0}. Hence

for all

�

D

p, in conflict with our calculations.

Remark 6.16 We offer the following more conceptual proof of Addendum 6.12.

Let

0

A* -L. B* ____g.:._ C* - 0

o· l 0 - A. -

1�·

1�·

gi C* f -1 -0 *1 i B*1 ---be a commutative diagram of chain complexes with exact rows. It is not hard to prove that the diagram �

HP( C*) __L._ HP+ 1 ( A* )

· 17

o! ·

HP ( Ci ) � HP+l ( Ai ) is commutative. In the situation of Addendum 6. 1 2 consider the diagram

o 0

-

D*(U) -L 0:"(U, ) 9 D*(U2 )

R ! ! .Q*(U) -L n*(Ul) E17 .Q"'(U2)

-

' R

-

with R(w1,w2) addendum.

=

-

D*(U1

n

U2 )

D*(Ul

n

U2)

1-Ro

-

-

0 0

(Riw2, R2w1 ). This gives equation (2) of the proof of

the

47

7.

APPLICATIONS OF

DE

RHAM COHOMOLOGY

Let us introduce the standard notation

Dn = { X E

sn-

1

Rn

l llxll $ 1 } l llxll = 1 }

= {x E Rn

A fixed point for a map

f: X

--t

(the n-ball) (the ( n - 1)-sphere)

X is a point x E X, such that f(x) = x.

Theorem 7.1 (Brouwer's fixed point theorem, 1912) Every continuous map J : vn ---7 vn has a .fixed point.

Assume that j(x) � X for all X E vn. For every X E the point g(x) E sn-l as the point of intersection between from j(x) through x.

vn

sn-1

Proof.

o"

we can define and the half-l.ine

f(x)

X

g(x)

We have that

g(:c) = x + tu,

where

t = -x -u + ·

u=

���=��:�11• and

J1 - llxll2 + (x · u)2 .

Here x · u denotes the usual inner product. The expression for g(.r,) is obtained by solving the equation (x + 't'u) - (x + tu,) = L There are two solutions since the line determined by f (x) and X intersects sn- l in two points. We are interested in the solution with t 2:: 0. Since g is continuous with 9iSn-1 = idsn- 1 , the theorem 0 follows from the lemma below. Lemma 7.2 There is no continuous map g: Dn

--t

sn-1 ,

with .9!S"-'

=

idsn- 1 .

Proof. We may assume that n 2:: 2. For the map T : Rn - {0} --t Rn - {O},T(x) = x/111:11. we get that idR"-{O } � ·r, because Rn - {0} always contains the line segment between x and T(x) (see Example 6.5). If g is of the indicated type, then g(t · T(x)), 0 :::; t ::; 1 defines a homotopy from a constant map to 1'. This shows

48

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

that Rn - {0} is contractible. Corollary which contradicts Theorem 6.13.

6. 10

asserts that Hn-l(Rn - {0})

=

0,

0

The tangent space of sn in the point X E sn is Txsn = {X} 1.. ' the orthogonal complement in Rn+l to the position vector. A tangent vector field on sn is a continuous map v: sn -+ Rn+l such that v(x) E Txsn for every X E sn. Theorem 7.3 The sphere S" has X E 5n if and only if n is odd. Proof.

Such a vector field by setting

v

a tangent vector field v with v(x)

-:fi

0 for all

can be extended to a vector field w on Rn - {0}

We have that w(x) # 0 and w(x) · x = 0. The expression F(x, t) = (cosn t)x + (sin 1r t)w(x)

defines a homotopy from fo = idcw•+' -{o}) to the antipodal map h , h (x) = -x. Theorem 6.8.(i) shows that fi is the identity on Hn(Rn+l - {0} ) , which by Theorem 6.13 is I -dimensional. On the other hand Lemma 6 . 1 4 evaluates !{ to be multiplication with (-l)n+ I . Hence n is odd. Conversely, for n = 2m - 1, we can define a vector field v with

In I 962 J. F. Adams solved the so-called "vector field problem": find the maximal number of linearly independent tangent vector fields one may have on sn. (Tangent vector fields v 1 , . , vd on sn are called linearly independent if for every X E 5n the vectors VI (X), . , Vd(X) are linearly independent.) .

.

.

.

For n = 2rn - 1, let 2m = (2c + 1)24a+b, where 0 :5 b :5 3. The maximal number of linearly independent tangent vector fields on sn is equal to 2b + 8a - l. Adams' theorem

(Urysohn-Tietze) If A � Rn is closed and J: A -+ Rm continuous, then there exists a continuous map g: Rn Rm with 91 A = f.

Lemma 7.4

-+

Proof.

We denote Euclidean distance in Rn by d(x , y) and for x E Rn we define

d(x, A) = yEA in£ d(x, y).

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

49

For p E Rn - A we have an open neighborhood Up � Rn - A of p given by

Up =

{ x E Rn I d(x, p)

<

� d (p, A) }·

These sets cover Rn - A and we can use Theorem A. I to find a subordinate partition of unity p · We define g by

g(x)

{=

f(x)

rA r/>p(x)J(a(p))

pe

if X E A

if X E Rn - A

where for p E Rn - A, a(p) E A is chosen such that

d(p, a(p)) < 2d(p, A ) . Since the sum is locally finite on Rn - A, g is smooth on Rn - A. The only remaining problem is the continuity of g at a point x0 on the boundary of A. If x E Up then

1 d(x0, p) :5 d(xo, x) + d(x, p) < d(xo, x) + 2 d(p, A)

:5

1 d(xo, x) + 2 d(p, xo).

Hence d(xo,p) < 2d(xo, x) for x E Up. Since d(p, a(p)) < 2d(p, A) :5 2d(xo,p) wegetforx E Up that d(xo, a(p)) :5 d(xo,p)+d(p,a(p)) < 3d(xo,P) < 6d(xo,x). For x E Rn - A we have

g(x) - g(xo) =

L

pER"-A

p(x)(f(a(p)) - f(xo))

and (I )

JJg (x) - g(xo)!l :5

L p(x) Jlf(a (p)) - f(xo)JJ, p

where we sum over the points p with x E Up. For an arbitrary <: > 0 choose 8 > 0 such that J l f(y) - f (xo )ll < <: for every y E A with d(x0, y) < 68. If x E Rn - A and d(x , xo) < 8 then, for p with x E Up, we have that d(xo, a(p)) < 68 and llf(a(p)) - f(xo)ll < <:. Then (I) yields

JJg(x) - g(xo)ll :5 Continuity of g at x0 follows.

L r/>p(x) · <: = t. p

0

The proof above still holds, with marginal changes, when Rn is replaced by a metric space and Rm by a locally convex topological vector space. Remark 7.5

7.

50

Lemma 7.6 Let A �

APPLICATIONS OF DE RHAM COHOMOLOGY

B be a � nm be closed sets and let ¢: A homeomorphism. There is a homeomorphism h of Rn+m to itself, such that

Rn and

B

�

h(x , Om) = (On,¢(x)) for all x

E A.

Proof. By Lemma 7.4 we can extend ¢ to a continuous map JI : Rn

homeomorphism h1 : Rn

x

Rm

�

Rn x Rm is defined by

-t

Rm . A

ht ( x, y) = (x, y + h (x)). The inverse to h1 is obtained by subtracting h (x) instead. Analogously we can extend ¢- 1 to a continuous map fz: Rm -t Rn and define a homeomorphism h2: Rn X Rm � Rn X Rm by h2 (x, y) = (x + h(y), y). If h is defined to be h

=

h21 h1, then we have for x E A that o

h(x, Om) = h21(x, h (x)) = h21(x, ¢(x )) = (x - h(¢(x)) , cp(x)) = (On , ¢(x)).

0

We identify Rn with the subspace of Rn+m consisting of vectors of the form (xt, . . , Xn , 0, . . . , 0). .

If¢: A -t B is a homeomorphism between closed subsets A and B R2n. of Rn, then cp can be extended to a homeomorphism ¢: R2n Corollary 7.7

-t

Proof. We merely have to compose the homeomorphism

the homeomorphism of R2n = Rn

x

h from Lemma 7.6 with

Rn to itself that switches the two factors.

D

Note that J by restriction gives a homeomorphism between R2n- A and R2n - B. In contrast it can occur that Rn-A is not homeomorphic toRn-B. A well-known example is Alexander's "horned sphere" E in R3: E is homeomorphic to 52, but 1R3 - E is not homeomorphic to R3 - 52 . This and numerous other examples are treated in [Rushing]. Theorem 7.8 Assume that A =I=

and B are homeomorphic, then

Rn and B =I= Rn are closed subsets of Rn. If A

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

Proof. By induction on m Proposition

6. 1 1

yields isomorphisms

Hp+ m(Rn+m - A) � HP ( Rn - A) Hm (J �n+m - A) � Ho(Rn - A)/ R·l

for all

m 2 1. Analogously for B.

R2n - B

51

(for

p > 0)

From Corollary 7.7 we know that

are homeomorphic. Topological invariance (Corollary

R2n - A and

6.9)

shows that

they have isomorphic de Rham cohomologies. We thus have the isomorphisms

for

p >

0

and

For a closed set

A <; Rn

the open complement

U = Rn - A

will always

be

a disjoint union of at most countably many connected components, which all are open. If there are infinitely many, then

H0 ( U)

will have infinite dimension.

Otherwise the number of connected components is equal to dim

H0 ( U).

Corollary 7.9 If A and B are two homeomorphic closed subsets ofRn, then Rn-A and Rn - B have the same number of connected components. .4. =I=

Rn and B =I= Rn the assertion follows from Theorem 7.8 and the A = Rn and B =I= Rn then Rn+l - A has precisely 2 connected components (the open half-spaces), while Rn+l - B is connected. Hence this case Proof. If

remarks above. If

cannot occur.

0

Theorem 7.10 (Jordan-Brouwer homeomorphic to sn- l then

separation theorem) If E �

Rn (n 2: 2) is

Rn - E has precisely 2 connected components U1 and U2, where U1 is bounded and u2 is unbounded. (ii) E is the set of boundary points for both U1 and U2. We say U1 is the domain inside E and Uz the domain outside E. (i)

Proof. Since E is compact, E is closed in

Rn. To show (i), it suffices, by Corollary

7.9, to verify it for sn- l <; Rn. The two connected components of Rn - sn-l

!Jn = {x E Rn 1 11.-r ll < 1 } By choosing

1· = max llxll, xEl:

and

W = {x

the connected set

E Rn l llxll > 1 }

are

52

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

will be contained in one of the two components in Rn - E, and the other component must be bounded. This completes the proof of (i). Let p E E be given and consider an open neighbourhood V of p in Rn. The set A E - (:L n V) is closed and homeomorphic to a corresponding proper closed subset B of sn- l . It is obvious that Rn - B is connected, so by Corollary 7.9 the same is the case for Rn - A. For PI E U1 and P2 E U2 , we can find a continuous curve 1: [a, bJ � Rn - A with r(a) = P1 and r(b) = P2· By (i) the curve must intersect E, i.e. f' - 1 (E) is non-empty. The closed set ,- 1 (E) � [a,bJ has a first element q and a last element c2, which both belong to (a, b). Hence r(c1) E E n V and 'Y(c2) E E n V are points of contact for "Y([a, c1)) � U1 and 'Y((c2, b]) � U2 respectively. Therefore we can find t1 E [a, c1) and t2 E ( c2, b ] , such that 'Y(tl ) E U1 n V and r(t2) E U2 n V. This shows that p is a boundary point for both ul and u2. and proves (ii). 0 =

Theorem 7.11 If A � is connected.

Rn is homeomorphic to Dk, with k

� n, then

Rn - A

Proof. Since A is compact, A is closed. By Corollary 7.9 it is sufficient to prove the assertion for Dk � Rk � Rn. This is left to the reader. 0 Theorem 7.12 (Brouwer) Let U � nn be an arbitrary open set and f: U � Rn an injective continuous map. The image f(U) is open in Rn, and f maps U homeomorphically to f(U). Proof. It is sufficient to prove that f(U) is open; the same will then hold for f ( W) , where W � U is an arbitrary open subset. This proves continuity of the inverse function from f(U) to U. Consider a closed sphere.

D

=

{x E Rn l llx - xoll

� 6}

contained in U with boundary S and interior D = D - S. It is sufficient to show that f(D ) is open. The case n = 1 follows from elementary theorems about continuous functions of one variable, so we assume n :2: 2. Both S and E = f(S) are homeomorphic to sn-l . Let U1 and U2 be the two connected components of nn - E from Theorem 7.10. They are open; U1 is bounded and U2 is unbounded. By Theorem 7. 1 1 , nn - f(D) is connected. Since this set is disjoint from E, it must be contained in U1 or U2. As f(D) is compact, Rn - J(D) is unbounded. We must have nn - f(D) � U2. It follows that E u U1 = Rn - U2 � f(D). Hence u1 � J (D).

Since D is connected, f(D ) will also be connected (even though it is not known whether or not f (D) is open). Since f (D) � U1 u U2, we must have that ul = f (D). This completes the proof. 0

7.

53

APPLICATIONS OF DE RHAM COHOMOLOGY

Corollary 7.13 (Invariance of domain) If V � Rn has the topology induced by Rn and is homeomorphic to an open subset of Rn then V is open in Rn . 0

Proof. This follows immediately from Theorem 7.12.

Corollary 7.14 (Dimension invariance) Let U � Rn and V � Rm be non-empty open sets. If U and V are homeomorphic then m. Proof. Assume that

m<

n=

n.

From Corollary 7.13 applied to V, considered as a subset of Rn via the inclusion Rm � Rn, it follows that V is open in Rn. This contradicts that V is contained in a proper subspace. 0 Example 7.15 A knot in R3 is a subset E

� R3 that is homeomorphic to S1 . The

corresponding knot-complement is the open set U = R3 - E. We show: R if o :::; P :::; 2 HP(U) � 0 otherwise. According to Theorem 7.8, it is sufficient to show this for the "trival knot" S 1 � R2 � R3 . First we calculate

{

(2)

Here D2 is star-shaped, while R2 - D2 is diffeomorphic to R2 - {0} . Using Theorem 3.15 and Example 5.4 it follows that (2) has dimension 2 for p = 0, dimension 1 for p = 1, and dimension 0 for p 2: 2. Apply Proposition 6. 1 1 . An analogous calculation of H*(Rn - E) can be done for a higher-dimensional knot E � Rn, where E is homeomorphic to Sk ( 1 :::; k ::; n - 2). See Exercise 7.2.

(n 2:

Proposition 7.16 Let E � Rn 2) be homeomorphic to U2 be the interior and exterior domains of E. Then

R ifp = HP (Ul) � { � and HP(U2) 0 otherwzse

Proof. The case

�

p = 0 follows from Theorem 7.10.

p > 0 there are isomorphisms HP (Ul) EB IJP(U2)

The inclusion map i : W inverse defined by

�

� �

sn-l and let U1 and

{ R ifp = � , n 0 otherwzse. Set W

1

= Rn - Dn. For

HP (Rn - E)

� HP ( Rn - sn-l ) HP(Dn) EB HP(JtV) � HP(W).

Rn - {0} is a homotopy equivalence with homotopy

llxll + x) = ( g llxll x. 1

54

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

The two required homotopies are given by Example 6.5. From Theorem 6.8.(iii) we have that HP(i) is an isomorphism. The calculation from Theorem 6.1 3 yields if p = 0, n otherwise.

1

We now have that HP(UI) = 0 and HP(U2) = 0 when p � {0, n - 1}. On the other hand the dimensions of Hn- I (Ul) and Hn- I (U2) are 0 or 1 , so it suffices to show that Hn-l ( U2 ) # 0. Without loss of generality we may assume that 0 E U1 and that the bounded set U1 u E is contained in Dn. We thus have a commutative diagram of inclusion maps

and apply Hn-l to get the commutative diagram sn- l (Rn -

H"-l(r 1

{0}) � R

1 Hn- (W) -- Hn- (U2)

where Hn- 1 (i) is an isomorphism. It follows that Hn-1(U2) =I= 0.

0

Remark 7.17 The above result about H*(Ul) might suggest that U1 is contractible (cf. Corollary 6. 1 0). In general, however, this is not the case. In Topological Embeddings, Rushing discusses several examples for n = 3, where U1 is not simply connected (i.e. there exists a continuous map S1 --+ U1o which is not homotopic to a constant map). Hence U1 is not contractible either. Corresponding examples can be found for n > 3. If n = 2 a theorem by Schoenfiies (cf. [Moise]) states that there exists a homeomorphism h : U1 u E - D2 .

By Theorem 7.12, such a homeomorphism applied to homeomorphically to b2 .

hlut and

ji

h 2 will map U1

1960 shows that the conclusion in Schoenflies' theorem is also valid if n > 2, provided it is additionally assumed that E is flat in IRn, that is, there exists a 8 > 0 and a continuous injective map ¢: sn-l x ( -6, 6) - Rn with E = ¢( sn-l x {0}).

A result by M. Brown from

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

Example 7.18 One can also calculate the cohomology of

55

"Rn with m holes",

i.e. the cohomology of

V = Rn -

( U Kj ) . m

j=l

The "holes" Kj in Rn are disjoint compact sets with boundary Ej , homeomorphic to sn- l . Hence the interiors Kj = Kj - Ej are exactly the interior domains of I:j. One has (3)

HP(V)

:::e

{ �m

if p = 0 if p = n otherwise.

1

We use induction on m. The case n = 1 follows from Proposition 7 . 1 6. Assume the assertion is true for V1

m-1

=

(

)

Rn - U Kj . j= l

Let V2 = Rn - Km . Then V1 u V2 = Rn and V1 n V2 = V. For p 2: the exact Mayer-Vietoris sequence

0 we have

If p = 0 then H0(Rn) � R and I* is injective. We get H0(VI) � R by induction and H0(V2) � R from Proposition 7 . 1 6. The exact sequence yields HO(V) � R. If p > 0 then HP(Rn) = 0 and the exact sequence gives the isomorphism HP(Vl ) EfJ HP(Vz) � HP(V). Now (3) follows by induction.

57

8. SMOOTH MANIFOLDS A topological space

X

has a countable topological base, when there exists a

countable system of open sets written in the form

Uie/ ui.

V

=

{Ui

I i E N}, such that every open set can be

I � N.

where

For instance Rn has a countable basis for the topology given by

V = {Dn (x;�;) where

b

(x; �;)

I

x = (XI, . . , xn ), Xi E Q; �; E Q, .

is the open ball with center at

A topological space

X

x

and radius

t:.

t: > O }

is a Hausdorff space when, for arbitrary distinct

Ux

Ux II Uy = 0.

x, y E X,

and

Uy

with

Definition 8.1 A topological manifold

M

is a topological Hausdorff space that

there exist open neighborhoods

has a countable basis for its topology and that is locally homeomorphic to The number n is called the dimension of

M.

Rn.

Remark 8.2 Every open ball bn(o, t:) in Rn is diffeomorphic to Rn via the map

q> given by

?f.. (

'K y) = (Smoothness of q> and

{ tan(7r llvll/2�;) · Y/IIYII 0

q>-1

at

0 can be shown

# 0, if y = 0. if y

IIYII < f

by means of the Taylor series at

0

for tan and Arctan.) Thus in Definition 8.1 it does not matter whether we require that

Mn

is locally homeomorphic to Rn or to an open set in Rn.

Definition 8.3

(U, h) on an n-dimensional manifold is a homeomorphism h: U --+ U', where U is an open set in M and U' is an open set in Rn. A system A = {hi: ui --+ Uf I i E J} of charts is called an atlas, provided {Ui I i E J} covers M.

(i) A chart (ii)

(iii) An atlas is smooth when all of the maps

hJ;· · - hJ· o hi-1 · h;· ( i II U·) J --+ hJ· ( U f � n U·) ; J .

are smooth. They are called chart transformations (or transition functions) for the given atlas.

Note in Definition 8.3.(iii) that Two smooth atlases

A 1 , A2

hi(Ui n Uj)

is open in Rn.

are smoothly equivalent if

A1 U A2 is a smooth atlas.

M. is an equivalence class A of smooth atlases on M.

This defines an equivalence relation on the set of atlases on on

M

A smooth structure

8.

58

SMOOTH MANIFOLDS

Definition 8.4 A smooth manifold is a pair ( M, A) consisting of a topological

manifold M and a smooth structure

A on M.

Usually A is suppressed from the notation and we write M instead of (M, A).

{ x E Rn+ l l ll x ll = 1 } is an n­ dimensional smooth manifold. We define an atlas with 2(n + 1) charts (U±i , h±i) Example 8.5 The n-dimensional sphere

sn

=

where

u i = {X E sn I Xi < 0} and h±i: u±i - iJn is the map given by h±i(X) = (xi, . . . ' Xi, . . . ) Xn+l )· The circumflex over Xi denotes that Xi is omitted. The inverse map is u+i = {X E sn I Xi > 0},

.. )

(

h±� (u) = u1 , . . . , Ui- 1, ±V1 - llull2, 1Li, . , un

It is. left to the reader to prove that the chart transformations are smooth. Example 8.6 (The projective space

relation:

X

"'

y

¢>

RPn) On sn we define an equivalence

X=

y

or X =

-y.

The equivalence classes [xJ = {x, -x} define the set lllPn. Alternatively one can consider RPn as all lines in Rn+l through 0. Let 1r be the canonical projection We give RPn the quotient topology, i.e.

