Morse Theory. (Am-51)

MORSE T H E O R Y BY

J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1963

CONTENTS

Copyright 0 1963. by Princeton University Press All Rights Reserved L C Card 63-13729

PART I .

NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD

. .

. $2.

. . . . . . . . . . . . . . . . . . . . . . D e f i n i t i o n s and Lemmas . . . . . . . . . . . . . . . . . . $3 . Homotopy Type i n Terms of C r i t i c a l Values . . . . . . . . $ 4 . Examples . . . . . . . . . . . . . . . . . . . . . . . . . $5 . The Morse I n e q u a l i t i e s . . . . . . . . . . . . . . . . . .

$1

I n t r o d u c t i o n.

0 6 . Manifolds i n Euclidean Space: Non-degenerate Functions . $7

.

PART I1

.

The k f s c h e t z Theorem on

1

4 12

25 28

The Existence of

. . . . . . . . . . . . . . . . Hyperplane Sections . . . . . . .

12

39

A WID COURSE I N RIEZ"NIAN GEOKETRY

.

Covariant D i f f e r e n t i a t i o n

.

The Curvature

$8 $9

$10

. Geodesics

. . . . . . . . . . . . . Tensor . . . . . . . . . . . . . . . .

and Completeness .

43 51

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

55

.

PART I11 THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS

. §12 . $11

$13

.

. 515 .

$14

Printed in the United States of America

. . . . . . . . . . . Path . . . . . . . . . . . . . . . . . . .

The Path Space of a Smooth Manifold

67

The Energy of a

70

The Hessian of t h e Energy Function a t a C r i t i c a l Path Jacobi F i e l d s :

The Index Theorem

. .

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

The Null-space of

E*++ .

§16

.

A F i n i t e Dimensional Approximation t o

817

.

The Topology

$18

.

EXistence of

§I9

.

Some R e l a t i o n s Between Topology and Curvature

74

77 a3

. . . . . . . . aa of the Pull Path Space . . . . . . . . . . . 93 Non-conjugate Points . . . . . . . . . . . . 9a

V

QC

......

loo

CONTENTS PART IV. APPLICATIONS TO LIE GROUPS $20.

$21. $22.

$23. $24.

APPmIX.

I

IC SPACES

. . . . . . . . . . . . . . . . . . . . Symmetric Spaces . . . . . . . . . . . . Lie Groups a s Symmetric Spaces Whole Manifolds of Minimal Geodesics . . . . . . . . . The Bott P e r i o d i c i t y Theorem f o r t h e Unitary Group . . The P e r j o d i c i t y Theorem f o r t h e Orthogonal Group . . .

THE HGMOTOPY TYPE OF A MONOTONE UNION

109

. . . .

. . . . . . . .

112

118 124

13’1

PART I

149

NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.

$1.

Introduction.

I n this s e c t i o n we w i l l i l l u s t r a t e by a s p e c i f i c example t h e s i t u a t i o n that we w i l l i n v e s t i g a t e l a t e r for a r b i t r a r y manifolds. sider a torus

M,

tangent t o t h e plane

Diagram

Let above t h e

f(x)

5

a.

V

f:

M+R

Let u s conas i n d i c a t e d i n D i a g r a m 1 .

V,

1.

(R always denotes t h e r e a l numbers) be t h e height I@‘ be the s e t of a l l p o i n t s x E M such that

plane, and l e t

Then the following t h i n g s a r e t r u e : (1)

If

a




E~

cpt(q)

PROOF:

for q

q

-

t

c(t)

M

= k(E0/2)

dc

by t h e i d e n t i t y

dc =(f)

=

't

h .

k

~

2

= 'E0/2

O

'pE0/2

.

only necessary t o r e p l a c e (Compare $ 8 . )

Now l e t

cp

cp

be a 1-parameter group of diffeomorphisms, generated by t h e v e c t o r f i e l d

X.

i t follows that this d i f f e r e n -

K,

qt(q)

i s defined for a l l v a l u e s of

Eo/2

t.

It1

for

cpt

q

t h e curve

completes the proof of Lemma

t

+

RFMARK: cannot be omitted.