U � RPn open ¢> n-1(U) � sn open.

With the conventions of Example 8.5, n(U-i) 1r(U+i). We define Ui n(U±i) � RPn, and note that 7r- 1 (Ui) = U+i U U-i with U+i n U_i = 0. An equivalence class [x] E Ui has exactly one representative in U+i and one representative in U-i· Hence 11': U+i ---+ Ui is a homeomorphism. We define =

·i =

1,

. . . , n.

=

hi = h+i 0 Jr- 1 : ui - iJn , This gives a smooth atlas on RPn.

Definition 8.7 Consider smooth manifolds M1 and M2 and a continuous map f: M1 - M2. The map f is called smooth at x E M1 if there exist charts h1: U1 - U{ and h2: U2 - U2 on Mt and M2 with x E Ut and j(x) E U2,

such that 1 1 h2 o j o h1 : h1 (f- (U2))

- U2

is smooth in a neighborhood of h1(x). If f is smooth at all points of M1 then f is said to be smooth.

8.

SMOOTH MANIFOLDS

59

Since chart transformations (by Definition 8.3.(iii)) are smooth, we have that Definition 8.7 is independent of the choice of charts in the given atlases for M1 and M2. A composition of two smooth maps is smooth. A diffeomorphism f: M1 � Mz between smooth manifolds is a smooth map that has a smooth inverse. In particular a diffeomorphism is a homeomorphism.

As soon as we have chosen an atlas A on a manifold M, we know which functions on are smooth. In particular we know when a homeomorphism f: V � V' between an open set V c M and an open set V' <;; Rn is a diffeomorphism. We can therefore define a new maximal atlas, Amax • associated with the given smooth structure:

M

Amax

=

{J: V

�

V' I V � Mn open, V' � Rn open, f diffeomorphism } .

The inverse diffeomorphisms j - 1: V'

V will be called local parametrizations. From Remark 8.2 it follows that every point in a smooth manifold lvfn has an open neighborhood V � Mn that is diffeomorphic to Rn. �

From now on chart will mean a chart in the maximal atlas. Definition 8.8 A subset N c Mn of a smooth manifold is said to be a smooth submanifold (of dimension k), if the following condition is satisfied: for every

xE N

there exists a chart x

(l) where Rk

c

h: U

EU

�

and

U' on M such that

h(U n N ) = U' n Rk ,

Rn is the standard subspace.

It is easy to see that a smooth submanifold N of a smooth manifold M is a smooth manifold again. A smooth atlas on N is given by aJl h: U n N � U'n Rk , where (U, h) are charts on M satisfying ( 1 ).

Example 8.9 The n-sphere sn is a smooth submanifold of Rn+l. In fact the charts (U±i, h±·i) from Example 8.5 can easily be extended to diffeomorphisms satisfying ( 1 ).

Definition 8.10 An embedding is a smooth map f: N is a smooth submanifold and f: N

�

�

M such that j(N)

f(N) is a diffeomorphism.

c

M

Theorem 8.11 Let Mn be a smooth manifold of dimension n. There exists an embedding of Mn into a Euclidean space Rn+k . This result will be proved below for a compact M, but let us first note that satisfies the following condition: Nn =

f(Mn )

60

8.

p

U'

SMOOTH MANWOLDS

For every E Nn there exists an open neighborhood V � Rn+k, an open set � Rn and a homeomorphism

g: U' --+ N n V,

U'

such that g is smooth (considered as a map from Dxg: Rn --+ Rn+k is injective.

to V) and such that

This is the usual definition of an embedded manifold (regular surface when n = 2). Theorem 8.11 tells us that every smooth manifold is diffeomorphic to an embedded manifold. Conversely, if N � Rn+k satisfies the above condition, then the implicit function theorem shows that it is a submanifold in the sense of Definition 8.8. A theorem by H. Whitney asserts that the codimension k in Theorem 8. I l can always be taken to be less or equal to n + 1. On the other hand k cannot be arbitrarily small. RP2 cannot be embedded in R3.

8.12

Let Mn be an n-dimensional smooth manifold. For exist smooth maps Lemma

p

E

M there

/p: M --+ Rn

¢p: M --+ R,

such that ¢p(P) > 0, and /p maps the open set M - ¢; 1(0) diffeomorphically onto an open subset o f Rn.

h: E C00(Rn, R)

V --+ V' with

p

E V. By Lemma A.7 we can find a function 7/J with compact support suppR (?/J) � V', such that 7/J is constantly equal to 1 on an open neighborhood U ' c V' of ( ) .The smooth map Proof. Choose a chart

/p

..

/p(q) = { 07f;(h(q))h(q) q h-1(U') /p U U U'. U' 7/Jo(h(p)) 0, qEv ¢>p(q) { 7f;o(h(q)) 0 M - ¢; 1 (0) U, 8.11 (M p M M

can now be defined by

hp

if E V otherwise.

h

the function coincides with and On the open neighborhood = therefore maps diffeomorphically onto Choose '1/Jo E C00(Rn, R) with compact support suppR (?/Jo) � and > and let ..

if otherwise.

=

Since

�

0

the final assertion holds.

choose ¢p and /p as in compact). For every E Proof of Theorem Lemma 8.12. By compactness can be covered by a finite number of the sets M - ¢; 1(0). After a change of notation we have smooth functions

¢i : M --+ R,

fr M --+ Rn

satisfying the following conditions:

(i) The open sets

(ii)

fj\Ui

maps

Uj

Uj

4>-;t

(0)

(1 ::; j ::; d)

Mcover Af. diffeomorphically onto an open set =

Uj � Rn.

8.

We define a smooth map

f: M

61

SMOOTII MANIFOLDS

._

Rnd+d by setting

f(q) = (h (q) , · · · , fd (q ) , rf>I (q) , · · · , cPd(q )) . f(ql) = j (q2) , we can by (i) choose j such that q1 E Uj. ¢j (q2) = cPj (q1 ) =I= 0, q2 E Uj, and by (ii), q1 = Q2 · Hence f is injective.

Assuming

M

is compact, f is a homeomorphism from

M

to

f(M).

Then Since

Let

n coordinates and the last n(d - 1) + d coordinates, respectively. By (ii) 1r1 o f = h is a diffeomorphism from U1 to U�. In particular 1r1 maps f(UI) bijectively onto Uf. Hence f(U1) is the graph of the smooth map be the projections on the first

g

1 ·. U1'

._

Define a diffeomorphism

Rn(d-l) +d '. h1

from

1 1r1 (Un

h1(x, y ) = (x , y - 9t (x)),

to itself by the formula

x E Uf, y

E

Rn(d- l)+n .

h1 maps f(U1) bijectively onto U{ x {0}. Since f(U1) is open in f(A1), f(UI) = f(M) n W1 for an open set W1 � Rnd+d, which can be chosen 1 to be contained in ?T1 (U{). The restriction h11w1 is a diffeomorphism from W1 onto an open set W{, and it maps f(M) n W1 bijectively onto W{ n Rn, as required by Definition 8.8. The remaining f (Uj) are treated analogously. Hence f(M) is a smooth submanifold of Rnd+d . Note also that fiU1 : U1 - f(U1) is a diffeomorphism, namely the composite of h 1u1 : U1 - U{ and the inverse to the diffeomorphism j(U1) - Uf induced by 1r1. The remaining U; are treated analogously. Hence f: M - f(M) is a diffeomorphism. 0 We see that

Remark 8.13

The general case of Theorem 8 . 1 1 is shown in standard text books

on differential topology.

To get

one uses Theorem 1 1 .6 below.

k = n+1

(Whitney' s embedding theorem)

In the proofs above one can change "smooth

manifold" to "topological manifold", "smooth map" to "continuous map" and "diffeomorphism" to "homeomorphism". This will lead to the theorem below, where the concept (locally flat) topological submanifold is defined in analogy to Definition 8.8, but with a homeomorphism instead of the diffeomorphism

h.

Theorem 8.14 Every compact topological n-dimensional manifold is homeomor­ phic to a (locally flat) topological submanifold of a Euclidean space Rn+k . 0 On a topological manifold functions jll[

._

Mn

C0(M, R) of continuous A on Mn gives a subalgebra

we have the R-algebra

R. A smooth structure

· c= (( M, A), R) � C0 (M, R )

8.

62

SMOO'fH MANIFOLDS

consisting of the maps M � R, that are smooth in the structure A on (and the standard structure on R). Usually A is suppressed from the notation, and the R-algebra of smooth real-valued functions on M is denoted by C00(M, R). This subalgebra of R) uniquely determines the smooth structure on M. This is a consequence of the following Proposition 8.15 applied to the identity maps idM

M

C0(M,

Proposition 8.15 If g: N � M is a continuous map between smooth manifolds N and M, then g is smooth if and only if the homomorphism

given by g*('lj;)

=

't/J

o

g maps C00(M, R) to C00(N, R).

Proof. "Only if'' follows because a composition of two smooth maps is smooth. Conversely if the condition on g" is satisfied, Lemma 8.12 applied to p :::: g (q) yields a smooth map f: Mn � Rn and an open neighborhood V of p in M, such that the restriction fw is a diffeomorphism of V onto an open subset of Rn. For the j-th coordinate function jj E C00(M, R) we have fi o g = g*(jj) E C00(N, R), so that j o g: N ---4 Rn is smooth. Using the chart fw on M, g is seen to be smooth at q. 0 Remark 8.16 There is a quite elaborate theory which attempts to classify n­ dimensional smooth and topological manifolds up to diffeomorphism and home­ omorphism. Every connected !-dimensional smooth or topological manifold is diffeomorphic or homeomorphic to R or S1 . For n = 2 there is a complete classification of the compact connected surfaces. There are two infinite families of them: Orientable surfaces:

Non-orientable surfaces: RP2 , Klein's bottle, etc. See e.g. [Hirsch], (Massey]. In dimension 3, one meets a famous open problem: the Poincare conjecture, which asserts that every compact topological 3-manifold that is homotopy equivalent to S3 is homeomorphic to S3. It is known that every topological 3-manifold has a smooth structure A and that two homeomorphic smooth 3-manifolds also

M3

8.

SMOOTH

MANIFOLDS

63

are diffeomorphic. In the mid 1 950s J. Milnor discovered smooth ?-manifolds that are homeomorphic to S1' but not diffeomorphic to S7. In collaboration with M. Kervaire he classified such exotic n-spheres. For example they showed that there are exactly 28 oriented diffeomorphism classes of exotic ?-spheres. In 1960 Kervaire described a topological 1 0-manifold that has no smooth structure. During the 1960s the so-called "surgery" technique was developed, which in principle classifies all manifolds of a specified homotopy type, but only for n � 5. In the early 1 980s M. Freedman completely classified the simply-connected com­ pact topological 4-manifolds; see [Freedman-Quinn]. At the same time S. Don­ aldson proved some very surprising results about smooth compact 4-manifolds, which showed that there is a tremendous difference between smooth and topolog­ ical 4-manifolds. Donaldson used methods originating in mathematical physics (Yang-Mills theory). This has led to a wealth of new results on smooth 4manifolds; see [Donaldson-Kronheimer]. One of the most bizarre conclusions of the work of Donaldson and Freedman is that there exists a smooth structure on R4 such that the resulting smooth 4-manifold "R4 " is not diffeomorphic to the usual R4 . It was proved earlier by S. Smale that every smooth structure on Rn for n =!= 4 is diffeomorphic to the standard Rn.

65

9. DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

In this chapter we define the de Rham complex

Q*(M) of a smooth manifold Mm

and generalize the material of earlier chapters to the manifold case.

For a given point p E Mm we shall construct an m-dimensional real vector space

TpM

M N Tf(p)M known as the tangent map of f at p.

called the tangent space at p. Moreover we want a smooth map f:

to induce a linear map Dpf; TpM

._

-t

Remarks 9.1 (i) In the case p E tangent space to

U U

� Rm,

where

at p with

Rm.

U

is open, one usually identifies the

Better suited for generalization is the

following description: Consider the set of smooth parametrized curves TI

-t

U

with

'Y(O) =

p, defined on open intervals around 0.

equivalence relation on this set is given by the condition

'Yi (0)

There is a 1-1 correspondence between equivalence classes and

An

'Yz (O).

Rm, which

')'1(0) E nm. Consider a further open set V £; Rn and a smooth map F: U V. The Jacobi matrix of F evaluated at p E U defines a linear map DpF: Rm Rn. For 1: I U, 'Y(O) = p as in (i) the chain rule implies that DpF(r1(0)) = ( F o 1)1(0). Interpreting tangent spaces as given by ['Y]

to the class (ii)

of 'Y associates the velocity vector

=

._

-t

-t

equivalence classes of curves we have

(I) In particular the equivalence class of F o

1

depends only on

['Y] ·

(U, h) be a smooth chart around p E Mm. On the set of smooth curves ex: I ._ M with cx(O) = p defined on open intervals around 0 we have an

Let

equivalence relation

(2) This equivalence relation is independent of the choice of is another smooth chart around p, one finds that

(h o cxl ) (0) I

=

(h o cx2 ) (0) I

{:}

·

(h o a1)'(0) �

=

(U, h).

In fact, if

(U, h)

(h o a2) I (0) �

by applying the last statement of Remark 9.l .(ii) to the transition diffeomorphism

F

=

h o h,- l

and its inverse.

66

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

Definition 9.2 The tangent space

TpMm is the set of equivalence classes with M, a(O) = p.

respect to (2) of smooth curves a: I

---+

We give TpM the structure of an m-dimensional vector space defined by the following condition: if (U, h) is a smooth chart in M with p E U, then

is a linear isomorphism; here [a]

E

TpM is the equivalence class of a.

By definition <J?h is a bijection. The linear structure on TpM is well-defined. This can be seen from the following commutative diagram, where F = h o h,-1 , q = h(p)

Nn be a smooth map and p E M. (i) There is a linear map Dpf: TpM ---+ Tf(p) N given in terms of representing

Lemma 9.3

Let f: Mm

---+

curves by

Dp!([aJ) = [f o a]. (ii) If (U, h) is a chart around p in M and (V, g) a chart around f(p) in N, then we have the following commutative diagram:

Tf(p)M

l�Rm�h Dh(p) (gofoh-1) �1Rn�9

= g o f o h-1, defined on the open set h(U n j-1(V)), shows that the bottom map in the diagram is linear and given

Proof. Remark 9 . l .(ii) applied to F

on representing curves as stated there. Since h and 9 are linear isomorphisms, there exists a linear map Dpf making the diagram commutative. The formula in 0 (i) follows by chasing around the diagram. ·

Note that the linear isomorphism <J?h in Definition 9.2 can be identified with Dph through the identification discussed in Remark 9.l .(ii). From now on we write Dph: TpM ---+ Rm for this linear isomorphism, and similarly Dh (p) h-1: Rm TpM for its inverse. ---+

9.

67

DIFFERENTIAL l''ORMS ON SMOOTH MANIFOLDS

Suppose Mm � R1 is a smooth submanifold with inclusion map i: M ---... R1 . Definition 8.8 implies that Dpi: TpMm ---+ TpRl "' Rl is injective. In this case we usually identify TpM with the image in R1, consisting of all vectors a'(O) where a: I ---+ M � R1 is a smooth parametrized curve with a(O) = p. For a composite rp o f of smooth maps f: M ---+ N, rp: N ---+ we have the chain rule, immediately from Lemma 9.3(i),

P and a point p E M

Dp (rp o f ) = Df(p)'P o Dp'P·

(3)

Remark 9.4 Given a smooth chart (U, h) around the point p E Mm we obtain a basis for TpM

(8�;

where )p is the image under Dh (p) h- 1 : Rm ---+ TpM of the i-th standard basis vector ei = (0, . . . 1, . . . E Rm. A tangent vector Xp E TpM can be written uniquely as

, 0)

(4 ) where a = (a1, . . . , a ) E Rm. If Xp m a representing smooth curve, we have a =

=

[a],

where

a: I ---+ U

with

a(O) = p

is

(h o a)'(O).

Given f E C00(M, R) we have the tangent map (5) The directional derivative Xpf E R is defined to be the image in R of Xp under 1 (5), i.e. Xpf = (f o a)'(O). In terms of f o h- we have by the chain rule

d Xpf = dt (fh-1

In particular

(6)

o

m Bfh-1 (h(p))ai. ha(t) ) lt=O = 2:=

(�) f OXj p

i::=l

=

ox·t

8fh-1 ( (p) . h ) OXj

Under the assumptions of Lemma 9.3.(ii) there is a similar basis (8/ByJ f(p)' 1 � j � n, for Tf(p) N and the matrix of Dpf with respect to our bases for TpM

9.

68

VlFFERENTIAL FORMS ON SMOOTH MANIFOLDS

and Tf(p)N is the Jacobian matrix at h(p) of g o f o h- 1 . Specializing this to the case N = M, f = idM we find that

(�)

(7 )

OXi p

=

f. ocpi (h(p)) (_!___)

j=1

OXi

OYj p

.

This holds when (U, h) and (V, g) are two charts around p with transition function (cpi . . . . ' 'Pm) = cp = g o h-

1

expressing the Yj-coordinates in terms of the Xi-coordinates. Suppose X is a function that to each p E M assigns a tangent vector Xp E TpM. Given the chart (U, h), formula (4) holds for p E U with certain coefficient functions ai: U - R. If these are smooth in a neighborhood of p E U, X is said to be smooth at p. From (7), this condition is independent of the choice of smooth chart around p. If X is smooth at every point p E M, X is a a smooth vector .field on M. Let us next consider families w = {wp } pE M of alternating k-forms on TpM, where wp E Altk (TpM). We need a notion of w being smooth as function of p. Let g: W --+ M be a local parametrization, i.e. the inverse of a smooth chart, where W is an open set in Rm. For x E W,

is an isomorphism, and induces an isomorphism

We define g*(w): W

--+

Altk(Rm) to be the function whose value at x is

Definition 9.5 A family w = { wp hE M of alternating k-fonns on TpM is said

to be smooth if g*(w) is a smooth function for every local parametrization. The set of such smooth families is a vector space D,k (M). In particular, fl0(M) = C00(M,R). Lemma 9.6 Let gi: Wi --+ N be a family of local parametrizations with N = U 9i(Wi)· If gi(w) is smooth for all i, then w is smooth.

N be any local parametrization. and z E W. We show that g*(w) is smooth close to z. Choose an index i with g(z) E 9i(Wi). Close to z

Proof. Let

g: W

--+

69

DD'F .ERENTIAL FORMS ON SMOOTH MANIFOLDS

9.

we can write g = 9i o gi1 o g = 9i o h , where h = gi1 o g: g-1(gi(Wi)) is a smooth map between open sets in Euclidean space. Thus

---t Wi

g"'(w) == (gi o h)*(w) = h"'(gi(w))

in a neighborhood of z, and the right-hand side is

a

smooth k-form by assumption.

0

The exterior differential

d: Qlc (M) ---t Qlc+l (M ) can be defined via local parametrizations g: W is a smooth k-form on M then

---t

M as follows. If w = {wp }pEM

(8) It is not immediately obvious that dpw is independent of the choice of local parametrization, but this is indeed the case: Given a local parametrization g, then any other locally has the form g o ¢, with ¢: U - W a diffeomorphism. Let 6, . . . , elc + l E TpM. We choose VI , . . . , vk+ l E Rn so that Dx(go ¢)(vi) = ei· We must show that

dyg* (w) (wb . . . , wk+l) = dx(9 o ¢) "'(w )(v1 , . . . , V!c+I)

where ¢(x) = y and Dx¢(vi)

Wi ·

=

This follows from the equations

(g o ¢)*(w) == ¢* (g* (w)) d¢"'(r) = ¢"'d(r),

where r = g*(w); see Theorem have defined a chain complex ·

We have

· ·

D,k(M) = 0

• •

---t

• .

T E D,k(N);

A:

1\


nk(M ) x n1 (M) - nH1 (M).

d(w 1\ T) = dw 1\ T + ( -1) kw 1\ dr T

for k = 0.

r)P = wp 1\ Tp ,

One shows by choosing local parametrizations that
for w E D,k (M),

0. Hence we

= 0 when k > dim TpM. N induces a chain map ¢* : S1"'(N) ---t D.* (M) ,

One defines a bilinear product w 1\ T by (w

( 1 1)

=

if k > dimM, since Altk (TpM)

4>* (r )p = Altk (Dp¢) (r4J(p) ) ,

(10)

obvious that d o d

- nk-1(M) � Ok(M) � n,k+1(M) ---t

A smooth map ¢: M (9)

3.12. It is

W I\ T = ( -1)klT I\ W

E rl1 (M).

1\ T are smooth.

9.

70

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

Definition 9.7 The p-th (de Rham) cohomology of the manifold M, denoted

HP (M), is the p-th cohomology vector space of Q*(M).

The exterior product induces a product HP(M) x HQ(M) - HTHQ(M) which makes H*(M) into a graded algebra. Note that HP(M) = 0 for p > n = dim M or p < 0.

The chain map ¢* induced by a smooth map ¢: M

-

N induces linear maps

and the de Rham cohomology becomes a contravariant functor from the category of smooth manifolds and smooth maps to the category of graded anti-commutative R-algebras.

Definition 9.8 (i) A smooth manifold Mn of dimension n is called orientable, if there exists an w E nn(Mn) with Wp -1= 0 for all p E M. Such an w is called an orientation form on M. (ii) Two orientation forms w, T on M are equivalent if T = f w, for some f E Q0(1vf) with f(p) > 0 for all p E M. An orientation of M is an equivalence class of orientation forms on M. ·

On the Euclidean space Rn we have the orientation form dxl 1\ . . . 1\ dxn. which represents the standard orientation of Rn. Let Mn be oriented by the orientation form w. A basis b1, . . . , bn of TpM is said to be positively or negatively oriented with respect to w depending on whether

the number

is positive or negative. (It cannot be 0, because wp :j:. 0.) The sign depends only on the orientation detennined by w . If w and T are two orientation forms on Mn,

then T = f . w for a uniquely determined function f E n°(M) with f(p) :j:. 0 for all p E M. We say that w and T determine the same orientation at p, if f(p) > 0. Equivalently, w and r induce the same positively oriented bases of TpM. If M is connected, then f has constant sign on M, so we have:

Lemma 9.9 On a connected orientable smooth manifold there are precisely 2 orientations. 0 If U is an open subset of an oriented manifold Mn, then an orientation of U is induced by using the restriction of an orientation form on M. Conversely we have:

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

71

Lemma 9.10 Let V = ( Vi) iEl be an open cover of the smooth submanifold Mn. Suppose that all Vi have orientations and that the restrictions of the orientations from Vi and Vj to Vi n Vj coincide for all i # j. Then M has a uniquely determined orientation with the given restriction to Vi for all i E I.

The proof is a typical application of a smooth partition of unity in the following form: Theorem 9.11 Let V = (Vi) iEl be an open cover of the smooth manifold Mn � R1 . Then there exist smooth functions
(i) SuppM(
Proof.

Since

M

1.

has the topology induced by Rl, we can choose an open set A.l to

Ui � R1 with Ui n M = Vi for each i E I. By applying Theorem U = Uiei Ui we get smooth functions 7/Ji: U [0, 1] with (i) Suppu(Wi ) � Ui. (ii) Local finiteness. (iii) For every x E U, Lie! Wi( x) = 1 .

-

[0, 1] be the restriction of 't/Ji; conditions (i), (ii) and (iii) of the Let
Proof of Lemma

9.10. Let the orientation of Vi

be

Wi E nn (Vi), and choose smooth functions
w

E nn(M) by

given by the orientation form [0, 1] as in Theorem 9.1 1.

-

w = L
iE/

where 0, then Wi',p (bl , . . . , bn) > 0 for every other i' with p E Vi, and in the formula -

Wp(bl,

· · · ,

bn)

=

L
· · · ,

bn),

all terms are positive (or zero). Thus w is an orientation form on M, and b1, . . . , bn are positively oriented with respect to w. The orientation of M determined by w has the desired property.

72

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

If T E nn (Mn ) is another orientation form that gives the orientation of the required type, then T = f w and Tp(b1 , . . . , bn ) = f(p)wp(bl, . . . , bn )· Since both Tp and Wp are positive on b1 , . . . , bn we have f(p) > 0. Hence w and T determine the same orientation. 0 ·

Definition 9.12 Let ¢: Mf -+ M!f be a diffeomorphism between manifolds that are oriented by the orientation forms Wj E nn(Mj ). Then ¢*(w2) is an orientation form on Mf. We say ¢ is orientation-preserving (resp. orientation-reversing), when ¢*(w2) determines the same orientation of Mf as w1 (resp. -wl). Example 9.13 Consider a diffeomorphism ¢: U1 -+ U2 between open subsets U1 , U2 of Rn, both equipped with the standard orientation of Rn . It follows from Example 3.13.(ii) that ¢ is orientation-preserving if and only if det(Dx¢) > 0 for all X E ul. Analogously ¢ is orientation-reversing if and only if all Jacobi determinants are negative.

Around any point on an oriented smooth manifold Mn we can find a chart h: U -+ U' such that h is an orientation-preserving diffeomorphism when U is given the orientation of M and U' the orientation of Rn . We call h an oriented chart of M. The transition function associated with two oriented charts of M is an orientation-preserving diffeomorphism. For any atlas of M consisting of oriented charts, all Jacobi determinants of the transition functions will be positive. Such an atlas is called positive. Proposition 9.14 If {hi: ui --7 u; I i E I} is a positive atlas on Mn, then Mn has a uniquely detemtined orientation, so all hi are oriented charts. Proof. For i E I we orient Ui so that hi is an orientation-preserving diffeomor­ phism. By Example 9 1 3, the two orientations on Ui n Uj defined by the restriction from Ui and Uj coincide. The assertion follows from Lemma 9.1 0. 0 .

Definition 9.15 A Riemannian structure (or Riemannian metric) on a smooth manifold Mn is a family of inner products ( , )p on TpM, for all p E M, that satisfy. the following condition: for any local parametrization f: W -+ M and any pair v 1 , v2 E Rn,

is a smooth function on W. It is sufficient to have the smoothness condition satisfied for the functions

where

e1,

.. , en .

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

the.fzrstfundamentalform.

coefficients of

Rn .

is the standard basis of

For

73

These functions are called the the n

xEW

is synunetric and positive definite.

x n

matrix

C9ij(x))

A smooth manifold equipped with a Riemannian structure is called a Riemannian manifold. A smooth submanifold Mn � R1 has a Riemannian structure defined by letting ( , ) be the restriction to the subspace TpM � Rl of the usual inner P

product on

IR1 .

Proposition 9.16 If Mn is an oriented Riemannian manifold then Mn has a

uniquely determined orientation form

with

vol.M

volM(bJ, . . . , bn)

::::: 1

for every positively oriented orthonormal basis of a tangent space TpM. We call volM the volume form on M. Proof. Let the orientation be given by the orientation form w two positively oriented orthonormal bases tangent space

TpM .

b� and

wp E AltnTpM

=

satisfies

wp (b�, . . . , b�)

(12)

b1, . . . : bn

n x n

There exists an orthogonal n

and

E f211( Mn) . Consider

b; , . . . , b� =(

in the same

C-ij) such that

matri x C

L Cij bj,

j= l

= (det C)wp(bt, . . . , bn)·

det C > 0;

det C == 1. Hence there exists a function p: M � (0, oo ) such that p(p) = wp(bt, . . . , bn) for every positively oriented orthonormal basis b 1 , . . . , bn of TpM. We must show that p is smooth; then volu = p-1w will be the volume form. Mn and set Consider an orientation-preserving local parametrization f : lV Positivity ensures that

Xj (q) =

(e�j )

q

=

but then

Dqj(ej) E Tf(q) M

These form a positively oriented basis of

for

---+

1 � j � n and q E W.

Tf (q) M. An

application of the

Gram-Schmidt orthonormalization process gives an upper triangular matrix

A(q )

=

(aij(q)) of smooth functions n

( 1 3)

on

j= l

W

with

.is a positively oriented orthonormal basis of

(14)

po

J( q)

aii(q)

Tf(q)M.

WJ(q)(bJ (q), . . . , bn (q)) = (detA (q)) * = (detA(q))(J w)q(e1, . , en)· =

This shows that p is smooth.

..

>

0, such that

Then

WJ(q)(XI (q),

.

. . , Xn.(q)) 0

74

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

Addendum 9.17 There is the following formula for volM in local coordinates:

(15 ) where f: W

._

Mn is an orientation-preserving local parametrization and

Proof. Repeat the proof above starting with w = volM, so that p(p) p E M. Formula (14) becomes

=

1 for all

(16) The inner product of ( 1 3) with the corresponding fonnula for bk ( q) yields (with a Kronecker delta notation) 8ik = (bi(q), bk(q)) f(q)

=

n

n

L L aij(Q) 9jt(q) akt (q). j=I l=I

This is the matrix identity, I = A(q) G(q) A(ql, where G(q) = (9jt(q)) . In particular (det A(q))2 detG(q) = 1 . Since det A(q) = ITi aii (q) > 0, we obtain (det A(q)) -1 Example 9.18 Define

= Jdet G(q).

an (n - 1)-fonn Wo E

0

nn- I(Rn) by

Wox(WI, . . . , Wn-1) = det(x, W1,..., W - I) E Altn- 1(Rn), n I i for x E R. Since wox(e1, . . . , ei, . . .,en) = (-l) - Xi, we have

(17)

( 1 8) If X E sn- 1 and WI, ... ' Wn- I is a basis of Txsn- I then x,WI,. . .,Wn1 becomes a basis for Rn and ( 17) shows that Wox # 0. Hence woiSn-l = i* (wo) is an orientation fonn on sn-l. For the orientation of sn-l given by Wo, the basis w1, ... , Wn- I of Txsn-1 is positively oriented if and only if the basis x, w1, . . . ,W -1 for Rn is positively oriented. n We give sn-1 the Riemannian structure induced by Rn . Then ( 17) implies that volsn-1 = woiS"-1. We may construct a closed (n - 1)-fonn on Rn - {0} with w!Sn-1 = volsn- 1 by setting w = r* (volsn-1 ), where r: Rn - {0} ._ sn- I is the map r(x) = xfllxll· For x E Rn - {0}, Wx E Altn I(Rn) is given by Wx(VI, . . . , Vn- d

=

Wor· (x)(Dxr(vi), . . . , Dxr(vn-d) = !lxll- 1 det(x,Dxr(vt), . . . , Dxr(vn-1)).

9.

75

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

Now we have if v E Rx . if v E (Rx)l so that Dxr (v) = Hx1J-1 w , where w is the orthogonal projection of v on (Rx)l. . Letting Wi be the orthogonal projection of Vi on (Rx )1. we have Wx(v1 , . . . , Vn -1) = IJxl / -n det(x, w1, . . . , Wn- 1 ) = /lxll-nwox(�!l , · · · , Vn- 1 ) ·

Hence the closed form

w=

(19)

w

=

/)xll-n det(x, VI , . . . , �Jn-1 )

i s given by

�

n 1 i-1 "' ( -1) Xi dx1 1\ n llx/l

-

. . . I\ dxi 1\ . . . 1\ dx n .

Example 9.19 For the antipodal map

A : sn- l - sn- l ;

Ax = -x

we have

A*(volsn-1) = (-ltvolsn-1 , and A is orientation-preserving if and only if n is even. In this case we get an orientation form T on RPn-1 such that 1r* ( T ) = volsn-1, where 1r is the canonical map 1r: sn-l - RPn- l. For x E sn- 1, Txsn- 1 D�A TA x sn- 1

is a linear isometry. Hence there exists a Riemannian structure on IRPn-1 characterized by the requirement that the isomorphism np n-1 T Sn- 1 D,;;r - Tn(x)" x

is an isometry for every x E sn-l. If n is even and RPn-l is oriented as before, one gets 1r* (volRP"-1 ) = volsn-1. Conversely suppose that RPn-1 is orientable, n 2:: 2. Choose an orientation and let volRP"-1 be the resulting volume form. Since Dx1r is an isometry, 1r* ( volRpn-1 ) must coincide with ±volsn- t in all points, and by continuity the sign is locally constant. Since sn- 1 is connected the sign is constant on all of sn- 1 . We thus have that

where

6

1r* ( volRp"-t ) = bvolsn- 1 ,

= ±1. We can apply A* and use the equation 1r o A = 1r to get

(-lt6 vols.. -1

This requires that n is even.

n

=

oA*(vols.. -1) = A*1r*(volRP"-1)

= (1r o A)*(volRpn-t) = 1r*(volRP"-1)

=

b vols,.. - t .

is even and implies that RPn-l is orientable if and only if

76

9.

Remark 9.20

DIFFERENTIAL FORMS ON SMOOTII MANIFOLDS

For two smooth manifolds

Mm

and

Nn

the Cartesian product

A1m x Nn is a smooth manifold of dimension m+n. For a pair of charts and

k: V ---+ V' of M M

as a chart of

pE M

and q

If M and N

E

x

and N, respectively, we can use

h x k: U x V

N. These product charts form a smooth atlas on

h: U ---+ U' U' x V'

---+

M

N there is a natural isomorphism

x

N. For

(U, h) and (V, k). The tran­ V , h x k) satisfy the condition

are oriented, one can use oriented charts

sition diffeomorphisms between the charts

(U x

product orientation of M x N. If the orientations are specified by orientation forms w E nm(M) and CJ E nn (N), the product orientation is given by the orientation form prM (w) 1\ pr/v( ) , where prM and prN are the projections of M x N on M and N. of Proposition 9 . 1 4.

Hence we obtain a

CJ

In the following we shall consider a smooth submanifold

Mn �

Rn+k of

p E M we have a normal vector space TpMJ. normal vector .field Y on an open set W � M is a Rn +k with Y (p) E TpMJ. for every p E W. In the case smooth map Y: W k = 1, Y is called a Gauss map on W when all Y(p) have length I . Such a map dimension n.

of dimension

At every point

k.

A smooth ---+

always exists locally since we have the following:

� Rn+k there exists an open neighborhood W of Po on M and smooth normal vector .fields }j (1 ::; j ::; k) on W such that Y1(p), . . . , Y (P) form an orthonormal basis ojTpMJ. for every p E W. k

Lemma 9.21 For every Po E Mn

Proof.

On a coordinate patch around Po

X1,

E M,

there exist smooth tangent vector

. . , Xn , which at every point p yield a basis of TpM, cf. Remark 9.4. X ( Choose a basis vl' . . . ' vk of Tpo M j_ . Since the ( k) determinant fields

is non-zero at on

M.

n + k)

.

p0,

n+

it also non-zero for all p in some open neighborhood

W

of p0

Gram-Schmidt orthonormalization applied to the basis

X1 (p), . . . , Xn (P), V1 ,

.

. , Vk .

(p E W)

of JRn+k gives an orthonormal basis

where the first

n vectors span TpM. The formulas of the Gram-Schmidt orthonor­

malization show that all desired properties.

Xi

and Yj are smooth on

W, so that Y1. . . . , Yk

have the

0

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOI,DS

Proposition 9.22 Let Mn

�

77

Rn+l be a smooth submanifold of codimension

1.

(i) There is a I-1 correspondence between smooth normal vector.fields Y on M and n-forms in nn(M). It associates to y the n-form w = Wy given by

Wp(Wl , . . . , Wn ) = det(Y(p), W1 , . . . , Wn ) M, Wi E TpM.

for p E (ii) This induces a I-1 correspondence between Gauss maps Y: M orientations of M.

--+

sn and

Proof. If p E M then Y(p) = 0 if and only if wp = 0. Since wy depends linearly on Y, the map Y - wy must be injective. If Y is a Gauss map, then wy is an orientation form and it can be seen that wy is exactly the volume form associated to the orientation determined by wy and the Riemannian structure on lvf induced from Rn+l. If M has a Gauss map Y then (i) follows, since every element in nn(M) has the form f . wy = WJY for some f E C00(M, R). Now M can be covered by open sets, for which there exist Gauss maps. For each of these (i) holds, but then the global case of (i) automatically follows.

An orientation of M determines a volume form volM, and from (i) one gets a Y with wy = volM. This Y is a Gauss map. D

Theorem 9.-23 (Tubular neighborhoods) Let Mn � Rn+k be a smooth submani­ fold. There exists an open set V � IRn+k with M � V and an extension of idM to a smooth map r: V --+ M, such that

(i) For x E V and y E M, JJx - r(x )JI :::; JJx - yJJ, with equality if and only if y == r(x). (ii) For every p E M the fiber r- 1 (p) is an open ball in the affine subspace p + TpM J. with center at p and radius p(p) , where p is a positive smooth function on M. If A1 is compact then p can be taken to be constant. E(p) p(p) for all p E M then (iii) If f.: M ._ R is smooth and

Se

==

0< <

{x E V l llx - r(x)JJ

is a smooth submanifold of codimension

1

==

E(1·(x) )}

in Rn+k.

We call V(= Vp) the open tubular neighborhood of M of radius p

Proof. We first give a local construction around a point Po E M. Choose normal vector fields Y1 , . . . , Yk as in Lemma 9 . 21, defined on an open neighborhood W of Po in M for which we have a diffeomorphism f: Rn --+ W with f(O) = PO· Let us define

k

+ I:>jYj(f(x)) j= l

78

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

The Jacobi matrix of <J? at 0 has the columns

of of !:1(0) , . . . , � ( 0), Yl (Po), . . . , Yk (Po). UXl UXn

k

..L The first n form a basis of Tp0 M and the last a basis of Tpo M . By the inverse function theorem, <J? is a local diffeomorphism around 0. There exists a (possibly) smaller open neighbourhood Wo of Po in M and an to > 0 , such that
=p+

k

'L ti Yj (p)

j=l

defines a diffeomorphism from W0 x t0Dk to an open set V0 � Rn+ k. The map 1 ro = prw0 o <1?0 defines a smooth map ro: Vo ._ Wo, which extends idw0, so ..L that the fiber r0 1 (p) is the open ball in p + TpM with center at p and radius to for every p E Wo. By shrinking to and cutting Wo down we can arrange that the following condition holds:

(20)

For x E Vo and y

with equality if and only if

EM

we have ll x - ro(x)ll :S llx - Yll

y = ro(x).

This can be done as follows.

By Definition 8.8 there exists an open neighborhood W of Po in Rn+k such that M n W is closed in W. In the above we can ensure that V0 � W where M n Vo remains closed in Vo. Choose compact subsets K1 � K2 of W0 so that (in the induced topology on M) Po E intK1 � K1 � intK2, where intKi denotes the interior of Ki. The set

B = (Rn+k - Vo) u (M n Vo - intK2) is closed in Rn+k and disjoint from K1. There exists an t E (0, to] such that lib - Yll 2: 2t for all b E B, y E K1. If we introduce the open set

V� = {x E Vo I ro(x) E intK1 and llx - ro (x)ll < €} ,

we get for x E V� and b E

M - K2 � B

that

llx - bll 2: lib - ro(x)ll - llx - ro (x)ll > t. Since llx - ro(x)ll < t, the function y ._ llx - Yll , defined on M, attains a minimum less than t somewhere on the compact set K2. Consider such a Yo E K2 with

llx - Yoll = min llx - Yll :S llx - ro(x) ll yE M

<

t :S to.

Hence x - Yo is a normal vector to M at Yo (see Exercise 9. 1 ), but then x E Vo and y0 = r0(x) . This shows that Condition (20) can be satisfied by replacing (Wo, Vo , to) by (intK1 > V�, t).

?.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

79

M can be covered with open sets of the type V0 with the associated smooth maps ro which satisfy Condition (20). If (Vi , r1 ) is a different pair then ro and r·1 will coincide on V0 n V1 . On the union V' of all such open sets (of the type M, which extends idM, such above) we can now define a smooth map r: V' that part (i) of the theorem is satisfied. We have rwo = ro for every local (Vo, ro) . If in the above we always choose Eo � 1 , then the fiber r- 1 (p) over a p E M will be an open ball in p + TpM .l. with radius p(p) for which 0 < p(p) � 1. Thus Now all of

---+

we have satisfied (i) and (ii), except that

The distance function from

M,

dM(x) =

p

in£ yEM

might be discontinuous.

llx - yjJ,

Rn+k. For x E V', (i) shows that dM (x) JJx - r(x)ll (x E V' ) . If p E M and x E p + TpM.l. has distance p(p) from p, we can conclude that dM(x) = p(p). In this case (i) excludes that x E V', so x lies on the boundary of V'. Hence the distance function d: M ---+ R; d(p) = in£ Jlp - zll is continuous on all of

=

satisfies

0 < d(p) � jj(p)

and is continuous.

B y Lemma A.9 the function function '1/J:

for all

V'

---+

x E V'.

R

zlj!V'

� d o r: V'

such that

---+

R

can be approximated by a smooth

II'I/J(x) - � d o r(x)ll � ! d o r(x)

In particular

! d(x) � '1/J(x) � � d (x) when x E M. Hence the restriction p = '!/JiM: M R is a positive smooth function with p(p) < jj(p) for all p E M. When M is compact, the same can be achieved for the constant function which takes the value p = � min d(p). If we define ---+

pEM

V = { x E V' J Jix - -r(x)ll < p(r(x)) } both (i) and (ii) hold for the restriction of -r to V. It remains to prove (iii). It is sufficient to show that S( n V0 is empty or a smooth submanifold of Vo of codimension 1 . The image under the diffeomorphism W0 x Eobk of S, n Vo is the set ci?0 1 : V0 S {(p, t) E Wo x Rk l lltl l = E(p) < Eo} . The projection of S on Wo is the open set U = {p E Wo I E(p) < co } s; M. U x Rk given by ¢(p, t) = (p, c(p)t) maps The diffeomorphism ¢: U x Rk X Sk-l to S. This yields (iii). 0 ---+

=

U

---+

9.

80

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

0

Remark 9.24 In Chapter 1 1 we will need additional information in the case where Mn � Rn+k is compact and p > 0 is constant. To any number £, < c < p, we define the closed tubular neighborhood of radius c around M by

Nt = {x E V l llx - r(x)l!

:S c .

}

This set is the disjoint union of the closed balls in p + Tp M..L with centers at p and radius c. Note that N( is compact and that S£ is the set of boundary points of Nt in Rn+k_ By Theorem 9.23.(i) we see for p E M that the real-valued function on St , x -+ llx - Pll, attains its minimum value £ at all points x E St with r(x) = p. It follows that (2 1 )

We end this chapter with a few applications of the existence of tubular neigh­ borhoods. Let (V, i, T) be a tubular neighborhood of M with i: M -+ V the inclusion map and T: V -+ M the smooth retraction map such that T o i = idM . In cohomology this gives

Hd(i) Hd(r) = idHd(M)' o

so that

Hd(i): Hd(V)

injective.