-a€--= Xcpt(q) '

where

p

=

cpt(q).

cpo(q)

=

q.

and l e t

But i t i s w e l l known that such a d i f f e r e n t i a l equation,

l o c a l l y , has a unique s o l u t i o n which depends smoothly on the i n i t i a l condi(Compare Graves, "The Theory of Functions of Real V a r i a b l e s i ' p . 166. ,un, t h e d i f f e r e n t i a l equaNote that, i n terms of l o c a l coordinates u' , tion.

i

2

E

for a l l

and

Further-

l t ( , 1 s ) , l t + s 1< c O . . Any

~

r

t

number

with




onto i t s e l f ,

of diffeomorphisms of M we define

cp

E

11

i

= 1

,...,n . )

Then

M.

X

(0,l)

does not

C R,

I.

12

NON-DEGENERATE FUNCTIONS

HOMOTQPY TYPE

$3. C:

R

+

$

i s a curve w i t h v e l o c i t y vector

M

13

note t h e i d e n t i t y

.

< dc E , g r a d f >d (=f o cT) $3.

Homotopy Type i n Terms of C r i t i c a l Values.

Throughout t h i s s e c t i o n , i f manifold

Let

i s a r e a l valued f u n c t i o n on a

f

M&

.

= f-’(- m , a l

= (p E M : f(p)

5 a)

.

1 / < g r a d f , grad f>

be a smooth f u n c t i o n which i s equal t o

throughout thecompact s e t f - l [ a , b ] ; and which vanishes Then t h e v e c t o r f i e l d

X,

defined by

f be a smooth r e a l valued f u n c t i o n on a manifold M. Let a < b and suppose that t h e s e t f - l [ a , b l , c o n s i s t i n g of a l l p E M w i t h a L f ( p ) L b , i s compact, and contains no c r i t i c a l p o i n t s of f. Then M& i s diffeomorphic t o Mb. Furthermore, i s a deformation r e t r a c t of Mb, so t h a t t h e i n c l u s i o n map Ma Mb i s a homotopy equivalence.

THEOREM 3 . 1

M+R

outside of a compact neighborhood of this s e t .

we l e t

M,

p:

Xq

Let

=

p ( q ) (grad

s a t i s f i e s the conditions of Lema 2 . 4 .

Hence

f)s

X

generates a 1-parameter

group of diffeomorphisms

vt: M For fixed

q E M

+

M.

consider the f u n c t i o n

t + f(cpt(q)).

If Tt(q)

-+

l i e s i n the s e t The i d e a of t h e proof is t o push n a l t r a j e c t o r i e s of t h e hypersurfaces

Mb

M&

down t o

f = constant.

f-’[a,bl,

then



(Compare D i a g r a m 2 . )

=

<X,

grad f >

=

+ 1.

Thus t h e correspondence

t

\

i s l i n e a r with d e r i v a t i v e

-*

f(vt(9))

as long as

+1

f ( c p t ( q ) ) l i e s between

Now consider the diffeomorphism

M&

diffeomorphically onto

Mb.

M + M.

\

I I

I

I

I

I

D i a g r a m 2.

I

Y

r t : M~

4

~b

and

b.

Clearly this c a r r i e s

T h i s proves the f i r s t half of

Define a 1-parameter family of maps

a

7.1.

I.

14

NON-DEGENERATE FUNCTIONS

HOMOTOPY TYPE

§ 3.

Choosing a s u i t a b l e c e l l in@;

e x C H,

15

a d i r e c t argument

( i . e . , push-

i n along the h o r i z o n t a l l i n e s ) w i l l show t h a t

M C - E ~ ex i s a deformation r e t r a c t of M ~ u- H.~ F i n a l l y , by applying 3 . 1 t o the f u n c t i o n F and the region F-1 [ C - E , C + E I we w i l l see that M ~ u- H~ i s a deformation r e t r a c t

of MC+& . T h i s w i l l complete t h e proof. Choose a coordinate system 90

=

holds throughout

c -

(U1l2-

...

Choose

>

E

(UX+l)2,

=

...

=

u”(p)

p

...

+

The region

p o i n t s other than

f-’[ c - E

MC-E

i s heavily shaded.

= f - l (-m,C-E

= 0

.