-+

Hd(M)

is smjective and

Hd(r): Hd(M)

-+

Hd(V)

is

Proposition 9.25 For any compact differentiable manifold Mn all cohomology spaces Hd ( M) are finite-dimensional. Proof. We may assume that Mn is a smooth submanifold of Rn+k by Theorem

8. 1 1 , and that (V, i, r) is a tubular neighborhood. Since M is compact we can find finitely many open balls U1, . . . , Ur in Rn+k such that their union U = U1 u . . . UUr satisfies M � U � V. Now we have a smooth inclusion i: M -+ U and a smooth map rw: u - M with TJU 0 i = id_;v[. The argument above shows that

is surjective, and the assertion now follows from Theorem 5.5.

0

Proposition 9.26 Let M1 and Af2 be smooth submanifolds of Euclidean spaces. (i)

If fo, ft: Mt

-+

M2 are two homotopic smooth maps, then

(ii) Every continuous map M1

-+ Mz is homotopic to a smooth map.

DIFFERENTIAL FORMS ON SMOOTH MANlFOLDS

9.

81

Proof. Choose tubular neighborhoods (Vv, iv, rv) of JV!v,

implies that i2 o fo o r1 so that

�

11 = 1, 2. Lemma 6.3 i2 o h o r1. Hence Hd(i2 o fo o r1) = Hd(i2 o fi o r1),

Since Hd(r1) is injective and Hd(i2) is surjective, we conclude that Hd(!0 ) = Hd( h ) . If¢: M1 - M2 is continuous, we can use Lemma 6.6.(i) to find a smooth map g: V1 - V2 with g � i2 o¢or1. For the smooth map f = r2 ogoi1: M1 - M2, Lemma 6.3 shows that f � r2 o (i2 o rp o r1) o i1 = rp. 0 Remark 9.27 As in the discussion preceeding Theorem 6.8, the de Rham coho­

mology can now be made functorial on the category of smooth submanifolds of Euclidean space and continuous maps. Theorem 6.8 and Corollary 6.9 are valid (with the same proofs) with open sets in Euclidean space replaced by smooth sub­ manifolds. By Theorem 8. 1 1 the same can be done for differentiable manifolds in general.

is a smooth submanifold and (V, i, r) an open tubular neighborhood, then Hd(i): Hd(V) - Hd(M) is an isomorphism with Hd(r) as its inverse. Corollary 9.28 If Mn � Rn+k

= idM and i o r idv, as V contains the line segment and r(x) for all x E V. By Proposition 9.26.(i) we can conclude that

Proof. We have r o i

between x Hd(r) and Hd(i)

are

Example 9.29 For

�

0

inverses.

n :2:: 1 we have Hd(sn)

�

{R 0

if d = 0, n otherwise.

Let i: sn - Rn+l {0} be the inclusion and define r: Rn+l {0} - sn by r(x) = Jfxrr Then r o i = idsn, i o r � idRn+l- {O} and Hd(i) is an isomorphism. The result follows from Theorem 6.13. -

-

Remark 9.30 Let U1 and U2 be open subsets of a smooth submanifold Mn � R1 . Using Theorem 9. 1 1 , the proof of Theorem 5 . 1 can be carried through without any significant changes. As in Chapter 5 this gives rise to the Mayer-Vietoris sequence

82

9.

DIFFERENTIAL FORMS ON SMOOTII MANIFOLDS

Example 9.31 We shall compute the de Rham cohomology of

RPn- 1 (n 2:: 2) .

With the notation of Example 9. 1 9 we see that

AltP (DxA): AltP (T.,.. (x) Rpn- 1 )

is an isomorphism for every X E

sn- 1 .

_.

AltP(Txsn- 1 )

Therefore

is a monomorphism, and we find that the image of DP (1r) is equal to the set of p­ forms w on sn- 1 such that A*w = w. Since A* = f!P (A): D.P(sn-1 ) _. flP(Sn-1 )

has order 2 we can decompose it into (± 1 )-eigenspaces

where

n� (sn- 1 ) = Im (� ( id ± flP(A))) . This in fact decomposes the de Rham complex of sn- 1 into a direct sum of two subcomplexes

(22) There is an isomorphism of chain complexes (23)

1r: sn- 1 RPn-t. From (22) we get isomorphisms HP (sn-1 ) � HP (n� (sn-1 )) EB HP ( n� (sn -1 ) ) � H� (sn- 1 ) EB H� (sn - 1 ) where H� ( sn- 1 ) is the ( ± 1) -eigenspace of A* on HP ( sn- 1 ) . Combining with

induced by

-+

(23) we find that (24) There is a commutative diagram with vertical isomorphisms (See Example 9.29)

il ·

Hn- 1 (Rn - { 0}) - Hn-l(Rn - {0})

�

Hn-1 (sn-t )

�

�l l

. i

Hn- (sn- I )

where the top map is induced by the linear map x _. -x of Rn into itself. Lemma 6.14 shows that the bottom map is multiplication by ( - 1 t. Using (24) and Example 9.29 we finally get (25)

HP (RPn- 1 )

�

{R 0

?

if p = or p otherwise.

= n - 1 with n even

83

10.

Let

INTEGRATION ON MANIFOLDS

Mn

be an oriented n-dimensional smooth manifold. We define an integral

JM:D.�(Mn) -+ R on the vector space of differential n-forms with compact support. Next we shalJ and Stokes' theorem will be proved. consider integration on subsets of Finally we calculate for an arbitrary orientable compact connected smooth manifold

Hn(Mn) Mn. Mn = Rn

Mn,

In the special case where (with the standard orientation) we can write w E uniquely in the form

D.�(Rn)

w where J

E

c=(Rn, R)

=

J(x)dx1 1\ . . . 1\ dxn,

has compact support. We then define

r J(x)dJ.Ln, r }Rn J(x)dxl l\ . . . 1\ dxn = }Rn

where dJ.Ln is the usual Lebesgue measure on Rn. The same definition can be used for an open set � since w and f are smoothly extendable when w E to the whole of by setting them equal to 0 on suppv(w).

D.�(V) Rn

V Rn,

Lemma 10.1 Let ¢>: V

Rn -

V

W be a diffeomorphism between open subsets and W of Rn, and assume that the Jacobi determinant det (Dx) is of constant sign 8 = ±1 for x E V. For w E D.�(W) we have that -+

i * (w) = · fw w. 8

Proof. If

with f

w

is written in the form

w = j(x)dx1 1\ . . . 1\ dxn

E Cg<>(W,

R),

it follows from Example 3.1 3.(ii) that

*(w )

= f((x)) det (Dx) dx1 1\ . . . 1\ dxn ==

8J ((x))!det (Dx)l dx1 1\ . . . 1\ dxn.

The assertion follows from the transformation theorem for integrals which states that

lw·{ . f (x)dJ.Ln = lv{ J((x)) ldet (Dx)1 dJ.Ln·

D

84

10. INTEGRATION ON MANIFOLDS Proposition 10.2 For an arbitrary oriented n-dimensional smooth manifold Mn there exists a unique linear map

J

M

: O�(Mn)

-4

IR

with the following property: If w E n�(Mn) has support contained in U, where (U, h) is a positively oriented C00-chart, then

JM w = 1(U) (h-1) *w

(1)

E n�( Mn) with "small" support, i.e. such that suppM (w) is contained in a coordinate patch. Then (U, h) can be chosen as above and Proof. First consider w

the integral is determined by ( 1 ). We must show !!_la.!_ the right-hand side is independent of the choice of chart. Assume that (U, h) is another positively oriented C00-Chart with SUPPM(w) � U. The diffeomorphism ¢: v w from v = h(U n U) to w = h(U n U) given by ¢ = h o h-1 has everywhere positive Jacobi determinant. Since -4

SUPPh(U) ((h-1)*w) � V,

SUPPii ( U) ((h- 1 )*w) � W

and ¢* (h-1)*(w) = (h-1)*(w), Lemma 10.1 shows that

(

r (h-1 ) *w = - (h-1 )*w. Jh(U) lii ( u)

So for w E D�(M) with "small" support the integral defined by (1) is independent of the chart. Now choose a smooth partition of unity (Pa- )a-eA on M subordinate to an oriented C00-atlas on M. For w E n�(M) we have that

w

=

L Pa-W,

a-EA

where every term Pa-W E n�(M) has "small" support, and where only finitely many terms are non-zero. We define

I(w) = L r Pa-W, o:EA M

j

where the term associated to a E A is given by ( 1 ), applied to a Ucr with SUPPM(Pa-) � Ucr. It is obvious that I is a linear operator on n�(M). If, in particular, SUPPM(w) � U, where (U, h) is a positively oriented C00-chart, the terms of the sum can be calculated by (1), applied to (U, h). This yields

!(w) =

J w, M

which shows that I is a linear operator with the desired properties. Uniqueness follows analogously. 0

10.

INTEGRATION ON MANIFOLDS

85

Lemma 10.3

(i) J� w changes sign when the orientation of Mn is reversed. (ii) lfw E n;;-(Mn) has support contained in an open set W c Mn, then

when W is given the orientation induced by M. (iii) If ¢: Nn -t Mn is an orientation-preserving diffeomorphism, then we have that

for w E n;;- (M). Proof.

By a partition of unity, we can restrict ourselves to the case where suppM(w) is contained in a coordinate patch. All three properties are now easy consequences of Lemma 10.1 and ( 1 ) . 0 Remark 10.4 In the above we could have considered integrals of continuous

n-forms with compact support on Mn. If the orientation of M is given by the orientation form a E nn(M), a continuous n-form can be written uniquely as fa, where f E C0(M, R). The support of fa is equal to the support of f . The integral of (1) extended to continuous n-forms gives rise to a linear operator

This linear operator is positive, i.e. Ia(!) � 0 for f � 0. By a partition of unity it is sufficient to show this when supp(J) � U, where (U, h) is a positive oriented C00-chart. Then we have that Ia (J) =

f f o h- 1 (x)¢(x)dJ-Ln, jh(U)

where ¢ is determined by (h- 1)*(a) = ¢(x)dx 1 1\ . . . 1\ dxn. Since ¢ is positive we get Ia(J) � 0. According to Riesz's representation theorem (see for instance chapter 2 of [Rudin]) Ia determines a positive measure /-La on M which satisfies

The entire Lebesgue integration machinery now becomes available, but we shall use only very little of it in the following.

86

INTEGRATION ON MANIFOLDS

10.

If Mn is an oriented Riemannian manifold, the volume form volM will determine a measure 1-tM on Mn analogous to the Lebesgue measure on Rn. For a compact set J( the volume of J( can be defined by Vol (]()

=

f volM E R. }I<

Definition 10.5 Let Mn be a smooth manifold. A subset N � Mn is called a

domain with smooth boundary or a codimension zero submanifold with boundary, if for every p E M there exists a C00-chart (U, h) around p, such that h(U n N)

(2) where R�

= {(xl, . . . Xn) E nn I XI l

= h(U) n R� , :::; 0}.

Note that (2) is automatically satisfied when p is an interior or an exterior point of N (one can choose (U, h) with p E U, such that h( U) is contained in an open half-space in Rn defined by either x1 < 0 or x1 > 0). If p is a boundary point of N then h(p) has first coordinate equal to zero. Let (U, h) and (V, k) be smooth charts around a boundary point p E oN. The resulting transition diffeomorphism ¢ = k o h- 1 : h(U n V)

---+ k(U n V)

induces a map h(U n V) n R� ---+ k(U n V) n R� , which restricts to a diffeomorphism \II :

h(U n

v) n oR�

(�(

---+

k(U n V) n me.

The Jacobi matrix at the point q = h(p) E oR� for ¢ = (¢1, . . . , ¢n ) has the form q)

Dq¢ =

: *

0

0

}

We must have (o¢dox1)(q) =f:. 0, as D9¢ is invertible. Since ¢ maps R� into R�. we have that (o¢dox1)(q) > 0. A tangent vector w E TpM at a boundary point p E oN is said to be outward directed, if there exists a c=-chart (U, h) around p with h(U n N) = h(U) n R� and such that Dph( w) E Rn has a positive first coordinate. This will then also be the case for any other smooth chart around p.

10. INTEGRATION ON MANIFOLDS

87

Lemma 10.6 Let N � Mn be a domain with smooth boundary. Then oN is an (n - !)-dimensional smooth submanifold of Mn. Suppose Mn (n 2: 2) is oriented. There is an induced orientation of oN with the following property: if p E oN and v1 E TpM is an outward directed tangent vector then a basis vz, . . . , v for Tp8N is positively oriented if and only if the n basis VI, vz, . . . , Vn for TpM is positively oriented. Proof. Every smooth chart

(U, h)

in M that satisfies h(U n N)

be restricted to a chart (U n oN, h1) on oN:

= h(U) n R� can

These charts have mutual smooth overlap according to the above. This yields a smooth atlas on 8N. Suppose Mn is oriented. Then, possibly changing the sign of x2, we can choose a positively oriented chart (U, h) of the considered type around any p E oN. The resulting smooth charts (Un8N, h1 ) on oN have positively oriented transformation diffeomorphisms, and they determine an orientation of 8N that satisfies the stated 0

property.

Remarks 10.7 (i) We want to integrate n-forms w E n�(M) over domains N with smooth boundary. In view of Remark 1 0.4 we can set

where l N is the function with value 1 on N and zero outside N. Alternatively, one can prove an extension of Lemma 10. 1 which uses the following version of the transformation theorem: Let ¢: V ---+ W be a diffeomorphism of open sets in Rn that maps R� n V to R� n W, and let f be a smooth function on W with compact support. Then

f j(x)dp,n = f j¢(x)Jdet Dx¢ld!J-n· }R-:_nw JR:, nv (One could approximate both integrals by integrals over W and V upon multiplying f by a sequence of smooth functions 7/Ji with values in [0, 1] and converging to lw� .) (ii) In the case n = 1, Lemma 10.6 holds in the following modified form. An orientation of 8N consists of a choice of sign, + or -, for every point

p E oN. Let VI E TpM be outward directed. Then p is assigned the sign + if v1 is a positively oriented basis of TpM , otherwise the sign is -.

10.

88

INTEGRATION ON MANIFOLDS

A 0-form on aN is a function f: aN

we define

!aoN

f ::=

L

pEoN

-t

R. When f has compact support

sgn(p) f (p).

These conventions are used in the case n = 1 of Stokes' theorem below.

0 Theorem 10.8 (Stokes' theorem) Let N � Mn be a domain with smooth boundary in an oriented smooth manifold. Let aN have the induced orientation. For every w E nn- l (M) with N n SUPPM(w) compact we have

where

i: oN

r i*(w) = r dw, ./N ioN

-t

M is the inclusion map.

Proof. We assume that changes for the case n

n

=

� 2 and leave it to the reader to make the necessary

1.

It is clear that ·i *(w) has compact support. We can choose f E D�(M) with value constantly equal to 1 on N n suppM (w). Since fw coincides with w on N, both integrals are unchanged when w is replaced by fw, so we may assume that w has compact support. Choose a smooth atlas on M consisting of charts of the type of Definition 10.5 and a subordinate smooth partition of unity (Pa)aEA. The formulas

r w = r PaW, JaN L a ioN

reduce the problem to the case where w E n� - 1 (Jvf), suppM (w ) � U and (U, h) is a smooth chart with h(U n N) = h(U) n R�. Furthermore the chart (U, h) is assumed to be positively oriented.

10.

89

INTEGRATION ON MANIFOLDS

Let K E n�-1(Rn) be the (n - 1)-form that is (h- 1 )*(w) on h(U) and 0 on the rest of R11• By diffeomorphism invariance we then have that

[ [ w = h(U)naR� (h-1 )*(w) = law� laN

1

and

K

(h -1)*(dw) = [ dK. [ dw = [ jR� jh(U)nR':_ }N

Hence the proof reduces to the special case where M = Rn , N = R� and w E n;;-1(Rn). This case is treated by direct calculation. We define n w = L fi(x) dx1 1\ . . . 1\ &i 1\ . . . i=1

1\

and choose b > 0 such that suppR..fi � [-b, bt, 1 :S i :S

dxn n.

Using Theorem 3.12,

Hence

{ W = J h (0, X2, . . · , Xn)dJ.Ln-1· laR�

(3)

By Theorem 3.7 we have

Hence (4)

For 2 � i :S n we get

1

= ofi � (XI , . . . , Xi-11 t, Xi+b . . . , Xn)dt

vx� =/i(x1, . . . , Xi-11 b, Xi+1 • · · · , xn) - /i(x1 , · · - ,Xi- l , - b, xi+1 · · · · ,Xn) =0, _00

and then by Fubini's theorem (5 )

90

INTEGRA'flON ON MANIFOLDS

10.

i

When

= 1,

one gets

10 aahXI (t, X2, . . . , xn)dt = h(O,x2, . . . , Xn ) - h ( -b, x2, . . . , xn) -oo

= fl(O, X2, . . . , Xn ) ,

a dJ.Ln = h(O, x2, . . . , xn) df.L -I · a n l XIfl I

and by Fubini's theorem

(6)

R�

By combining Equations Taking

N= M

Corollary 10.9

the desired formula follows.

(3-6)

in Theorem

10.8

0

we have

If Mn is an oriented smooth manifold and w

E

JM dw = 0. Let

Remark 10.10

cohomology class

w be a closed d-form on Mn.

[w] E Hd(M)

h f* (w)

10.9

yields

If

#0

M from a d-dimensional compact f : Qd w = dr for some r E nd- 1 (Af ) , then Corollary

for a suitably chosen smooth map

Qd.

One way of showing that the

is non-zero is to show that

(7)

oriented smooth manifold

---+

h f*(w) = h d(f*(r)) =

This, in essence, was the strategy from Examples

0.

1.2

and 1 .7. It can be shown

(albeit in a very indirect way via cobordism theory) that all integrals of the form of

Example 10.11

Rn -

In Example

{0}, w =

(7)

1 ll xlln

�

�

[w] =

we considered the closed (n - 1)-form on

i-I - 1 ) xidx1 1\ . . . 1\ dxi 1\ . . . 1\ dxn.

Since the pre-image of w under the inclusion of snwhich has positive integral over

I

is the volume form volsn-1,

sn-l, we can conclude from Remark 10.10 that

Hn- 1 (Rn - {0} ) . If n � 2 then, by Theorem 6.13, Hn- 1 (Rn - {0} ). We thus have an isomorphism Hn-l(Rn - {0} ) � R (n � 2) [w]

# 0 in

defined by integration over the volume

0 if and only if

vanish.

9. 1 8

(

n�- 1(M ) then

sn- l .

The image of

[w]

[w] is a basis of

under this isomorphism is

10.

INTEGRATION ON MANIFOLDS

91

Example 10.12 The volume of sn-I can be calculated by applying Stokes'

theorem to Dn with the standard orientation of Rn and the (n Rn given by

Since woiS>"-1 = volsn-1 and dwo

==

ndx 1 1\

. . .

-

1)-form on

1\ dxn we have that

By induction on m and Fubini's theorem, it can be shown that Vol ( D2m)

7rm

=

-,

m!

1 m 22m+ l m.7r Vol (D2m+l ) = (2m + 1 ) !

This yields Vol (S2m- l )

=

27rm , (m - 1 ) !

Vol (S2m ) =

t m 22m+ l m.1r

(2m)!

We conclude this chapter with a proof of the following: Theorem 10.13 IfMn is a connected oriented smooth manifold,

then the sequence

(8)

is exact. For a connected compact smooth manifold Mn, integration over M induces an isomorphism Corollary 10.14

0

In (8) it is obvious that the integral is non-zero and hence surjective. It follows from Corollary 10.9 that the image of d is contained in the kernel of the integral. We show the converse inclusion.

10.

92

Lemma 10.15

INTEGRATION ON MANIFOLDS

Theorem 10.13 holds for M = R'\ n 2: 1.

. ..

w E D�(Rn) be a differential n-forrn with JR w = 0. We must find l "' E n�- (Rn), such that d!'- = w. We can write w = j(x)dx1 1\ . 1\ dxn. and let

Proof. Let

A simple calculation gives

( ) = L -. ]= 1 n

d!'-

.

&Jj

f) XJ

.

dx1 1\ . . 1\ dxn·

Hence we need to prove the following assertion: (Pn): Let f E C�(Rn ) be a function with J f(x)df.kn = 0. There exist functions n fi, . , fn in C� (R ) such that

..

We prove (Pn) by induction. For n = 1 we are given a smooth function f E C�(R) with f�oo f(t)dt = 0. The problem is solved by setting fi(x) =

��

J(t)dt.

Assume (Pn- 1) for n 2: 2, and let f E C�(Rn, R) be a function with J f(x)df.kn = 0. We choose C > 0 with supp (!) � [-C, ct and define

(9) (The limits can be be replaced by -C and C, respectively). The function g is smooth, since we can differentiate under the integral sign. Furthermore supp(g) � [-c,cr- 1 . Fubini's theorem yields J g dttn-1 = J fdttn = 0. Using (Pn-d we get functions 91 , . . . , 9n-1 in C� (Rn-I , R) with n- 1

f)

g· I: _ J = g. &xj

(10)

j=l

We choose a function C�(Rn, R),

(11)

p

E C�(R, R) with f�oo p(t)dt

- 1,

and define /j E

/j(Xl, . . . , Xn-1, Xn) = 9j(Xl, . . . , Xn_I )p(xn) , 1 � j � n - 1 .