,C+E

1

i s compact and contains no c r i t i c a l

The image of

U

under the diffeomorphic

. . . ,un) :

imbedding

U -Rn

“u Now d e f i n e

We w i l l introduce a new f u n c t i o n

coincides w i t h t h e h e i g h t f u n c t i o n

f

except that

F

F: M


0 . Then, f o r a l l s u f f i c i e n t l y s m a l l E , t h e s e t MC+E has t h e homotopy type of MCFE with a k - c e l l a t t a c h e d .

i n a neighborhood

that the i d e n t i t y f

Diagram 3 .

u 1 ,..., un

c’

f

C+E

I Diagram 5.

f . C i E

‘f.c

16

u1

=

...

ux

=

and G;

b a l l of r a d i u s

.

and f - ' ( c + E )

respectively;

= 0

un

= 0

F

and

i s h e a v i l y shaded; t h e region

MC-€

z o n t a l dark l i n e through

p

f-' ( c - E )

Within this e l l i p s o i d we have

x

We must prove t h a t

formation r e t r a c t of M ' + ~ . Construct a new smooth f u n c t i o n

F: M

-

R

MCeE

u

The h o r i -

PROOF:

C+E

a r e t h e same as those of

F

f.

Note that

so that e x e x i s a de-

as follows.

c-c+q < c+ ls+q 2 -


are t h e s t a n d a r d v e c t o r f i e l d s on t h e co-

should be c o n s t a n t .

that

PROOF:

Choose p a r a l l e l v e c t o r f i e l d s

are orthonormal a t one p o i n t of

c

t i v e l y (where vi

w

lows that

V



= =

1

by t h i s formula, i t i s n o t d i f f i c u l t t o v e r i f y

that c o n d i t i o n s ( a ) , (b), and ( c ) A vector f i e l d



along

i f the c o v a r i a n t d e r i v a t i v e

DV

are s a t i s f i e d .

c

i s s a i d t o be a p a r a l l e l v e c t o r f i e l d

Theref o r e i s i d e n t i c a l l y zero.

which completes the p r o o f .

a

viwi

P,,

. . . , Pn

along

and hence a t e v e r y p o i n t of

t h e g i v e n f i e l d s V and W can be e x p r e s s e d as

Conversely, d e f i n i n g

I n o t h e r words,

of' p a r a l l e l v e c t o r f i e l d s

P, P '

LEMMA 8 . 3 . Suppose that t h e connection i s compatible w i t h the m e t r i c . L e t V, W be any two v e c t o r f i e l d s a l o n g c . Then

(or a n a p p r o p r i a t e open

I t f o l l o w s from ( a ) , ( b ) , and ( c )

o r d i n a t e neighborhood.

c,

c

1

viPi

and

2

c. wjPj

i s a r e a l valued f u n c t i o n on R) . and t h a t

c

which Then respec-

It f o l -

COROLLARY 8 . 4 . For any v e c t o r f i e l d s vector Xp E TMp:

xp < Y , Y ' >

<xp

=

COVARIANT DIFFERENTIATION

§ 8.

11. RIEMANNIAN GEOMETRY

48

Y,Y'

on

M

three quantities

and any

I- a j , a k > ,

k y,y;>

+

.

I- Y ' >

0 s o that, for each (To,To) E W the differen-

numbers I1vII




E

s o that:

~0

f o r each

p

E

and each

U

57

v

TMp w i t h

E

t h e r e i s a unique geodesic ( - 2 ~ ~ , 2 - +~M ~ )

yv:

s a t i s f y i n g the required i n i t i a l conditions. has a unique s o l u t i o n t - < ( t ) which i s defined for It\ < E, and s a t i s f i e s t h e i n i t i a l conditions

To o b t a i n t h e sharper statement i t i s only necessary t o observe that the d i f f e r e n t i a l equation for geodesics has t h e following homogeneity property.

c

Let

be any c o n s t a n t .

I f t h e parametrized curve

Furthermore, t h e s o l u t i o n depends smoothly on t h e i n i t i a l conditions. I n other words, t h e correspondence

Go,Yo,t ) from w x ( - E , E ) 2n+1 v a r i a b l e s .

to

is a

R~

3t ) ern f u n c t i o n

i s a geodesic, then the parametrized curve

+

of a l l

t

+

Y(Ct)

w i l l a l s o be a geodesic.

PROOF:

Introducing t h e new v a r i a b l e s

second order equations becomes a system of

v

i

=

dui

t h i s system of

Now suppose that

n

(tl

?n f i r s t order equations:




(2)

p(p;,q)

PA

W e claim that

=

(r

-

i s equal t o

to) -

7(t0

6'

.

+ sl).