INTEGRATION ON MANIFOLDS

10.

Let h E C�(IRn)

93

be the function

n- 1 afi h = f - "' � axJ· ' j=l

( 1 2)

C�(Rn ) with ofn/oxn = h is given by

A function fn

E

( 1 3)

fn(Xt, . . . , Xn-1,Xn) =

!Xn -

oo

h(x1, . . . , Xn-1, t)dt.

It is obvious that fn is smooth, but we must show that it has compact support. To this end it is sufficient to show that the integral of ( 1 3) vanishes when the upper limit Xn is replaced by oo. Now ( 1 0), ( 1 1 ) and (12) yield that

h(x l > . . . , Xn-1, t) = j(x 1 , . . . , Xn-1. t) -

n-l

L j= l

!:l

ug ·

J (x1, . . . , Xn-l)P(t) axJ

= f(xl> . . . , Xn-b t) - g(x1, . . . , Xn-dP(t). Finally from (9) it follows that

/00 = 1: -

oo

h(x1 . . .

·.

,

Xn-1, t)dt

j(x1, . . . , Xn-1, t)dt - g(x1, . . . , Xn-d

= 0.

1:

p(t)dt 0

Lemma 10.16 Let (Ua)aeA be an open cover of the connected manifold M, and

let p, q E

M.

There exist indices a1 , . . . , ak such tho.t

(i) p E Ua1 and q E Uak (ii) Ua; n Uai+l =j:. 0 when 1

s; i s;

k - 1.

Proof. For a fixed p we define V to be the set of q E M, for which there exists a finite sequence of indices a1, . . . , ak from A, such that (i) and (ii) are satisfied. It is obvious that V is both open and c.losed in M and that V contains p. Since M is connected, we must have V = M. 0

be an open set diffeomorphic to Rn and let W � U be non-empty and open. For every w E n�(M) with suppw s;;; U, there exists a "' E n�- 1 (JV!) such that supp "' � U and supp (w - d"') � W. Lemma 10.17 Let

U�

M

Proof. It suffices to prove the lemma when M =

U,

in variance it is enough to consider the case where M =

and by diffeomorphism

U = Rn.

10.

94

INTEGRATION ON MANIFOLDS

fRn WI = 1. Then where a = r w. }Rn

Choose WI E n�(Rn) with suppwl � w and

r }R� (w - awl) = 0,

By Lemma 10.15 we can find a "' E n�-1(Rn) with

w - aw1 = d"'.

Hence w - d"' = aw1 has its support contained in W.

0

Lemma 10.18 Assume that Mn is connected and let W � M be non-empty and open. For every w E S1�(M) there exists a "' E n�-1(M) with supp(w - d"') � w. Proof. Suppose that suppw � U1 for some open set U1 � M diffeomorphic to Rn. We apply Lemma 10.16 to find open sets U2, . . . , Ub diffeomorphic to Rn, such that Ui 1 n Ui #- 0 for 2 .:::; i ::; k and Uk � W. We use Lemma 10.17 to successively choose "'I' "'2 ' . . . , "'k -I in n�- 1 (M) such that

( - t d!
supp w

t=l

<;

U, n U;+ I (I

<; j :S k -

I).

The lemma holds for "' = E:::l "'i. In the general case we use a partition of unity to write

w=

m

L Wj,

j=l

where Wj E S1�( M) has support contained in an open set diffeomorphic to nn. The above gives Kj E n�-1(M), 1 ::; j ::; m, such that supp(wj - d'Kj) � W. For ;;;_

m

1 = L "-i E n�- (M)

j=l

we have that m

w - d'K = L (wj - d'Kj )· j=l Hence supp(w - d'K) £; U;: 1 supp(wj - d'Kj) � W.

0

Proof of Theorem 10.13. Suppose given w E Sl�(M) with fM w = 0. Choose an open set W � M diffeomorphic to nn. By Lemma I 0. I 8 we can find a "' E n�-1(M) with supp(w - d"') � W. But then by Corollary 10.9,

fw (w - d"') JM (w - d"') - JM d"' = =

=

0.

10.

INTEGRATION ON MANU'OLDS

Lemma 10.15 implies that Theorem 10.13 holds for W, i.e. there exists n�- t (W) that satisfies

95

a

ro E

Let r E n�- 1(M) be the extension of ro which vanishes outside of suppw(ro). Then w - dK = dr, so r + "' maps to w under d. 0

97 11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

Let f: Nn ---+ Mn be a smooth map between compact connected oriented mani­ folds of the same dimension n. We have the commutative diagram

(1) where the vertical isomorphisms are given by integration over M and N respec­ tively; cf. Corollary 10.14. The lower horizontal arrow is multiplication by the real number deg(J) that makes the diagram commutative. Thus for w E nn(M),

j�

(2)

f*(w)

= deg(J)

JM w.

This formulation can be generalized to the case where N is not connected: Proposition 11.1 Let f: Nn ---+ Mn be a smooth map between compact n­ dimensional oriented manifolds with M connected. There exists a unique deg(J) E R such that (2) holds for all w E nn(M). We call deg(J) the degree of f.

N as a disjoint union of its connected components N1, . . . , Nk and denote the restriction of f to Nj by fi. We have already defined deg(fj ); Proof. We write

we set

k

deg(J)

(3)

= L deg(fj). j=l

Thus for w E nn (M),

l

f'(w) =

k

J;; l,

Jj (w)

=

k

J;;

deg(f;)

L

w = deg(f)

L

w.

0

Corollary 11.2 deg(f) depends only on the homotopy class off: N ---+ M.

Proof. By (3) we can restrict ourselves to the case where N is connected. The assertion then follows from diagram (I), since Hn(J) depends only on the homotopy class of f. 0

98

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

f Mn 9 pn are smooth maps between nSuppose N -+ dimensional compact oriented manifolds and that M and P are connected. Then Corollary 11.3

n

-+

deg(gf) = deg( f )deg(g). Proof. For

w E nn(P), deg(g f)

r J"'(g*(w)) r (gf)*(w) jN r w = jN }p = deg(J) r g*(w) = deg(J)deg(g) r w. }M jp =

0

Remark 11.4 If f: Mn

-+ Mn is a smooth map of a connected compact orientable manifold to itself then deg(J) can be defined by chasing an orientation of M and using it at both the domain and range. Change of orientation leaves deg(J) unaffected.

We will show that deg(J) takes only integer values. This follows from an important geometric interpretation of deg(J) which uses the concept of regular value. In general p E M is said to be a regular value for the smooth map f: Nn -+ Mm if

D9f: T9 N

-+

TpM

is surjective for all q E f-1(p). In particular, points in the complement of f(Nn) are regular values. Regular values are in rich supply:

For every smooth map f: Nn regular values is dense in Mm. Theorem 11.5 (Brown-Sard)

-+

Mm the set of

When proving Theorem 1 1 .5 one may replace Mm by an open subset W � Mm diffeomorphic to Wt, and replace Nn by f 1 ( W) . This reduces Theorem 1 1 .5 to the special case where Mm = Rm. In this case one shows, that almost all points in Rm (in the Lebesgue sense) are regular values. By covering Nn with countably many coordinate patches and using the fact that the union of countably many Lebesgue null-sets is again a null-set, Theorem 1 1 .5 therefore reduces to the following result: -

Theorem 11.6 (Sard, 1 942)

set U � Rn and let

Let f: U

S=

-+

Rm be a smooth map defined on an open

{x E U I rank D f < m}. x

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

99

Then f(S) is a Lebesgue null-set in Rm.

x,

Note that x E U belongs to S if and only if every m x m submatrix of the Jacobi matrix of f, evaluated at has determinant zero. Therefore S is closed in U and we can write S as a union of at most countably many compact subsets K � S. Theorem J 1.6 thus follows if f(K) is a Lebesgue null-set for every compact subset K of S. We shall only use and prove these theorems in the case m = n, where they follow from

x

Proposition 11.7 Let f: U -+ Rn be a C1 -map defined on an open set U � Rn, and let K � U be a compact set such that det (Dxf) = 0 for all E K. Then f(K) is a Lebesgue null-set in Rn.

Proof. Choose a compact set L � U which contains K in the interior, K � L. Let C > 0 be a constant such that (4) Here fi is the j-th coordinate function off, and II II denotes the Euclidean norm. Let n T = IT !ti,ti + aJ i=l

be a cube such that K � T, and let t > 0. Since the functions {}fJj{}xi are uniformly continuous on L, there exists a ti > 0 such that (5)

lx-

Yll � ti ::::;.

lar ar 1 x : ( ) a ox� x

(y) �

(1 $ i,j $ n and x, y E L).

€,

We subdivide T into a union of Nn closed small cubes Tz with side length and choose N so that (6)

x

N•

For a small cube Tt with T1 n K =I= 0 we pick E Tt n K. If y E T1 the mean value theorem yields points �j on the line segment between x and y for which

(7) Since �j

E T1 �

L,

the Cauchy-Schwarz inequality and (4) give

l ()()

11.

DEGREE. UNKJNG NUMBERS AND INDEX OF VECTOR FIELDS

and by (6),

llf ( y) - f (x) ll

(8)

�

c diam(T1) Cvn

==

anC

N

.

Formula (7) can be rewritten as

J(y) = f(x) + Dxf(y - x) + z,

(9)

where z = ( zl> . . . , z ) is given by n

By (6), ll�i - x ll S 8, so that (5) gives l zil ::; (10)

l l z ll

S f.

n;n

a

m

N.

Hence

.

Since the image of Dxf is a proper subspace of Rn , we may choose an affine hyperplane H � Rn with

f(x)

+ Im(Dxf) � H.

By (9) and ( 1 0) the distance from f(y) to H is Jess than f.an#. Then (8) implies that f(Tt) is contained in the set Dt consisting of all points q E Rn whose orthogonal projection pr(q) on H lies in the closed ball in H with radius and centre f(x) and llq - pr(q)JI ::; we have J.1,n(Dt) =

7f

e.anp. For the Lebesgue measure J.1,n on Rn

( c)

r,;; anv •• an 2c � N

n-1

Vol(Dn-1 )

=E

c

Nn

,

where c = 2 annn+icn- lVo1(Dn-l). For every small cube Tt with Tt n K ::/= 0 we now have J.1,n (j(T1)) ::; E Jn . Since there are at most Nn such small cubes Tt, J.1,n (J(K)) ::; EC. This holds for every c > 0 and proves the assertion. 0 Lemma 11.8 Let p E Mn be a regular value for the smooth map f: Nn -+ Mn, with Nn compact. Then J-1 (p) consists offinitely many points Ql , . . . , qk. Moreover, there exist disjoint open neighborhoods Vi of qi in Nn, and an open neighborhood U of p in Mn, such that

( i) J- 1(U) =

U:::t Vi

(ii) fi maps Vi diffeomorphically onto U for 1 ::; i $ k.

11. DEGREE, LINKING NUMBERS AND INDEX OF VECI'OR FIELDS

f-1(p), D9f: f-1(p). .. f f( i) p ( f (Wi)) f (

101

T9N ----t TqM is an isomorphism. From the E inverse function theorem we know that f is a local diffeomorphism around q. In particular q is an isolated point in Compactness of N implies that consists of finitely many points q1 , . , Qk· We can choose mutually disjoint open neighborhoods Wi of Qi in N, such that maps diffeomorphically onto an open neighborhood W of in M. Let k k N-U U= n Proof. For each q

f-1(p)

Wi

i=l

i=l

Wi) .

Since N - u�l wi is. closed in N and therefore compact, N - Uf=l is also compact. Hence U is an open neighborhood of in M. We then set Vi = wi n J- 1 (U). o

f(

p

f:

Wi)

Consider a smooth map Nn - Mn between compact n-dimensional oriented manifolds, with M connected. For a regular value E M and q E define the local index n ; T9N ----t TpM preserves orientation 1 if I (ll) q = -1 otherwtse.

d(f ) {

p

f-1(p),

D9f.:

Theorem 11.9 In the situation above, and for every regular value

deg(J) = In particular

deg(f) is an integer.

L qEJ-l(p)

Proof. Let Qi , Vi, and

Ind(f; q).

p,

U be as in Lemma 1 1 .8. We may assume that U and Vi U is positively or negatively hence Vi connected. The diffeomorphism oriented, depending on whether lnd(f; Qi) is 1 or - 1 Let E nn(M) be an n-forrn with

Jrv.: -

w

.

suppM(w) k U, JM w = 1 . V1 Then suppN(f*(w)) k f - 1(U) Vk> and we can write k f* (w) = L Wi , i=l where Wi E nn(N) and supp(w ) £:; "\ti. Here Wij V; Ujv;) *(wiU) . The fonnula =

U ... U

=

i

is a consequence of the following calculation:

k

k

* (wiU) 1 f deg(f) deg(f) 1M w = 1N f* (w) = L lN Wi = L ( , v ;) i =

k

k

=

i=l

= l Vi

nd (f qi). L Ind(f;. Qi) 1 wiU = I i=l i=l

u

)

;

0

Jl. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

102

f-1(p) = 0 the theorem shows f*(w) = 0). Thus we have

In the special case where proof above we get

Corollary 11.10

that deg(f)

= 0 (in the

If deg(J) =/= 0, then f is surjective.

0

Proposition 11.11 Let F: pn+ l -t Mn be a smooth map between oriented smooth manifolds, with Mn compact and connected. Let X � P be a compact domain with smooth boundary Nn = oX, and suppose N is the disjoint union of submanifolds Nf, . . . , NJ:. If fi = FjN,, then

k

L deg(fi) = 0. i=l

Proof.

Let

f =

FJ N

so that

k

deg(J) = L deg(fi). i=l

On the other hand, if

g

de (J )

w E nn(M)

=

has

JM w = 1,

then

f f*(w) = f dF*(w) = f F*(dw) = 0 JN lx lx

where the second equation is from Theorem

0

1 0.8.

We shall give two applications of degree. We first consider linking numbers, and then treat indices of vector fields.

Definition 11.12 Let Jd

and

K1 be

two disjoint compact oriented connected

smooth submanifolds of Rn+ l , whose dimensions d 2: 1 , l 2: 1

Their

linking number

lk(J, K)

w(x, y) = JXK

d + l = n.

= deg ( w J,I<)

where

Here

satisfy

is the integer

y-x IIY - xll "

is equipped with the product orientation (cf. Remark

9.20) and sn is

oriented as the boundary of Dn+l with the standard orienation t of Rn+ l . We note that lk( J, K) changes sign when the orientation of either .J or

K

is reversed.

11.

DEGREE, LINKING NUMBERS AND lNDEX OF VECTOR FIELDS

1.03

Proposition 11.13

(i) lk( K1 , Jd) = ( l ) (d+l)(l+ I )lk ( Jd , K1 ) (ii) If J and K can be separated by a hyperplane H c Rn+I -

then lk(J, K)

0.

(iii) (iv)

=

Let 9t and ht be homotopies of the inclusions go: J ---+ Rn+l and ho: /{ ---+ Rn+I to smooth embeddings 9I and h1. such that 9t ( J) n ht (K) = 0 for all t E [0, lJ. Then 1k(J, K) = l k(g1(J), h1 (I<)) . Let : pl+l ---+ Rn+l - J be a smooth map with P oriented. Given a compact domain R � P with smooth boundary oR, let Q1, . . . , Qk be the connected components of oR. Suppose each I Qi is a smooth embedding. If Ki = (Qi). then k

L lk(J; Ki) = 0. i=l

Proof.

We look at the commutative diagram JX

f{

lrJ

/{ X

WJ,K

sn

WK,J

sn

lA

where T interchanges factors and A is the antipodal map Av follows from Corollary 11.3 upon using that deg(T)

=

1 (-l)d ,

=

-v. Then (i)

deg(A) = (-lt+1 = ( - l )d+l+l .

In the situation of (ii) the image of \lJ will not contain vectors parallel to H, and the assertion follows from Corollary 11.1 0. Assertion (iii) is a consequence of the homotopy property, Corollary 11.2. Indeed, a homotopy J X J( X [0, lj Sn is given by --t

( ht(Y) - 9t(x)) I llht(Y) - 9t(x)]j .

Finally (iv) follows from Proposition 11.11 applied to the map F: J x P

---+

sn with

F(x , y) = ((y) - x) I II (y) - xjl,

and to the domain X = J x R with boundary components J fi = FjJxQ, has degree deg( fi ) = lk( J, Ki ) · Here is a picture to iilustrate (iv):

x

Qi.

Indeed, D

1 04

11. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS Figure

I

If lk (J, K) :j:. 0 then (ii) and (iii) of Proposition 1 1 . 1 3 imply that J and A- cannot be deformed to manifolds separated by a hyperplane. We shall now specialize to the classical case of knots in R3 where J and K are disjoint oriented submanifolds of R3 diffeomorphic to 81. Let us choose smooth regular parametrizations a:

R - .J,

/3: R

-K

with periods a and b, respectively, corresponding to a single traversing of J and K, respectively, agreeing with the orientation. For p E S2, consider the set I (p)

= { ( ql,q2) E J X J( I q2 - ql = ).p, A > 0}.

Let v(q1) and w(q2 ) denote the positively oriented unit tangent vectors to J and in q1 and q2, respectively.

!(

Theorem 11.14 With the notation above we have: (i) (Gauss)

r t det (a(u) - f](v ) , a'(u1, {3'(v)) 47f Jo lo lla(u) - ,B (-v )ll

lk(J, K) = � (ii)

There exists a dense set of points p

E

du. dv .

S2 such that

det (ql - q2, v(qt), w(q2) ) =1- 0 for (q1 , q2) E I (p).

(iii)

For such points p, lk(J, K) = l:(q�.q2)El(p) J(ql , q2 ), where o(ql, qz ) is the sign of the determinant in (ii).

11.

DEGREE, LINKING NUMBERS AND lfll'l>EX OF VECTOR FIELDS

l05

Proof. We apply formula (2) to the map \II = \ll J,K and the volume form

w

=

vols2 (with integral 41T) to get

lk(J, K) = deg(\ll )

( 12) We write

\ll

=

= 1· o f with

.2_ .J{JxJ( 41T

\ll * (vols2) .

.f : .J x K -+ R3 - {0}; f (qt , qz) = qz - q1 r: R3 - {0} S2; 1· (x) = xfllxll· --+

For

x

E

R3 - {0}, r*(vols2 )x E Alt2 (R3) is given by r-* ( vols2 )x ( v, w) = det (x, v , w)/llxll3

(cf. Example 9.18). The tangent space T(q1,q2) ( J x K) has a basis {v ( ql ) , to(q2 ) } and Therefore ( 1 3)

w* (vols2 )(ql ,q2)(v(ql)w(qz)) = r*(vols2 )q2 -ql ( -'v (ql), w(q2 ))

= llq1 - qzll- 3 det (ql - en, v(qt), w(qz)). The integral of (12) can be calculated by integrating (a x .8) *w*(vols2 ) over the

period rectangle [0, a] x [0, b]. This yields Gauss's integral. For p E S2 , I(p) is exactly the pre-image under w. Thus p is a regular value of w if and only if the determinant in ( 1 3) is non-zero for all (q1 , qz) E I (p), and the sign o (q1, qz) is determined by whether D(91 ,q2) w preserves or reverses orientation. Assertions (ii) and (iii) now follow from Theorems 1 1 .5 and 1 1 .9. 0 Remark 11.15 In Theorem 1 1 . 14.(ii), after a rotation of R3, the regular value p

can be assumed to be the north pole (0, 0, 1). The projections of J and I< on the x1, x2-plane may be drawn indicating over- and undercrossings and orientations, e.g. Figure 2 /

/

/

....... ...... --..--........... . � . ,.

I

J I

'\

'··

'·

..

........ ....

.._ • ____

•

....

....___ .4

..

�-

/ /

/

I 06

11.

DEGREE, LINKlNG NUMBERS AND INDEX OF VECTOR FIELDS I(p) for every place where K crosses over (and not under)

There is one element in

J. The corresponding sign 6 is determined by the orientation of the curves and

of the standard orientation of the plane as shown in the picture Figure 3 ' J.

K

2, lk(J, K) lk(J, K2 )

=

/

K

-L In Fig. l ,

= lk(J, K4 ) = 0 ,

The sum of these linking numbers i s l

'\..

.

/

0:1

In Fig.

"

lk(J, K3)

0.

=1,

lk(J, KI)

= -1.

This is i n accordance with (iv) of Lemma

1 . 13.

We now apply the concept of degree to study singularities of vector fields. Consider a vector field F E C00(U, Rn ) on the open set

U

� Rn , n

us assume that 0 E U is an isolated zero for F. A zero for singularity for the vector field. We can choose a p > 0 with

l l lxll

S p} � U

and such that sn - 1 by

0 is the only

zero for F in

pDn.

Define a smooth map

=

{x E

Fp(x)

=

and Theorem

Definition 11.16

Fp: sn-l

--+

F(px) I I F(px))i

Fp is independent of the choice of p , I 1 .9, degFp E Z is independent of p.