I n f a c t the t r i a n g l e

i n e q u a l i t y states t h a t 0,

choose a neighborhood

denote a s p h e r i c a l s h e l l of r a d i u s

6


dt.

A(t),A(t)

=

c

for a l l

t.

Hence each

W(ti)

=

at2V.

(ol[ti,ti+ll

Then

C',

even a t the p o i n t s

ti.

from the uniqueness theorem f o r d i f f e r e n t i a l equations that

shows

path

/

-!f ( t )


=

Conversely, suppose t h a t Choose a subdivision

WII [ti-,,til

Tn7

.

The second

0 =

to




.

0

=

(Since DV iTt; = o ) ,

0



0

for W

Then W

If

For if

T




0

M

S

x

to

y

x

E

and

S


i s constant along 7 . I n f a c t ,

=

p

IV. APPLICATIONS

110

@ven

P

q

= 7(0),

carries

p

to

q.

= 7 ( c)

,




Thus

remains t r u e everywhere along

i s parallel.

R(U,V)W

7.

=

eiUi

Any v e c t o r f i e l d W along

may be

y

expressed uniquely as

i s constant f o r every p a r a l l e l



111

the condition R(V,Ui)V

I t c l e a r l y follows that

X.

17(c/2)Ip which

i s an isometry, this q u a n t i t y i s equal t o

.

R(Up,Vp)Wp,%>

=

Then

>


=

Therefore we may choose an orthonormal b a s i s %(Ui)

where

el,

...,en

along

7

by p a r a l l e l t r a n s l a t i o n .

*

=

a r e t h e eigenvalues.

...,Un

U,,

eiUi

Mp

s o that

,

Extend the

Then since

. for

M

should n o t be confused w i t h t h e R i c c i t e n s o r

Ui

t o vector f i e l d s

is l o c a l l y symmetric,

Of

$19.

=

ei

=

ci s i n

wi(t)

(5 t),

O

*

t

If

= 0.

f o r some constant

a r e a t the m u l t i p l e s of

t




0

then

ci.

.

n/Gi

-

= 0

vanishes only a t

<W,KJ(W>

=

wi(t) Then t h e zeros of

THEOREM 2 0 . 5 . The conjugate p o i n t s t o p along 7 are the p o i n t s y ( c k / G i ) where k i s any non-zero i n t e g e r , and ei i s any p o s i t i v e eigenvalue of %. The m u l t i p l i c i t y of 7 ( t ) a s a conjugate p o i n t I s equal t o t h e number of ei such t h a t t i s a mult i p l e of c / G i . PROOF:

=

then

Thus i f

ei

1. 0 ,

wi(t)

LIE GROUPS

§21.

IV. APPLICATIONS

112

113

LEMMA 2 1 . 2 . The geodesics y i n G with ~ ( 0 =) e a r e p r e c i s e l y t h e one-parameter subgroups of G .

PROOF:

Lie Groups as Symmetric Spaces.

521.

G

I n this s e c t i o n we consider a Lie group

the map

with a Riemannian metric

so

which i s i n v a r i a n t both under l e f t t r a n s l a t i o n s

Let

I y ( t ) I e takes

G

+

L,(u)

G,

T d

=

~

and right t r a n s l a t i o n , certainly exists.

as follows:

R,(u)

If

the Haar measure on new i n n e r product

If

UT.

i s commutative such a metric

G

GI and L e t

be any Riemannian metric on

G.

Then

>



on

1

=

i s r i g h t and l e f t i n v a r i a n t .

I.I

G

denote

I.I

Define a

by

>

LU,RTy(W)



gent v e c t o r of Hence

and only if

GxG

Then

geodesic through y 7'

e

at

e.

Ie: G

-

G

L,

and

Iex: TGe

-+

-

Ie*: Gu

1,

Finally, defining I,

Q

t'

i s rational so that n'

n" then

and

y(nt)

n't

=

+ t ")

y(t'

= y(t)n

t"

and

n"t

=

= y(t)n'+n"

=

i s a homomorphism.

be a

1-parameter subgroup.

Let

at

7'

be the

7'

i s t h e tan-

e

i s a 1-parameter sub-

7'

on a Lie group

=

Xa.b [X,YI

G

i s called l e f t invariant

for every a and i s also.

b

in

If X

G.