The homotopy class of

1 1 .2

2, and let

F is also called a

Rn

pDn

2:

The degree of Fp is called

and b y Corollary

local index of F at 0, and is denoted

t,(F; 0). Lemma 11.17

Suppose F E C00(Rn, Rn) has the origin as its only zero. Then

induces multiplication by t(F; 0) on Hn- 1(Rn - {0}) � R.

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

107

Let i: sn- l Rn - {0} be the inclusion map and r: Rn- l - {0} --+ sn-l the retraction r(x) = x/llxl l · We have t.,(F; 0) = degFt, where F1 = r o F o i. The lemma follows from the commutative diagram below, where Hn-1 (i) and Hn-1 ( r) are inverse isomorphisms: Proof.

--+

Hn-1 (Rn -

l

{0 } ) Hn- l (F ) Hn-1 (R - {0})

l

Hn-l(r ) H'' - I (i) Hn-1 (1·) 1 Hn- l (sn - 1) H"- (Fl ) Hn- I (sn - 1)

0

Given a diffeomorphism ¢: U V to an open set V � Rn and a vector field on U, we can define the direct image ¢*F E c=(v, Rn) by --+

If F E c=(u, Rn) has 0 as an isolated singularity and ¢: U --+ V is a diffeomorphism to an open set V � Rn with ¢(0) = 0, then

J_,emma 11.18

By shrinking U and V we can restrict ourselves to considering the case where 0 is the only zero for F in U, and where there exists a diffeomorphism 1/J: V Rn. The assertion about ¢ will follow from the corresponding assertions about 1/J and 1/J o ¢, since

Proof.

--+

1/;*(¢* F)

= (1/; o ¢) *F.

Thus it suffices to treat the case where ¢: U Rn is a diffeomorphism and where Y = ¢*F E c=(Rn, Rn) has the origin as its only singularity. Let U0 � U be open and star-shaped around 0. We define a homotopy --+

:

Uo X [0, 1]

--+

Rn ;

t(X)

{

(Do¢)x = (x, t) = c/J(t x)/t

For x E Uo,

where cPi

E

c=(Uo, Rn) is given by cPi (x) =

1 1 8¢ (tx) dt. �

0 UXi

if t = 0 if t # 0.

108

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

It follows that

<J? (x, t) =

n

L XicPi(tx ) i= l

,

and in particular that has a smooth extension to an open set W with U0 x [0, 1]

W � Uo

x

R.

For each t E [0, 1]. ;- 1 of Y restricted to t(U0):

The function Xt(x) is smooth on W. Now X1 = where A = Do.

Choose p

�

FI Uo

and Xo

=

Rn.

(A-1)*Y.

> 0 such that pDn � Uo. The homotopy sn- 1 X [0, 1] - sn-1 given by Xt(Px ) / IIXt(px)i! ,

and Corollary

1 1 .2

0 :::; t � 1,

shows that

Since A: Rn - Rn is linear we have (A- 1 ) * Y = yields the commutative diagram

A-1 o Y o A: Rn - Rn.

This

Rn - { 0 } (A-t).Y Rn - {0}

lA

Rn - { 0 }

--L

lA

Rn - { 0}

Now use the functor Hn- l and apply Lemma 1 1 .17 to both Y and get 1-((A-1) * Y;O) = 1-(Y;O). Hence 1-(F;O) = 1-(Y;O).

(A- 1 )* Y

to

0

Definition 11.19 Let X be a smooth tangent vector field on the manifold Mn, n 2: 2 � it p0 E M as an isolated zero. The local index 1-(X; Po ) E Z of X is defined by

where (U, h) is an arbitrary smooth chart around Po with h(po) =

0.

We note that Lemma 1 1 . 1 8 shows that the local index does not depend on the choice of (U, h). One can picture vector fields in the plane by drawing their integral curves, e.g.

JJ.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

'=

Let X be a smooth

-1

Figure 4

'=

tangent vector field on Mn

109

+2

and let Po

E

Mn be a zero. Let

for a chart (U, h) with h(po) = 0. If DoF: Rn ---+ Rn is an isomorphism, then Po is said to be a non-degenerate singularity or zero. Note that by the inverse function theorem F is a local diffeomorphism around 0, such that 0 is an isolated zero for F'. Hence p0 E Mn is also an isolated zero for X . Lemma 11.20

If Po is a non-degenerate singularity, then t.(X,po) = sign(det DoF) E {±1}.

Proof.

B y shrinking U we may assume that h maps U diffeomorphically onto an open set U0 � R'\ which is star-shaped around 0, and that F is a diffeomorphism from Uo to an open set. As in the proof of Lemma 1 1. 18 we can define a homotopy

G : Uo

x

[0, 1]

---+

Rn ;

G(x, t ) =

{

DoF F(tx )j t

if t = 0 if t # 0,

where G can be extended smoothly to an open set W in Uo x R that contains Uo X [0, 1]. Choose p > 0 so that pDn � Uo. We get a homotopy G: sn- l X [0, 1] -

sn- 1

'

G(x, t) = G(px, t) / l!G(px, t)IJ,

between the map Fp in Defi nition 1 1 . 1 6 and the analogous map Ap with A = DoF. It follows from Corollary 1 1 .2 that t.(X ; Po ) = [,(F; 0) = deg(Fp) = degAp = t(A ; 0). {0} ---+ Rn - { 0} induced by A operates on Hn-l (Rn - {0}) by multiplication by t(X;po); cf. Lemma 1 1 . 17. The result now follows from Lemma 6.14. 0

The map fA: Rn

-

110

11.

DEGREE, LJNKING NUMBERS AND INDEX OF VECTOR FIELDS

Definition 11.21 Let X be a smooth vector field on Mn , with only isolated singularities. For a compact set R to be

R £; M

we define the total index of X over

Index (X ; R ) = I >(X;p),

where the summation runs over the finite number of zeros p E compact we write Index (X) instead of Index (X; M).

R for X . If M

is

Theorem 11.22 Let F E C00(U, Rn) be a vector field on an open set U £; Rn, with only isolated zeros. Let R £; U be a compact domain with smooth boundary oR, and assume that F(p) =/= 0 for p E aR. Then Index(F; R)

where f: aR

-t

= F(x)

sn-I is the map j(x )

Proof. Let p1 , . . . , Pk be the zeros in

D1 £; R - aR,

= deg j,

R for F,

with centers Pi · Define

fj : aDj

-t

I II F(x )ll. and choose disjoint closed balls

fj (X) = F(x) I IIF(x) l l ·

sn-I ;

We apply Proposition 1 1 . 1 1 with X = R- uj Dj. The boundary ax is the disjoint union of aR and the (n - !)-spheres aD1 , . . . , aDk· Here aDj, considered as boundary component of X, has the opposite orientation to the one induced from Dj. Thus

k

deg(f) +

L

-deg(fi)

i=l FinaJly deg(Ji)

= 0.

= t( F; Pi) by the definition of local index and Corollary 1 J .3.

0

Corollary 1 1.23 In the situation of Theorem 1 1.22, Index(F; R) depends only on the restriction of F to aR. 0 Corollary 1 1.24 In the situation of Theorem 1 1.22, suppose for every p E aR that the vector F(p) points outward. Let g: aR -t sn-I be the Gauss map which to p E oR associates the outward pointing unit normal vector to aR. Then Index(F; R)

= deg g.

Proof. By Corollary 1 1 .2 it sufficies to show that f and g are homotopic. Since j(p) and g(p) belong to the same open half-space of Rn, the desired homotopy can be defined by

(1 - t)j(p) + tg(p) 11( 1 - t )f(p) + tg (p) [l

(0 � t � 1).

0

11. DEGREE, LINKING NUMBERS AND INDEX OF VECI'OR FIELDS

III

Lemma 11.25 Suppose F E C00(Rn, Rn) has the origin as its only zero. Then there exists an F E C00(Rn, Rn), with only non-degenerate zeros, that coincides

with F outside a compact set.

Proof.

We choose a function ¢ E C00(Rn, [0, 1]) with

{

1 if llxll � 1 ¢(x) - 0 if llxll � 2. _

We want to define F(x) = F(x) - ¢(x)w for a suitable w E we have F(x) F(x). Set

IRn.

For llxll > 2

=

c=

inf IIF(x)fl > 0 l �llxll�2

and choose w with llwll < c. For 1 � llxll � 2, IIF(x)ll � c - llwll > 0. Thus all zeros ofF belong to the open unit ball bn. Since F coincides with F -w on bn, We can pick w as a regular value of F with llwll < c by Sard's theorem. Then = DpF will be invertible for all p E .F- 1 (0) , and F has the desired 0 properties. DpF

Note, by Corollary 1 1.23, that (14)

t(F;O) =

L t(.F,p)

peF-1(p)

Here is a picture of F and F in a simple case: Figure 5

The zero for F of index -2 has been replaced by two non-degenerate zeros for F, both of index -1.

112

11.

DEGREE, LINKING NU�mERS AND INDEX OF VECTOR FIELDS

Coronary 1 1.26 Let X be a smooth vector field on the compact manifold Mn with isolated singularities. Then there exists a smooth vectorfield X on M having only non-degenerate zeros and with Index(X) = Index(X).

Proof. We choose disjoint coordinate patches which are diffeomorphic to Rn around the finitely many zeros of X, and apply Lemma 1 1 .25 on the interior of each of them to obtain X. The formula then follows from ( 1 4).

Let Mn � Rn+k be a compact smooth submanifold and let N€ be a tubular neighborhood of radius E > 0 around M. Denote by g: 8Ne. -t sn+k- l the outward pointing Gauss map. If X is an arbitrary smooth vector field on Mn with isolated singularities, then Theorem 1 1.27

Index(X)

=

deg g.

Proof. By Corollary 1 1 .26 one may assume that X only has non-degenerate zeros. From the construction of the tubular neighborhood we have a smooth projection 1r: N -t M from an open tubular neighborhood N with NE � N � Rn+k, and can define a smooth vector field F on N by F( q) = X(1r (q) ) + ( q - 1r( q)) .

(1 5)

Since the two summands are orthogonal, F(q) = 0 if and only if q E M and X(q) = 0. For q E oN, , q - 1r(q) is a vector normal to TqfJN< pointing outwards. Hence X(1r(q) ) E Tq8Nf., and F(q) points outwards. By Corollary 1 1 .24

lndex(F; N€) = deg g.

and it suffices to show that �(X;p) :::::: �(F;p) for an arbitrary zero of X. In local coordinates around p in M, with p corresponding to 0 E R'l, X can be written in the form ( 1 6) where fi ( 0)

(17)

By

X=

=

a 2:: ]i(x)�, x�

n

i::: l 0, and by Lemma 1 1 .20 t( X; p) is the sign of det

( ![; (0)) .

differentiating (16) and substituting 0 one gets

a x (o) = x aj

t

8 /i (o)�. 8xj axi i=l lt follows from (15) that DpF: Rn+k -t Rn+k is the identity on TpMJ., and by ( J 8) DpF maps TT>M into itself by the linear map with matrix (8f:,/OXj (0)) (with respect to the basis (8j ax i)0). It follows that p is a non-degenerate zero for F and that detDpF has the same sign as the Jacobian in (17). 0

( 1 8)

113

THE POINCAR.t-HOPF THEOREM

12.

..... In the following, Mn � Rn+k will denote a fixed_pn60th submanifold. If the cohomology of Mn is finite-dimensional (e.g. -when Mn is compact), the i-th Betti number is given by ./

( 1)

The

Euler characteristic

of Mn is defined to be x(Mn) =

(2)

n

L ( -l) i bi( M). i=O

This chapter's main result is: Theorem 12.1 (Poincare-Hopf) Let X be a smooth vector field on a compact manifold M. If X has only isolated zeros then

lndex(X) = x(M) . By the final result of Chapter 1 1 it is sufficient to show the formula for just one such vector field X on M. We shall do so by making use of a Morse function on Mn. Given f E C00(M, R), a point p E M is a critical point for f if dpf = 0. Proposition 12.2 Suppose that p E M is a critical point for f E C00(M, R).

(i)

(ii)

There exists a quadratic form d�f on TpM characterized by the equation

where a: ( -15, 15) - M is any smooth curve with a(O) = p. Let h: U - Rn be a chart around p and let q == h(p). Then the composition

is the quadratic form associated to the symmetric matrix

I 14

Proof. Set h

12.

o a(t) =

THE POINCARi-HOPF THEOREM

'Y(t)

calculation yields

(! o a)'(t)

=

- ('Y1(t), . . , 'Yn (t)) .

(¢> o 'Y) ' (t) =

Since p is critical, (8¢>/oxi)('Y(O)) = substituting t = 0, we get

(! o a)" (O) This is the value at 'Y'(O) Both (i) and (ii) follow.

n

=

n

0.

and ¢> = f

t :: .

i=l

o

h-1. A direct

('Y(t))'Y�(t).

1

By differentiating once again and

82 ¢>

L L axl·ax . (q)'Y�(o)'Yj(o). J

. 1 ]= . 1

t=

= Dph(a'(O))

E

Rn

of the quadratic form from (ii).

0

Consider another chart h: [J --... Rn around p with q = h(p) and let F = h o h - l defined in a neighborhood of q. The last formula i n the proof above can be compared with

where ¢ = f o Tt-1 and i = h o o.. Let J denote the Jacobi matrix associated with F in q. Then i'(O) = J'Y'(O) for the column vectors i'(O) and 'Y'(O). By substituting this and comparing, one obtains the matrix identity

(3)

Definition 12.3 A critical point p

E M

of f E C00(M, R) is said to be non­ degenerate, if the m�trix in Proposition 12.2.(ii) is invertible. We call f a Morse function, if all critical points of f are non-degenerate. The index of a non­ degenerate critical point p is the maximal dimension of a subspace V s;:; TpM for which the restriction of d�f to V is negative definite. For smooth submanifolds Mn �

Rn+k

Theorem 12.4 For almost all Po

E Rn+k the function f: M

one can get Morse functions by:

1 f (p) = 2 I IP - Poll 2 is a Morse function.

--...

R defined by

12.

THE POINCAR.E-HOPF THEOREM

115

Proof. Let g: Rn --+ M be a local parametrization and Yj: Rn --+ nn+k, j = 1 , . . . , k smooth maps, such that Yi (x), . . . , Yk (x) is a basis of Tg(x) M.1 for all x E Rn. By Lemma 9.21 we know that M can be covered by at most countably many coordinate patches g(U) of this type. Therefore it suffices to prove the assertion for g( U) instead of M.

We define : Rn+k

(4)

-t

Rn+k by

k (x, t) = g(x) + I: tj Yj(x)

j= I

(x E Rn , t E Rk ) .

By Sard's theorem it suffices to prove that if Po is a regular value of , then f becomes a Morse function on g(U). Set k = f o g: Rn --+ R; we can show instead that k becomes a Morse function on Rn. We have

k (x) =

(5)

1 2

(g (x) - Po , g(x) - Po),

where ( ' ) denotes the usual inner product on nn+k . Since (agjaxi (x), Yv(x))

= 0, it follows by differentiation with respect to Xj that

(6) From (5) we have

(7) and therefore (8) Furthermore, by (4),

(9)

k a ag I: aYv = + tv ax·J ax·J v=l axJ· )

-

-

Assume that p0 is a regular value of and let x be a critical point of k. It follows .l from (7) that g(x) - Po E T_g(x)M . Hence there exists a unique t E Rk with

( 1 0)

g(x) - PO =

k

- I: tv Yv(x), v=l

1 16

12.

THE POINCARE-HOPF THEOREM

and (x, t) E -1(po). The n + k vectors in (9) are linearly independent at the point (x,t). At this point, the equations (8), ( 1 0), (6) and (9) give

Let A denote the invertible (n + k) x (n + k) matrix with the vectors from (9) as rows. Then ADxg takes the following form: o2k

&2k OXjbX I

�

o2k bx,.8x1 0

82k � 0

0

0

Since Dxg has rank n, so does ADxg. Hence the n x n matrix

is invertible. This shows that x is a non-degenerate critical point. Example 12.5

Let

f: Rn - R be

0

the function

2 2 + . . . Xn, f(X) = C - x12 - x22 - . . . - X,\2 + X>.+ l

where c E

R, A E 7L

and 0

:::; A :::; n.

Since

gradx(f) = 2( -Xt, . . . , -X>., X>.+1 , . . . , Xn), 0

is the only critical point of f. We find that

( [J2j ) oxilxj

(0)

. -2, . . . , -2, 2, . . . , 2) = dtag(

with exactly A diagonal entries equal to -2. Thus the origin is non-degenerate of index A. We note that the vector field grad(!) has the origin as its only zero and that it is non-degenerate of index ( -1) >. .

12.

THE POINCAR.t-HOPF THEOREM

1 17

Theorem 12.6 Let p E Mn be a non-degenerate critical pointfor f E C00(M, R). There exists a C00-chart h: U - h(U) � Rn with p E U and h(p) = 0 such that

n

f

o h - 1 (x) = f(p) + L OiXl , x E h(U), i=l

where Oi = ± 1 (1 :::; i ::; n) (By an additional pennutation of coordinates we can put f into the standard jonn given in Example 12.5.)

Proof. After replacing f with f - f(p) we may assume that f(p)

= 0.

Since the problem is local and diffeomorphism invariant, we may also assume that f E C00(W, R), where W is an open convex neighborhood of 0 in Rn and that 0 is the considered non-degenerate critical point with f(O) = 0.

We write f in the form

n

J(x ) = L Xi 9i (x) ; i:=l

Since 9i(O)

=

-3f(O)

=

0,

we may repeat to get

n

( ) 9i (X) = L Xj 9ij X ; 9tJ0 · (X) j=l

11 agi(sx) !:1 o

ux1·

d

8.

On W we now have that n

n

J(x) = L L Xi Xj 9ij(x), i:::l j=l

where 9ij E C00(W, IR). If we introduce hij = !(9ii + 9ii) then symmetric n x n matrix of smooth functions on W, and n

(11)

f(x)

(hij)

becomes a

n

= L L Xi Xj hij(x). i=l j=l

By differentiating (I I ) twice and substituting 0, we get

a2f 0Xi 0Xj

( 0)

= 2hij(O).

(hij(O)) is invertible. Let us return to the original f E C00(M, R). By induction on k, we attempt to show that the C00-chart h from the theorem can be choosen such that f 0 h- 1 In particular the matrix

is given by ( 1 1 ) with

1 18

12.

THE POJNCARt-IIOPF THEOREM

with D a (k - 1) x (k - 1) matrix of the form diag(±1, . . . , ±1), and E some symmetric (n - k + 1 ) x (n - k + 1 ) matrix of smooth functions. So suppose inductively that k- 1 (12) f(x) = L t5ixf + L L Xi Xj hij(x), n

i=l

n

i=k j=.:k

for x in a neighborhood W of the origin. We know that the minor E is invertible at 0. To start off we can perform a Jinear change of variables in Xk , . . . , Xn , so that our new variables satisfy (12) with hkk(O) # 0. By continuity we may assume that hkk (x) has constant sign 6k = ±1 on the entire W. Set q=

and introduce new variables:

� E C00 (W, R),

(

t ��)

Yk

= q(x)

Yj

= Xj for j # k, 1

xk

+

i=k+l

ik Xi z

::;

j

kk

::;

n.

The Jacobi determinant for y as function of x is easily seen to be 8yk foxk(O) q( O) # 0. The change of variables thus defines a local diffeomorphism w around 0. In a neighborhood around 0 we have for y = 'll(x): =

f

0

w-1(y) = f(x) n n k-1 n = OiXJ + x�hkk(x) + 2xk XiXjhij(x) Xj hjk (x) + i=l i=k+l j=k+l j=k+ l n k-l h. ( ) 2 OiXJ + hkk(x) Xk + = Xj 1 i=l j=k+l hkk ( ) n n n hj - hkk (x) Xj X;Xjhij(x) + kk (x) j=k+ i=k+l j=k+l n n k = + iY xixj hij(x) s l i=l i=k+l j=k+l n n k = S;y} + YiYj hij o w-1(y), i;:k+l j=k+J i=l

L

L

(L

(

:L

:L :L

:L

:L :L

L

L

))2

) �:

L L

L L

where hij E C00(W, R). This completes the induction step.

0

We point out that with the assumptions of Theorem 12.6 p is the only critical point in U. If M is compact and f E C00(M,R) is a Morse function then f has only finitely many critical points. Among them there will always be at least one local minimum (index ). = 0) and at least one local maximum (index ). = n).

·

12.

THE POINCARt-HOPF THEOREM

119

Definition 12.7 Let f E

C00(M, R) be a Morse function. A smooth tangent vector field X on M is said to be gradient-like for f, if the following conditions are satisfied: (i) For every non-critical point p E M, dpf (X(p)) > 0. (ii) If p E Mn is a critical point of f then there exists a C00-chart h : U --+ h(U) � Rn with p E U and h(p) = 0 such that f o h-1(x) = f(p) - xi - · · · - xl + x� + l + and h.Xtu = grad ( ! o h-1).

A

smooth parametrized curve a: I � M is an a'(t)

· · ·

+

x;,

integral curve

x E h(U),

for X, if

= X(a(t)) for t E I.

one gets (f o C¥)1(t) = da(t) f(X(a(t))). If a(I) does not contain any critical points, then f o C¥: I - R is a monotone increasing function by condition (i). Hence

Lemma 12.8 Every Morse function on

M admits a gradient-like vector field.