The Lie algebra

Q

[

of

if

and G

Y

i s the

I.

i s a c t u a l l y a Lie algebra because the Jacobi I d e n t i t y + "Y,Zl,Xl

+ "Z,Xl,YI

=

0

holds f o r a l l ( n o t n e c e s s a r i l y l e f t i n v a r i a n t ) v e c t o r f i e l d s

.

-1

THEOREM 2 1 . 3 .

RU- 1 I,',,

so i s certainly

X,Y

and

Z.

=

TU-~T,

u

E

G.

Since

the identity

I,

=

G

i s a C"

i t s tangent v e c t o r a t

e.

G

W

.

A s i n 5 0 we w i l l use t h e n o t a t i o n

PROOF:

e.

d e r i v a t i v e of

R,IeRil

homomorphism of R

and

1,

i s a n isometry which r e v e r s e s geodesics through of

be a Lie group with a l e f t and


b) R ( X , Y ) Z = ~[X,Yl,Zl C) = [X,YI,[Z,W1>

Now t h e i d e n t i t y

=

IT(u)

A 1-parameter subgroup G.

u

i s an isometry f o r any

G -1

r e v e r s e s t h e tangent space a t

shows that each

=

r e v e r s e s t h e tangent space of

TGe

an isometry on this tangent space.

shows that

Iy(t)Ie(~ = )y ( t ) u 7 ( t )

By induction it follows that

We have j u s t seen that

X

(La)*(Xb)

the bracket

Define a map

are isometries.

By Lemma 20.1

by Ie(a)

Then

R,

e.

T h i s completes t h e proof.

= 7.

"X,Yl,Zl By hypothesis

Now

2t).

=

v e c t o r space of a l l l e f t i n v a r i a n t v e c t o r f i e l d s , made i n t o a n a l g e b r a by

LEMMA 2 1 . 1 I f G i s a Lie group with a l e f t and right i n v a r i a n t metric, then G i s a symmetric space. The r e f l e c t i o n I, i n any p o i n t T E G i s given by t h e formula ~ , ( u ) = T o - ' , . PROOF:

7(O)

such that t h e tangent v e c t o r of

a r e l e f t i n v a r i a n t then

i s l e f t and r i g h t i n v a r i a n t .

G

7

7(u +

into

t l / t "

-

R

y:

A vector f i e l d

.

dp(d) dp(T)

If

By c o n t i n u i t y

Now l e t

group.



be t h e orthogonal complement of

* It follows

that t h e t r i - l i n e a r f u n c t i o n X,Y,Z .-, Q X , Y I , Z > i s skewsymmetric i n a l l t h r e e v a r i a b l e s . Thus one obtains a l e f t i n v a r i a n t d i f f e r e n t i a l 3-form on G, r e p r e s e n t i n g a n element of t h e de R h a m cohomology group H3(G). I n this way Cartan w a s a b l e t o prove that H3(G) # 0 i f G i s a non-abelian compact connected Lie group. (See E. Cartan, "La Topologie des Espaces Representatives des Groupes de Lie," P a r i s , Hermann, 1936.)

GI x Rk possesses a

Conversely it i s c l e a r that any such product l e f t and right i n v a r i a n t metric.

if

x,y

E

and

01

C

c

E

hence

[x,yI

E

0'.

Hence

o for a l l

.

Then

c

=

I t follows that G

gr

C

E

and l e t

c 1

i s a Lie sub-algebra.

For

then

< [X,YI,C > Lie a l g e b r a s .

=

B

<X,[Y,CI> g

=

0;

s p l i t s as a d i r e c t sum

S p l i t s as a C a r t e s i a n product

i s c m p a c t by 21.5 and. GI'

GI

x G";

f l l (3

C

where

of GI

1s Simply connected and a b e l i a n , hence isomorphic

$21

IV. APPLICATIONS

116

LIE GROUPS

117

by

T h i s completes the

t o some Rk. (See Chevalley, "Theory of Lie Groups.")

.

Ad V(W)

proof.

=

[V,WI

we have

%

THEOREM 2 1 . 7 ( B o t t ) . Let G be a compact, simply connected Lie group. Then t h e loop space n(G) has t h e homotopy type o f a CW-complex with no odd dimensional c e l l s , and with only f i n i t e l y many X-cells for each even value of h . Thus t h e X-th homology groups of

n(G)

i s zero f o r

A

The l i n e a r transformation

odd, and i s

+

G

(-a1O

REMARK

an example, i f

1.

G

T h i s CW-complex w i l l always be i n f i n i t e dimensional.

i s t h e group

that t h e homology group REMARK 2.