Proof. We can find a C00-atlas (Ua , ha)aEA for M which satisfies the following two conditions: (i) Every critical point of f belongs to just one of the coordinate patches Ua . (ii) For any a E A either f has no critical point in Ua or f has precisely one critical point p in Ua, ha(P) = 0, and f o h;;1 has the form listed in Example 12.5.

Let Xa be a tangent vector field on Ucr determined by Xcr = (h;; 1 ) * (grad (f o h;;1 ) ) . Choose a smooth partition of unity (Pcr)O<EA subordinate to (Ucr) QEA • and

define a smooth tangent vector field on M by X=

L Pcr Xcr,

0<E A

where PaXa is taken to be 0 outside Ua · If p E M is not a critical point for f then, for every a E A with p E Ucr and q = ha(P), we have Indeed, there is at least one C¥ with Pa(P) > 0 and

We see that X satisfies condition (i) in Definition 12.7.

120

12.

THE POINCARt-HOPF THEOREM

If p is a critical point of f then there exists a unique a E A with p E Ua. It follows from (a) that X coincides with Xa on a neighborhood of p, and condition (b) above shows that assertion (ii) in Definition 12.7 is satisfied. D The next lemma relates the index of a Morse function to the local index of vector fields as defined in Chapter 1 1 .

Lemma 12.9 Let f be a Morse function on M and X a smooth tangent vector

field such that dpf(X(p)) > O for every p E M that is not a critical point for f. Let Po E M be a critical point for f of index A. If X (po) = 0, then t-(X ; Po) = ( -1) A .

Proof. We choose a gradient-like vector field X. By Definition 12.7.(ii) and Example 12.5,

Let U be an open neighborhood of Po that is diffeomorphic to Rn and chosen so small that Po is the only critical point in U. The inequalities

dpj(X(p)) > 0, valid for p E U - {Po}, show that X (p) and half-space in TpM. Thus

X (p)

( 1 - t)X(p) + tX (p) (0 � t defines a homotopy between JC and X considered Rn - {0}, and t(X; Po) = t-(X; Po).

belong to the same open

1) maps from U

- {Po}

to

D

Remark 12.10 Given a Riemannian metric on M and f E C00(M, R), one can

define the gradient vector field gradf by the equation

(gradp(f) , Xp) = dpf(Xp) for all Xp E TpM. Then Lemma 12.9 holds for grad(!).

Theorem 12.11 Let Mn be a compact differentiable manifold and X a smooth

tangent vector field on Mn with isolated singularities. Let f E C00(M, R) be a Morse function and cA the number of critical points of index A for f. Then we have that Index(X) =

n

L

A=O

(-l)A cA .

!

·J ·i

t(X;po) = (-1) A .

!

12. THE POINCARE-HOPF THEOREM

121

Proof. It is a consequence of Theorem 1 1 .27 that any two tangent vector fields with isolated singularities have the same index. Thus we may assume that X is gradient-like for f. The zeros for X are exactly the critical points of f, and the

0

claimed formula follows from Lemma 12.9.

It is a consequence of the above theorem that the sum ( 1 3)

is independent of the choice of Morse function J E C00(M, R). Given Theorem 1 2 . 1 1 , the Poincar6-Hopf theorem is the statement that the sum ( 1 3) is equal to the Euler characteristic; cf. (2). We will give a proof of this based on the two lemmas below, whose proofs in turn involve methods from dynamical systems and ordinary differential equations, and will be postponed to Appendix C. Let us fix a compact manifold Mn and a Morse function f on M. For a E R we set M (a) = {p E M I f(p)

<

a}.

Recall that a number a E R is a critical value if f- 1 (a) contains at least one critical point. Lemma 12.12 If there are no critical values in the interval [a1 , a2]. then M(a1 ) and M(a2) are diffeomorphic. Lemma 12.13 Suppose that a is a critical value and that Pl, . . . , Pr are the critical points in f- 1 (a). Let Pi have index Ai. There exists an E > 0, and disjoint open neighbourhoods Ui of Pi. such that

(i) PI> . . . , Pr are the only critical points in f-1 ([a - E, a + E]). (ii) Ui is diffeomorphic to an open contractible subset of Rn. (iii) ui n M(a - €) is diffeomorphic to s).·- 1 X v;, where v; is an open contractible subset ofRn->.;+1 (in particular Ui nM(a - E) = 0 if Ai = 0). (iv) M ( a + E) is diffeomorphic to U1 u . . . u Ur u M(a - E). 0

Proposition 12.14 In the situation of Lemma 12.13 suppose that M( a - E) has finite-dimensional cohomology. Then the same will be true for M (a + E). and r

x( M( a + t:)) = x(M (a - t: )) + L ( -1)>.' . i=l

12. THE POINCARE-HOPF THEOREM

122

Proof. For U = U1 U . . .

u

Ur,

Lemma 12.1 3.(ii) and Corollary 6.10 imply that

HP(U) c:=

{

�f p # 0

0

W

If p = 0.

This gives x(U) = r. Condition (iii) of Lemma 12.13 shows that Ui n M(a is homotopy equivalent to S..\•- 1 , and Example 9.29 gives x ( Ui n M(a -

t) ) =

)

€

1 + ( - 1) ..\' -1.

Since U n M(a - €) is a disjoint union of the sets finite-dimensional de Rham cohomology, and

Ui

n

M(a

- €) ,

r

r

it has a

r

,\ x(U n M(a - t)) = 2:::: ( 1 + (-1) ' - 1) = r - L ( - 1 ),\' = x(U) - L (- 1 ),\'. i=1 i= 1 i= 1 The claimed formula now follows from Lemma 12.1 3.(iv) and the lemma below, 0 applied to U and V = M(a - t) . Lemma 12.15 Let U and V be open subsets of a smooth manifold. If U, V and UnV have.finite dimensional de Rham cohomology, x(U

the same is truefor Uu V, and

u V) = x(U) + x(V) - x(U n V).

Proof. We use the long exact Mayer-Vietoris sequence .

.

.

� HP- 1 (U n v)

�

HP(U u V)

� HP(U) G1 HP(V)

�

HP(U n V)

�

.

.

.

First we conclude that dim HP(U u V) < oo. Second, the altematirlg sum of the cf. Exercise dimensions of the vector spaces in an exact sequence is equal to 4.4. 0

zef; /_ -

Theorem 12.16 If f is a Morse function on the compact manif old Mn, then x(Mn)

n

= L ( - 1) ,\ C

..\=0

,\ ,

where c..\ denotes the number of critical points for f of index >.. < ak be the critical values. Choose real numbers bo < a1, bj E (aj, ai+d for 1 :S j :S k - 1 and bk > ak· Lemma 12.12 shows that the dimensions of Hd( M(bj)) are independent of the choice of bj from the relevant interval. If M(bj-1) has finite-dimensional de Rham cohomology, the same will be true for M(bj) according to Proposition 12.14, and Proof. Let

(14)

a1

< a2 < . . . <

ak_1

x(M(bj ) ) - x(M(bj-1 ) ) =

I: (-1)..\(p)

pE/- 1 (ai)

12.

THE

POJNCARE-HOPF THEOREM

123

Here the sum runs over the critical points p E j-1 (aj). and .A(p) denotes the index of p. We can start from M(bo) = 0. An induction argument shows that dim Hd(M(bj )) < oo for all j and d. The sum of the fonnulas of (14) for 1 ::; j ::; k gives

x(M)

= x(M(bk)) = 2::::: ( - 1 ) -A(p) p

where p runs over the critical points.

0

The Poincare-Hopf theorem 12. I follows by combining Theorems 1 2. 1 1 and 12.16.

Corollary 12.17

If A1n

is

compact and of odd dimension n then x(Mn )

= 0.

Proof. Let f be a Morse function on M. Then -f is also a Morse function, and - f has the same critical points as f. If a critical point p has index .A with respect to f, then p has index n - .A with respect to -f. Theorem 12.16 applied to both f and -f gives

x(M)

n

n

-A=O

.A=O

= 2::::: ( -1) -A c.A = 2::::: ( - 1 t-.A c.A·

The two sums differ by the factor (-l )n, and the assertion follows.

0

Example 12.18 (Gauss-Bonnet in R3). We consider a compact regular surface S � R3, oriented by means of the Gauss map N: S --+ S2 . The Gauss curvature of S at the point p is K(p) = det (dpN) ; Sard's theorem implies that we can find a pair of antipodal points in 52 that are both regular values of N. After a suitable rotation of the entire situation we can assume that P± = (0, 0, ± 1 ) are regular values of N.

Let f E C00(S, R) be the projection on the third coordinate axis of R3 . The critical points p E S of f are exactly the points for which Tp S is parallel with the x1 , x2-plane, i.e. N(p) = P± · At such a point p, the differential of the Gauss map dpN is an isomorphism. Hence K(p) :j:. 0. A neighborhood of p in S can be parametrized by (u, v, f(u, v)), and in these local coordinates the Gauss curvature has the following expression: K=

(!+ (��) (��) ) 2

2

+

-l

( �� 82!

det

8u8v

124

12.

THE POlNCARt-HOPF THEOREM

(see e.g. [do Carmo] page 1 63). Since K(p) =I= 0, the determinant in the expression does not vanish at p, so p is a non-degenerate critical point for f. If K (p) > 0 the determinant is positive and p has index 0 or 2. If K(p) < 0 the determinant is negative and p has index 1 . We apply Theorem 12.16 to get X ( S) # { p E s I N (p) = P±' J((p) > 0 } - # { p E s I N (p) = P±' [( (p) < 0} . ===

Since P+ is a regular value for N, we have by Theorem 1 1 .9

1

# { p E N- 1 (P+ ) I K(p) > 0} - # { p E N (p+ ) I K(p) and analogously with P- instead of P+· It follows that

deg(N)

=

-

x(S)

(15)

=

<

0}

2 deg(N).

The map Alt.2 (dNp): Alt2 (Trv(p)S2)

= Alt2 (TpS)

is multiplication by det(dpN) = K(p) , so N*(vols2)

{ Kvols = {s2 N*(vois2)

Js

J

Combined with (15) this yields the

=

(degN )

� =

Alt2 (TpS)

K(p)vols. Hence

{ vol82

Js2

=

4?TdegN.

Gauss-Bonnet fonnula

� is Kvols = x(S).

( 1 6)

2

Consider the torus T in R3. The height function f: T Ill is a Morse function with the four indicated critical points. The Gauss curvature ofT is

Example 12.19

�

p

s

positive at p and s, but negative at q and r (cf. Example 12.18). Hence f is a Morse function on T. The index at p, q, r and s is 0, 1, 1 and 2, respectively. Theorem 12.16 gives x(T) = 0. Since we know that dim H0(T) = dim H2 (T) = 1 , we can calculate dim H1 (T) 2. =

12.

THE POINCARE-HOPF THEOREM

125

RPn) Real functions on RPn are equivalent f: sn - R, f(x) = f( -x) for all x E sn. Let us try n f(x) L: aixt i=O for X = (xo, Xt> . . . ' Xn ) E sn � Rn+ l , and real numbers ai. The differential

Example 12.20

(Morse function on i.e. to even functions

=

of f at

x

is given by

n dxf(vo, . . . , vn) = 2 L aiXiVi, i=O where v = (vo, VI, . . . , vn ) E TxSn so that n L XiVi = 0. i=O

x is a critical point for f if and only if the vectors x and (a0xo, a1 x1, . . . , anxn ) are linearly independent. If the coefficients ai are distinct, this occurs precisely for x = ±ei = (0, . . , ± 1, . . , 0), and f has ex­ actly 2n + 2 critical points. The induced smooth map j: RPn - R then has n + 1 critical points [ei]· In a neighborhood of ± eo E sn we have the charts h with Thus

.

.

h±1(u1, . . . , u,) = (±VI - 2:::1 u; , and in a neighborhood of 0

).

n

UJ. . . . , U

E Rn,

The matrix of the second-order partial derivatives for f o matrix

h;±1 (at 0) is the diagonal

diag(2(al - ao), 2(a2 - ao), . . . , 2(an - ao)). Hence ±eo are non-degenerate critical points for f; the index for each is equal to the number of indices i with 1 :::; i :::; n and � < ao. An analogous result holds for the other critical points ±ej. For simplicity, suppose that ao < a1 < a2 < . . . < an . Then the two critical points ±ej for f: sn R have index j. The induced function /: RPn - R is a Morse function with critical points [ej] of index j. We apply Theorem 12.16 to

j.

-

Since

cA

=

1

if n i s even if n is odd. This agrees with Example 9.3 1.

for 0 :::; >. :::; n, we get

127 13.

POINCARE DUALITY

Given a compact oriented smooth manifold is the statement that

Mn of dimension n, Poincare duality

(1) where Hn-p(M)* denotes the dual vector space of linear forms on Hn-P (M). The proof we give below is based upon induction over an open cover of M. Thus we need a generalization of ( I ) to oriented manifolds that are not necessarily compact. The general statement we will prove is that

(2) where the subscript

c

refers to de Rham cohomology with compact support.

For a smooth manifold M we let 0.�(M) be the subcomplex of the de Rham complex that in degree p consists of the vector space 0.� ( M) of p-forms with compact support. The cohomology groups of (0.�(M), d) are denoted H�(M), i.e.

Ker(d: 0.�(M) -+ n�+ 1 (M)) Im(d: n�- 1 (M) -+ n�(M)) · Note that when M is compact 0.�(M) = 0.*(M), so that H;(M) HP (M ) c

=

this case.

=

H*(M)

in

The vector spaces H�(M) are not in general a (contravariant) functor on the category of all smooth maps. However, if c.p: M -+ N is proper, i.e. if c.p-l ( K) is compact whenever K is, then the induced form c.p* (w) will have compact support when w has. Indeed

and c.p* becomes a chain map from 0.�(N) to 0.�(M). Hence by Lemma 4.3 there is an induced map

Hf (c.p): Hg(N) -+ Hf(M) and Hg(-) becomes a contravariant functor on the category of smooth manifolds and smooth proper maps. A diffeomorphism is proper, so Hf(M) � Hf(N) when M and N are diffeomorphic.

13.

l28

POINCARE DUALITY

Remarks 13.1

(i) The vector space H2(M) consists of the locally constant functions f: M - R with compact support. Such an f must be identically zero on every non-compact connected component of M. In particular H2(M) = 0 for non-compact connected M. In contrast H0(M) R for such a man­ ifold. (ii) If Mn is connected, oriented and n-dimensionaJ, then we have the iso­ morphism from Theorem 10.1 3, =

f

JM

: H�(M) � R

whereas by (i) and (2), Hn(M) Lemma 13.2

Hq(Rn) c

=

=

0 if M is non-compact.

{R

if q = n 0 otherwise.

The above remarks give the result for q = 0 and q = n, so we may assume that 0 < q < n. We identify Rn with sn - {Po}, e.g. by stereographic projection, and can thus instead prove that Proof.

Now the chain complex rl� ( sn - {Po}) is the subcomplex of rl* ( sn) consisting of differential forms which vanish in a neighborhood of p0. Let w E ng(sn - {Po}) be a closed form. Since Hq(Sn) 0, by Example 9.29, w is exact in rl*(Sn), so there is a T E nq-l(Sn) with &r w. We must show that T can be chosen to van.ish in a neighborhood of PO· Suppose W is an open neighborhood of {p0}, diffeomorphic to Rn, where w l w = 0. If q == 1, then T is a function on sn that is constant on W, say T l w = a. But then K = T - a E S1� (Sn - {Po}) and dK = W. If 2 :::; q < n then we use that Hq- l ( W) � Hq- 1(Rn) = 0, and that T l w is a closed form, to find a CJ E nq- 2 ( W) with dCJ = T l w. Now choose a smooth function
=

Let V c U be open subsets of a smooth manifold M, and let i: V - U be the inclusion. There is an induced chain map

13.

POINCARE DUALITY

129

defined by setting i* (w) l v = w, for w

i*(w)lu-supp(w} = 0.

E n� (V). We get a linear map

(3) which is called the direct image homomorphism. Given a second inclusion j: W -+ V, (i o j)*(w) i* o j*(w), so that (Hr(-),i*) becomes a covariant functor on the category of open subsets and inclusions of a given manifold. There is also a Mayer-Vietoris theorem for this functor. Indeed, if U1 and U2 are open subsets of M with union U, and iv: Uv -+ U, jv: U1 n U2 -+ Uv are the inclusions, then the sequence =

is exact, where

We leave the verification of the exactness to the reader, with the remark that surjectivity of Iq uses a smooth partition of unity on U subordinate to the covering {Uv } ; cf. Theorem 5 . 1 . Theorem 13.3 (Mayer-Vietoris) With the above notation there is an exact se­ quence

Proof. This follows from Theorem 4.9 applied to (4).

0

In comparing Theorem 13.3 with Theorem 5.2 the reader will notice that the directions of all arrows have been reversed. 1 For later use, let us explicate Definition 4.5. o*[w) E Hg+ (U1 n U2) is defined as follows: write w = WI + w2 with Wv E n�(U) and SUPPu(wv) c Uv. Then dwl and - dw2 agree on U1 n U2 and the common value T = dwdu1nU2 = -dw2l u1nU2 is a closed form in n�+l(U1 n U2) that represents o*[w). Proposition 13.4 Suppose {Ucr I a E A} is a family of pairwise disjoint open subsets of the smooth manifold M with union U. Then there are isomorphisms

(i) Hq(U) -+ llcreA Hq (Ua); (ii) EBaeA Hg (Ua ) -+ Hl (U) ;

[w] f--t [i�(w)] ([wa]) aEA f--t LaEA ia* ([waJ)

130

13.

POINCARE DUALITY

where ia: Ua - U denotes the inclusion. Proof. There are isomorphisms

9 : nq(U) -

II nq(Ua);

etEA

q (w) = (i� (w) )aEA

aEA which define isomorphisms of chain complexes when we give differential

fi D.* (Ua)

the

d((-ra )aEA) = ( d-ra)aEA and view EB n� (Ua) c n n�(Ua) c n D.* (Ua ) as a subcomplex.

0

The contravariant functor that sends a vector space A to its dual vector space A* = HomR(A, R) is exact, i.e. if A �

B!C

is an exact sequence of vector

spaces, then C* � B * � A* is exact. It is clear that
· · · -t

with

Hg+1(Ul n U2)* .! HZ(U)* L Hg(UI)* El1 H2(U2)'" � HJ(UI n U2)

(

)

-t · · ·

I'(a) = ii(a),i�(a) and i(cr1,cr2) = ji(a1) - j�(a2) where we have written ii: Hg (U)* - H% (U1 )* for the vector space dual of ir., etc.

The dual of a direct sum is a direct product, so Proposition 1 3.4.(ii) implies the isomorphism

(6)

etEA

For an oriented n-dimensional manifold the exterior product defines a bilinear map

flP(M) x n�-P(M) - n�(M)

since supp (w1 /\ w2) � supp(wl) n supp(w2). It induces a bilinear map

HP(M ) X H�-P(M) - H�(M)

and we may compose with integration (cf. Remark 13.1 .(ii)) to obtain a bilinear pairing

HP(M)

x

H�-P(M) - R;

which in tum defines a linear map

([wl], [w2J ) t-+

JM w1

/\ w2

D�: HP(M) - H�-p(M)*.

13.

131

POINCARE DUALITY

Theorem 13.5 (Poincare duality) For an oriented smooth n-dimensional manifold,

nPM is an isomorphism for all p.

The proof is based upon a series of lemmas.

Lemma 13.6 Suppose

V � U � Mn are open subsets. Then the diagram IfP(U) W(i� HP(V )

l n�

commutes. Proof. Let w E classes

[w]

Jn�

H�-p(U)* � H�-p( V )*

and

·'

QP(U), 7 E n�-p(V) .be closed forms representing cohomology [7] . Then

D� o HP(i)([w])([7]) = D�([i*(w )])[7] = i! o D�([w]) ( [7] ) = D�([w])[i* (7 )] = Since

suppu(w 1\ i* (7))

C

V.

1\

suppu(i* (7)) = suppv(7)

second integral just integrate over agree on

fv i*(w) 1\ 7 fu w i* (7).

V.

But the n-forrns

we may

i*(w) 1\ 7

Lemma 13.7 For open subsets U1 and U2 of Mn with union

HP(U1 n U2)

lD�1nu2

H�-p(U1 n U2)*

�

( l)p+lfj'. -:_

as

well in the

and

w 1\ i* (7)

0

U the diagram

HP+l(U )

l n1 �+'

H�-p- (U)*

is commutative. Here 8* is the boundary in the Mayer-Vietoris sequence of Theorem 5.2, and fi is from (5).

QP(Ul n U2) and 7 E n�-p-l(U) be closed forms. We write w = Ji(wl ) - J2(w2) with Wv E 0P (Uv ) and )v: ul n u2 ---+ Uv the inclusions. Let 1 K E QP+ (U) be the (p + 1)-form with i:(K) = dwv, where iv : Uv ---+ u are the inclusions. Then K represents &* ([w]) so that Proof. Let w E

1 1 D�+ &*([w])([7]) = D�+ ([K])[7] = It was pointed out after the proof of Theorem

&*[7] E H�-p(U1 n U2) 7 = 71 + 72 with

l K 1\ 7.