0

Ad V =

(Ad

.

V)

i s skew-symmetric; that i s

-




Ad V

t a k e s t h e form

of u n i t quaternions, then we have seen

S3

i s i n f i n i t e c y c l i c f g r a l l even values of i.

HiO(S3)

It follows that t h e composite l i n e a r transformation

T h i s theorem remains t r u e even f o r a non-compact group.

(Ad V) (Ad V) 0

has

matrix

I n f a c t any connected Lie group contains a compact subgroup as deformation (See K. Iwasawa, On some types of topological proups, A n n a l s of

retract.

Mathematics 50 ( 1949), Theorem 6 . ) PROOF of 2 1 . 7 .

Choose two p o i n t s

conjugate along any geodesic.

p

and

By Theorem 17.3,

to

each

q X.

of index

X.

G

O(G;p,q)

type of a CW-complex with one c e l l of dimension p

in

q

A

which a r e not

has t h e homotopy

Therefore t h e non-zero eigenvalues of

f o r each geodesic from

By 6 1 9 . 4 t h e r e a r e only f i n i t e l y many

Thus i t only remains t o prove that the index

X

= - &Ad

V)*

a r e p o s i t i v e , and

occur i n p a i r s .

A-cells for

I t follows from 20.5 that t h e conjugate p o i n t s of

of a geodesic is

occur i n p a i r s .

always even.

p

7

starting at

v

=

$(O)ETG

P

t h e conjugate p o i n t s o f

p

with velocity vector

p

on

y

a r e determined by the

eigenvalues of t h e l i n e a r transformation

q:

TGp - T G

P

'

defined by

%(W)

=

R(V,W)V

=

Defining the a d j o i n t homomorphism Ad V:

kl

-.,

geodesic from

' a .

$[V,Wl,Vl

.

.*

along

7

also

I n other words every conjugate p o i n t has even m u l t i p l i c i t y .

Together with the Index Theorem, this implies that the index

Consider a geodesic

According t o 6 2 0 . 5

%

p

to

q

i s even.

T h i s completes t h e proof.

X

of any

MANIFOLDS OF MINIMAL GEODESICS

$22.

IV. APPLICATIONS

118

119

Thus we obtain: COROLLARY 2 2 . 2 . With t h e same hypotheses, isomorphic t o I X ~ + ~ ( M )f o r 0 5 i 5 X o -

Whole Manifolds of Minimal Geodesics

$22.

n(M;p,q) based on two p o i n t s

So f a r we have used a p a t h space

special position. let

p,q

A s an example l e t

be a n t i p o d a l p o i n t s .

d e s i c s from

p

to

q.

M

p,q

a smooth manifold of dimension

n

Rn2

is

and contain two conjugate p o i n t s , each of m u l t i p l i c i t y

2n.

=

Xo

For any non-minimal geodesic must wind one and a h a l f times around

S"";

n, i n i t s i n t e r i o r .

T h i s proves the following.

Then t h e r e a r e i n f i n i t e l y many minimal geo-

I n f a c t t h e space

)

2.

t h e ( n + l ) - s p h e r e . Evidently t h e hypotheses a r e s a t i s f i e d with

i n some

Sn+l , and

be t h e unit sphere

d

L e t u s apply t h i s c o r o l l a r y t o the case of two a n t i p o d a l p o i n t s on

However, Bott has pointed out

p,q E K which a r e i n "general p o s i t i o n . " t h a t very u s e f u l r e s u l t s can be obtained by considering p a i r s

ni(0

of m i n i m a l geodesics forms

COROLLARY 2 2 . 3 . (The Freudenthal suspension theorem.) The homotopy group ni(Sn) i s isomorphic t o n i + l ( S n + l ) f o r i 5 2n-2.

which can be i d e n t i f i e d with the equator

Sn C S n + l . We w i l l see t h a t this space of minimal geodesics provides a

Theorem

P

space

-

n

a l s o implies that t h e homology groups of the loop

22.1

ad

a r e isomorphic t o those of

f a c t follows from

22.1

for example Hu, p .

206.

i n dimensions

5

Lo

-

2.

This

together with the r e l a t i v e Hurewicz theorem. Compare a l s o

(See

J . H. C . Whitehead, Combinatorial

homotopy I , Theorem 2 . ) The r e s t of

$22

w i l l be devoted t o the proof of Theorem 2 2 . 1 .

The

proof w i l l be based on t h e following l e m a , which a s s e r t s t h a t t h e condition " a l l c r i t i c a l p o i n t s have index

2 Xot'

remains t r u e when a f u n c t i o n i s

jiggled slightly. 4

Let

f a i r l y good approximation t o t h e e n t i r e loop space Let

M

n(Sn+').

be a complete Riemannian manifold, and l e t

points w i t h distance

p(p,q)

=

be a compact subset of the Euclidean space Rn;

K

a neighborhood of

p,c, E M

K;

be

U - R

f:

be a smooth f u n c t i o n such that a l l c r i t i c a l p o i n t s of

>

U

and l e t

be two

a.

let

f in

K

have index

lo.

THEOREM 2 2 . 1 . I f the space ad of minimal geodesics from p t o g i s a t o p o l o g i c a l manifold, and i f every non-minimal geodesic from p t o q has index 2 L o , then t h e r e l a t i v e homotopy group n i ( n , n d ) i s zero f o r o 5 i < h0.

LEMMA

If

22.4.

i s "close" t o

g: f,

-

U R i s any smooth f u n c t i o n which i n t h e sense t h a t

I t follows that t h e i n c l u s i o n homomorphism ni(O

is a n isomorphism for i group

ni(n)

5

Xo

i s isomorphic t o

-

d 2.

)

-

ni(n)

ni+l(M)

f o r a l l values of

S. T. Hu, "Homotopy Theory," Academic P r e s s , 1959, p. 817.1.)

uniformly throughout

But i t i s w e l l known that t h e homotopy

111;

i.

f o r some s u f f i c i e n t l y small constant

then a l l C r i t i c a l P o i n t s of

(Compare

together with

K,

(Note that the a p p l i c a t i o n , points.)

f

g

g

in

K

have index

2

E,

Xo.

1s allowed t o have degenerate c r i t i c a l p o i n t s .

In

w i l l be a nearby f u n c t i o n without degenerate c r i t i c a l

IV. APPLICATIONS

120

PROOF of 2 2 . 4 .

The f i r s t d e r i v a t i v e s of

g

a r e roughly described

U; which vanishes p r e c i s e l y a t t h e c r i t i c a l p o i n t s of

d e r i v a t i v e s of

g

121

To complete the proof of 2 2 . 4 , i t i s only necessary t o show t h a t

the i n e q u a l i t i e s ( * ) w i l l be s a t i s f i e d providing t h a t

by the s i n g l e r e a l valued f u n c t i o n

on

MANIFOLDS OF MINIMAL GEODESICS

§ 22.

can be roughly described by

n

g.

for s u f f i c i e n t l y small

The second

T h i s follows by a uniform c o n t i n u i t y argument

E .

which w i l l be l e f t t o t h e reader (or by the footnote above ) .

continuous f u n c t i o n s

We w i l l next prove an analogue of Theorem 2 2 . 1

for r e a l valued

functions on a manifold.

as follows.

Let

Let 1

eg(x)

of

g

eg(x)

F...

n eigenvalues of t h e matrix

denote the x

5

2

has index

2

such that each

ei(x)

a2

( 6-$) . xi x j

i f and only i f the number

A

f:

Thus a c r i t i c a l p o i n t

MC

+

R be a smooth r e a l valued f u n c t i o n with minimum i s compact.

f-'[O,cl

=

0,

LIDlMA 2 2 . 5 . If the s e t Mo of minimal p o i n t s i s a manifold, and if every c r i t i c a l p o i n t i n M - Mo has index 2 A o , then nr(M,M 0 ) = 0 f o r 0 5 r < X o .

i s negative.

ek(x)

M

e x follows from the f a c t that the 63 * A-th eigenvalue of a symmetric matrix depends continuously on t h e matrix The c o n t i n u i t y of t h e functions

.

T h i s i n t u r n follows from the f a c t that the r o o t s of a polynomial depend

continuously on the polynomial.

U C M.

(See § 1 4 of K . Knopp, "Theory of Functions,

P a r t 11," published i n the United S t a t e s by Dover, 1 9 4 7 . ) LO m ( x ) denote t h e l a r g e r of the two numbers kg ( x ) and -eg ( x ) . g Similarly l e t mf(x) denote t h e l a r g e r of t h e corresponding numbers k f ( x )

Let

and index mf(x)

The hypothesis that a l l c r i t i c a l p o i n t s of

-e)(x).