1 3 .3

that a representative for

can be obtained by the following procedure:

write

132

13.

and let

a. (

[T]).

a = Ji(dTl) Hence

= -j2(dTz).

POINCARE DUALITY Then CJ

is

a closed (n - p)-form that represents

a'D�1nu2 ([w])([T]) �1 nu2 ([w))[a] { w 1\ (J. Ju1nu2 =

=

( -1 )P+ 1 .

We must show that the two integrals are equal up to the sign

{

K,

lu

1\ T =

since supp (Tv) Corollary

1 0.9,

{ K, 1\ Tl + 1 K, 1\ Tz = 1 u

lu

� Uv.

Now

U1

d(wv 1\ Tv)

=

dw1 1\ T1 +

{

lu2

We have

dwz 1\ T2

dwv 1\ Tv + ( -ltwv 1\ dTv ,

and by

so that

Corollary 13.8 (i)

Let

U1

Uz

be open subsets of

Mn,

and suppose that

U1, U2

and

Uz satisfy Poincare duality. Then so does U = U1 U U2. ' (Ua)aeA be a family of pairwise disjoint open subsets of Mn. If each U01 satisfies Poincare duality then so does the union U = Ua U01• Every open subset U � Mn that is diffeomorphic to Rn satisfies Poincare U1

n

and

(ii) Let (iii)

duality.

Proof. Consider the diagram

r H"(U) H"(Ut)'
� D

2 D�� e D� 1 1 - H:-"(U)" - H�'-P(UI)"eH:-p(U2)" l

I

_ r

� J

H"(U1nU2)

1 Du1nv2

_ a·

I0!

l)P+ H:-"(U1nU2) (-�

H"+1(U) ou+l

1 H;•-P-\U) -

133

13. POINCARE DUALITY

This is a commutative diagram according to the two lemmas above. Our assump­ tions are that D�1 EB D�2 and D�1 nu1 are isomorphisms for all p, and it follows by the 5-lemma (cf. Exercise 4. 1 ) that so is D�. The proof of (ii) uses the commutative diagram

IJc:.eEA HP (Ucx )

liTDP

Ua

H;}'-P(U)* - ITcxEA H;)'-P(Ucx)* The horizontal maps are isomorphisms by Proposition 13.4 and the right-hand vertical map is an isomorphism by assumption. To prove (iii), we use that HP(U) � HP(Rn) and H;)'-P(U) � H;}'-P(Rn) together with Theorem 3.15 and Lemma 13.2. We only need to check that D� : H0(U) -+ H�·(U)* is an isomorphism. The constant function 1 on U is mapped to the basis element

of H;}'(U)* � R.

0

Theorem 13.9 (Induction on open sets). Let Mn be a smooth n-dimensional manifold equipped with an open cover V = (V�J),aes· Suppose U is a collection of open subsets of M that satisfies the following four conditions

(i) 0 E U. (ii) Any open subset U � V,a diffeomorphic with Rn belongs to U. (iii) If U�, U2, U1 n U2 belong to U then U1 U U2 E U. (iv) If U1, U2, . . . is a sequence of pairwise disjoint open subsets with Ui E U then their union Ui Ui E U.

Then Mn belongs to U.

The proof is based upon the following lemma, where the term relatively compact means that the closure is compact.

Lemma 13.10 In the situation of Theorem 1 3.9, suppose U1, U2, . , . is a sequence of open, relatively compact subsets of J\tf with (i) n Uj E U for any finite subset J, jEJ (ii) ( Uj) . E N is locally finite. J

.

134

13. POINCARE DUALITY

Then the union U1

U

U2

U

. . .

belongs to U.

U Ujm E U for every set Proof. First we show by induction on m that Uj1 U of indices j1 , . . . ,Jm· The cases m = 1 , 2 follow from (i) and condition (iii) of Theorem 1 3.9, so suppose m 2: 3 and that the claim is true for sets of m - 1 indices. Then setting V = U32 u . . u Uim , .

.

.

.

Uj1 n V

=

m

U Uj1 n Uiv E U

v=2

by the induction hypothesis applied to the new sequence (Uj1 n Uj )j EN ' and condition (iii) of Theorem 13.9 implies that Uj1 u . . . UUj E U. Since UinUj E U by (i), we also have ,.

m

U (Uiv

(7)

v=l

n

Uj.,) E U

for any set of 2m indices 'il , Jl,

. . . : im, Jm· Inductively we define index sets lm and open sets Wm � M as follows: It

{1},

wl

=

ul and for m

lm

(8)

=

2: 2

{m} u {i I i > m, ui n Wm- 1 i= 0} -

m-1

u lj

j=l

If Im- 1 is finite, then Wm- 1 is relatively compact and (ii) implies that Wm_1 only intersects finitely many of the sets Ui. This shows inductively that Im is indeed finite for all m. Moreover, if m 2: 2 does not belong to any Ij with j < m then it certainly, by definition of Im, belongs to Im. Thus N is the disjoint union of the finite sets Im. Since we already know that finite unions are in U we have

lm = 0, Wm = 0 E U). Similarly, (7) shows that Wm n Wm+ 1 =

U

Wm E U (if

Ui n Uj E U.

(i,j)Elm X Im+l

Note also from (8) that Wm n Wk = 0 if k 2: m + 2. Indeed, if the intersection were non-empty, then there would exist ·i E lk with Wm n Ui i= 0, and by (8) i E lj for some j s m + 1. One now uses condition (iv) of Theorem 13.9 to see that the following three sets are in U:

W(O)

=

00

U

j=l

W j, W( l ) 2

=

U W2j-1 , W(o) n W(l) = U (Wm n Wm+d· j=l j=l 00

00

13.

POINCARE DUALITY

But then Theorem 1 3.9.(iii) implies that the union

135

U� 1 Ui = W(O) U W ( l ) E U. D

Proof of Theorem 13.9. Consider first the special case where an open subset. We consider the maximum norm on Rn

M = W � Rn is

llxlloc = l�t� max lxi j n

whose open balls are the cubes IIi= I (ai, bi). The argument of Proposition gives us a sequence of open II 11 00-balls Uj such that (i) w = Uf=l Uj = Uf=l Uj. (ii) (Uj )j EN is locally finite. (iii) Each u1 is contained in at least one

A.6

V13.

A finite intersection Uj1 n . . . n U1m is either empty or of the form [1� 1 (ai, bi ), so is diffeomorphic to Rn. Thus we can conclude from Lemma 13.10 that W E U. In the general case, consider first an open coordinate patch (U, h) of M, h: U W a diffeomorphism onto an open subset W � Rn. We apply the above special case to W with cover (h- 1 (V13)) /3EB ' and with uh the open sets of W whose images by h- 1 belong to the given U. The conclusion is that U E U for each coordinate �

patch. If M is compact then we apply the argument of Lemma 13.10 to a finite cover of M by coordinate patches to conclude that M E U. In the non-compact case we make use of a locally finite cover of M by a sequence of relatively compact coordinate patches; cf. Theorem 9. 1 1 . (One may alternatively imitate the proof of Proposition A.6 using relatively compact coordinate patches instead of discs to construct the desired cover). D

Proof of Theorem 13.5. Let

U = { U � Mn I U open, Df1 an isomorphism for all p}

and let V = (V13) /3EB be the trivial cover consisting only of M. Corollary 13.8 D tells us that the assumptions in Theorem 13.9 are satisfied, so M E U. We close the chapter with an exact sequence associated to a smooth compact manifold pair (N, M), i.e. a smooth compact submanifold M of a smooth compact manifold N. Let U be the complement U = N - M, and let

i: U - N,

j: M - N

be the inclusions. Then we have

Proposition 13.11 There is a long exact sequence

. . . - Hq-l(M) � Hg(U) !..; Hq(N) L Hq(M) - . . . The proof of this result is based upon the following:

13.

136

POINCARE DUAUTY

Lemma 13.12 (i) j"' = nq(j): n,q(N) � nq(M) is an epimorphism. (ii) If w E nq(M) is closed, there exists a q-form T E nq(N) such that j "' (T) == w and such that dT is identically zero on some open set in N containing M. (iii) If T E nq(N) has SUPPN (dT) n M = 0 and j"'(T) is exact, there exists a form u E n,q-l ( N) such that T - du is identically zero on some open set in N containing M. that N is a smooth submanifold of Rk . gives us tubular neighborhoods in Rk with corresponding smooth

Proof. We can assume by Theorem Theorem

9.23

8. 1 1

inclusions and retractions (VN , i N , rN), (VM, iM,rM) for N and M respectively; we may arrange that VM ; VN. Let
Let w E nq(M) be closed, and Jet w = rM(w) E nq(N n VM ) . Then (ii) follows upon defining T E D,q(N) to be equal
= rjy (T)IVM

E

D,q (VM ) .

The assumption that supp N ( dT) n M = 0 implies that drjy ( T) = rjy (dT) vanishes on a neighborhood of M. Hence VM may be chosen so small that dr = 0. Observe that

iM('T) = (iN o j)"'(r'jy(T)) = j* iN r'jy(T) = j* (T ) o

o

so [iM (7)] = 0 in Hq ( M). It follows from Corollary 9.28 that [TJ = 0 in Hq ( VM ). Pulling this back by the inclusion N n VM ---+ VM we find that TINnVM is exact.

Now choose uo E D,9-1( N n VM) with duo = TI NnVM and define u E n,q-1(N) to be
Proof of Proposition 13.1 1 . By Lemma l 3 . 1 2.(i) we have a short exact sequence of chain complexes

0 � D,"'(N, M)

�

D,"'(N)

� f2*(M) � 0

where D,q(N, M) is defined to be the kernel of D,9(j). Let us denote the cohomology of (D.*(N, M), d) by H* (N, M). Then Theorem 4.9 gives us a long exact sequence

..·

�

Hq-1(M)

�

Hq(N, M)

�

H9(N)

£. Hq(M)

� · · · .

POINCARE DUALITY

13.

The chain map i* : D�(U) show that

-t

137

D*(N) has image in U*(N, M), so it suffices to

i * : D�(U)

-t

Q* (N, l'v! )

induces an isomorphism in cohomology. H'1(N, M ) in the above exact sequence.

We can then substitute Hg(U) for

Consider an element [w] in the kernel of

- H9(N, M)

H'1(i * ) : HZ(U)

represented by a closed q-form w E ng(U). Then i* (w) = dr for some r E Q'1-1(N, M). Since j * (r) = 0 and suppN(dr) � U we may apply Lemma 1 3 . 1 2.(iii) to find r7 E Q'1-2(N) such that r - dr7 vanishes on an open set around M. This gives us K == (r - dr7) U E n�-1(U) with dn: = w, so (w] = 0.

J

Let [w] E H9(JV, M) be represented by the closed q-form w E Q9(N, M). We use Lemma 1 3 . 1 2.(iii) to find r7 E Q'1-1(N), with the property that w - d(J vanishes on an open set containing M. Observe that

d(j * (a))

=

.i*(da)

=

= 0.

j*(w)

By Lemma 1 3 . 12.(ii) we may choose r E nq-1(N) with .i * ((J) = _j * (r) and such that dr vanishes on a neighborhood of M. Thus (J-T E nq-l (N, M), and defining K ==

(w - d(a - r))!u

we obtain (w] = [w - d((J - r)J

(w - cla) 1u

=

= Hq(i*)[K).

+

dri U E D�(U),

0

Let us finaJly introduce the important signature invariant of oriented compact manifolds of dimension congruent to zero modulo 4. Given a 2k-dimensional compact smooth manifold M2k, its intersection form is the bilinear fonn J.L: Hk(M)

x Hk(M)

defined by

J.L([o.}, [,6'])

(9)

This is ( -l) k-symmetric, i.e.

0

=

R

-t

r a 1\ ,B. jM

k J.L([a], [,B] ) = ( -l) J.L([a] , [fJJ). (mod 2 ) , where J.l becomes symmetric. All such bilinear

We focus on k = forms can be diagonalized; i.e. there exists a basis

J.L(ei, e1·) =

( 1 0)

{0

O:i

eJ

if i I= j

, . . . , em

such that

1.f 't. = J. ·

Given a diagonalization of J.l, we define the signature by

(J (J.l)

=

#{i I Cl'i

> 0} - #{i I Ui

<

0}.

This number is independent of the chosen diagonalization. In the case of the intersection form we know that D'i f. 0 for all i, since by Poincare duality the adjoint D'M of J.l is an isomorphism.

138

13.

POINCARE DUALITY

Definition 13.13 The signature of an oriented closed 4k-dimensional manifold is the signature of its intersection form.

139

14.

THE COMPLEX PROJECTIVE SPACE cpn

The set of !-dimensional complex subspaces of cn+ l is denoted by CPn, and is called the complex projective n-dimensional space. For z = (zo, Z!, . . . ' Zn) E cn+I - {0} let 7r(z) = [zo, Zt, . . . ' Zn] denote the "point" Cz E CPn spanned by z. We give cpn the quotient topology with respect to 1r: a set U � cpn is open if and only if 1r- 1 (U) � cn+ l - {0} is

open. In particular there are open subsets of epn,

and the homeomorphisms hi: Ui - en, (1) with inverses

(2)

wtfwm

The transition functions h k o hj 1 have coordinate functions of the form or / m . The atlas H. = { (Uj , hi)} gives epn the structure of a complex (analytic or holomorphic) manifold, because the transition functions are holomorphic. In the following, however, we shall mainly use the underlying structure as a smooth 2 manifold, by interpreting H. as a C00-atlas upon identifying en with R n .

1

w

Example 14.1 (The Riemann sphere and Hopf fibration) Since e

identified with R3, the unit sphere

S2

can be written as

1)

x

R can be

and south pole P- = (0, - 1 ) The equatorial plane with north pole P+ = (0, C x {0} is identified with C. The stereographic projections 1/J±: S2 - {P±} - C map p to the points of intersection between the equatorial plane and the line through P± and p. A straightforward calculation gives, for (z, t) E S2 - {P±}. .

1P+ (z, t ) =

z 1 t ' 1/J-(Z, t) = -. 1+t z

The 1/J± are diffeomorphisms with inverses

w (w ) W±' Cwf + 1 ' ::::::) 1/;:;_ 1: o 1j;:;_ 1(w) = (w)- 1 . =

The transition function 1/J- o

±

C - {0} - C - {0} is easily calculated to be

1/J-

14.

140

THE COMPLEX PROJECTIVE SPACE CP"

If we orient C in the usual manner and consider S2 as the boundary of D3 with t and '1/J+ orientation­ standard orientation from R3, then 7/J- is orienation-preserving reversing (check at the poles!). This inspires us to replace '1/J+ by its conjugate 7/J+ : S2 - {P+} -t C. Then the transisition functions between '1/J+ and '1/J- are inversion in the multiplicative group of C, so the atlas on S2 consisting of '1/J+ and '1/J- gives S2 the structure of a } -dimensional complex manifold (Riemann sulface). The classical Hopf map 'T/: S3 -t S2 is given by

7J(zo, zi) = (2zo z l > lzol 2 - ! z1 l 2 ) .

(3)

Hopf discovered (in 1931) that

11

is not homotopic to a constant map.

The Riemann surface of Example 14. 1 is holomorphically equivalent to CP1 . Indeed, by (2), the compositions

are

h0 1 '1/J- (z, t) = [1, z/(1 + t)), h11 '1/J+ (z, t) = [z/( 1 - t), 1]. These expressions agree when t E ( 1 1 ) , so we can define a homeomorphism o

o

-

\11 (z, t)

(4)

,

=

{ [1

+

t, z] , (z,t) # (0, -1) [z, 1 - t], (z, t) # (0, 1).

Actually w is holomorphic with holomorphic inverse. The complex projective plane CP 1 is often identified with C U { 00} by letting [zo , ZI ) E CP1 correspond to z0 1 Z!t and assigning oo to z0 = 0. In this identification \11 : S2 -t C u { oo} becomes the stereographic projection '1/J- from the south pole p-, extended by mapping P- to oo. One may generalize the Hopf fibration to the map (5)

Its fiber 7r- 1 (p) is the unit circle in the complex line p E CPn. For n = 1 this is nothing but the Hopf fibration of Example 1 4. 1 . Indeed with the notation of (3), (4)

w- 1 ( [l , zQ' 1 zl ] ) = '!/': 1 (zQ'1zl ) = (2zozl , lzol2 - lzi i2 ). The unit circle S1 � C acts on the sphere sZn+l � cn+l by coordinatewise w -1 ([zo, zl])

=

multiplication, (6)

The action is free, and the set of orbits of this action is precisely cpn in the sense that the orbit space s2n+l I S1 is homeomorphic to cPn . .

14.

TilE COMPLEX PROJECTIVE SPACE

cpn

141

Theorem 14.2 The cohomology of CPn is

H2i(CPn) = R Hk(CPn) = 0 Proof. The embedding en

c cn+l

for 0 � j � n otherwise.

induces an embedding of

cpn-1

into

CPn,

(CPn, cpn- 1 ). We can assume the and that n � 2, since the cohomology of CP 1 = S2 was given

and we can use Proposition 1 3. 1 1 on the pair result for

cpn-1

in Example

9.29.

The complement

cpn - cpn-l = U n is by ( 1 ) and

(2) homeomorphic with R2n.

The exact cohomology sequence takes

the form

We know from Lemma

13.2

that

Hl(R2n)

is non-zero only for

is a copy of R, and the result follows easily.

q = 2n,

---+

---+

would be the identity, but this is impossible as Since

0

14.2 that the general Hopf map 1r: S2n+1 cpn cannot s2n+l : if s existed with 1f 0 s = id, then s: cpn H2(cPn ) � H2 ( s2n+1 ) £ H2(cPn )

It follows from Theorem have a section

when it

H2n(cPn)

=

R

H2(CPn) =!=

0 and H2(S2n+l) = 0.

we know from Exercise 10.4 that

CPn

is an oriented

manifold, and hence from Theorem 13.5 that the bilinear map

given by the wedge product (followed by the integration isomorphism) is a dual

(non-singular) pairing. In particular, the generator of H2P(CPn) and the generator of

H2n-2P(CPn)

has non-trivial product.

Theorem 14.3 The cohomology algebra H*(CPn) is a truncated polynomial

algebra

H*(CPn )

= R[c]/(cn+ l)

where c is a non-zero class in degree 2, and (cn+1 ) the ideal generated by cn+l . Proof. We use induction over n, so suppose the theorem proved for cpn- 1 . The inclusion

14.

142

THE COMPLEX PROJECTIVE SPACE

cpn

induces the map

The proof of Theorem 14.2 shows that j* is an isomorphism for i Hence cn-l i= 0 in H2n-2(CPn), and the pairing

H2(CPn) X H2n-2(CPn)

---t

<

2n - 2.

H2n(C Pn )

implies that en i= 0.

0

It is a consequence of the above theorem that cpn supports an orientation­ reversing diffeomorphism only if n is odd. Indeed, a map f: cpn ---t cpn induces a homomorphism

If !*(c)

ac,

then

!*(d) = ajd (0 � j � n). For J. = n we get deg(f) = an. If n is even then deg(f ) � 0 . In this case there are no orientation-reversing diffeomorphisms of cpn . . If n is odd, on the other hand, then complex conjugation is an orientation-reversing diffeomorphism of CPn:

f( [zo, z1 , . . . , Zn] ) = [zo, z1 , . . . , zn]

has a = - 1 as one sees by restricting to 82 = C P1.

We close this chapter by constructing closed differential forms which represent a basis for H2j (cpn). This requires some preparations. Consider a unit vector v E S2n+l c cn+l with image p = 1r(v) . Now iv is a unit vector tangent to the 8 1 -orbit of v (which was precisely the unit circle in the )-dimensional C-subspace Cv). The orthogonal complement (Cv) l. with respect to the usual hermitian inner product on cn+l is a real 2n-dimensional subspace of TvS2n+l, and is orthogonal to iv with respect to the real inner product on TvS2n+ l induced from en+I = R2n+2 . Lemma 14.4

(i) Let p E CPn and v E 1r -1(p) � S2n+l. There is an open neighborhood u around p in cpn and a smooth map s: u ---t s2n+ l such that s(p) = v and 1r o s = idu. (ii) Let v E S2n+ l and p = 1r(v). The differential Dv?T induces an R-linear isomorphism from (Cv)l. to Tpcpn_ (iii) There exists a well-defined structure on TpCPn as an n-dimensional C­ vector space with hermitian inner product, which makes the isomorphisms of (ii) into C-linear isometries.

14.

THE COMPLEX PROJECTIVE SPACE CP"

Proof. Choose u = Uj such thatp

given by

(7)

E U, cf. (1).

Sj([zo, . . . , Zj, . . . , zn] ) = (t lzkl2

k=O

)

143

Consider the map Sj: Uj

- l /2

�

s2n+l

(zo, . . . , Zj, . . . , zn )

where Zj = We let s be this map composed with multiplication by the unique S ). E 1 such that >.sj(P) = v. The chain rule implies that

1.

Dv7r: TvS2n+l

�

Tp CPn

is surjective. The kernel has real dimension 1 , and since it contains iv, assertion (ii) follows.

Let ¢: S2n+l � S2n+l be the diffeomorphism given by multiplication by ). E S 1 . The chain rule gives a commutative diagram

TvS2n+l D., T.xv S2n+l

/D>.u1r

D.,�

T,p cpn

Multiplication by >. can be considered as an R-linear map cn+ I � cn+l, and l. Dv