2

>

implies that

A.

0

for a l l x

Let

6

>

i s so c l o s e t o

0

f

E

AO

-ef ( x )

>

0

whenever

kf(x)

f

= 0.

in

K

PROOF:

have

F i r s t observe t h a t

Mo

I n f a c t Harmer has proved that any manifold

neighborhood r e t r a c t .

mf

on

K.

absolute neighborhood r e t r a c t s , A r k i v f o r Matematik, V o l . 389-408.)

Replacing

i s joined t o the corresponding p o i n t of Mo

U

by a unique m i n i m a l geodesic.

Let '1

Thus

h: (Ir,Tr) be any map.

can be deformed i n t o

U

denote t h e u n i t cube of dimension

g

that

(1950), pp.

1

by a smaller neighborhood i f necessary, we may

U

assume that each point of

I n other words

Now suppose that

i s an absolute

Mo

(See Theorem 3.3 of 0 . Harmer, Some theorems on

K.

denote t h e minimum of

i s a r e t r a c t of some neighborhood

We must show that

h

-

r




Computation shows t h a t the

a r e odd i n t e g e r s .

0

non-zero eigenvectors of the l i n e a r transformation

i s e a s i l y seen t o be



n

...

-I i f and only i f

i s equal t o

and only if a 1

...

0

The i n n e r product

=

LEMMA 2 4 . 2 . Any non-minimal geodesic from i n O(2m) has index 2 2 m - 2 .

T(exp

*.. ...

...

J*

The proof i s similar t o that of 2 3 . 2 .

c o s na,

\

-I t h i s implies that

Together with t h e i d e n t i t y

J.

proof.

0

equal t o

/

denotes t h e transpose of

f o r some & , , a 2 ,... am -> that a, = . .. = = 1;

-?2* ..

-a with

=

J*

135

i s skew-symmetric, t h e r e e x i s t s an element

A

(

so that

THE ORTHOGONAL G R O W

$24.

IV. APPLICATIONS

134

I)

for each i

2)

f o r each

i

<


a

=

1

(3+1-2)

+

0

(ai - a j -

2)

.

j

a2

=

aj)2/ 4.

This l e a d s t o the formula

=

... -- & m = ’

For a non-minimal geodesic we have

2

are

2m

-

a,

s o that

2 3;

so t h a t

2.

2

hence

A

i s a complex s t r u c t u r e .

Conversely, l e t

J

This completes t h e proof.

be any complex s t r u c t u r e .

nal we have

JJ*

=

I

Since

J

i s orthogo-

NOW l e t u s apply Theorem 2 2 . 1 .

The two lennnas above, together w i t h

IV. APPLICATIONS

136

the statement t h a t a , ( n )

0 24.

i s a manifold imply the following.

a,

PROOF of 2 4 . 4 .

-

0,( n )

=

fli+,

Any p o i n t i n O(n)

expressed uniquely i n t h e form

THEOREM 2 4 . 3 ( B o t t ) . The i n c l u s i o n map (n) R O(n) induces isomorphisms of homotopy groups i n dimensions < n-4. Hence fii

THE ORTHOGONAL GROW

sgmmetric matrix. J

O(n)

exp A,

c l o s e t o the i d e n t i t y can be

where

Hence any p o i n t i n O ( n )

can be expressed uniquely as

137

J exp A,;

A

is a "small,"

skew-

c l o s e t o t h e complex s t r u c t u r e

where again

i s s m a l l and

A

skew.

for i .( n-4. ASSERTION Now we w i l l i t e r a t e t h i s procedure, studying t h e space of geodesics from

J

to

high power of

Let commute

*,

i n n l ( n ) ; and s o on.

-J

J

r # s.

anti-commutes w i t h

J,,

. . ., J k - l

be f i x e d complex s t r u c t u r e s on Rn

anti-commutes with

If A I

which anti-

i n the sense that JsJr

=

exp(J-'A J ) exp A

( J exp A ) 2

Therefore

+

=

J,

then

J-lA J

=

J - ' ( e x p A ) J exp A

.

hence

-A

Since

denote t h e s e t of a l l complex s t r u c t u r e s J

which anti-commute with t h e fixed s t r u c t u r e s

J,, . . . , J k - l .

A

J- A J

s o that

C l e a r l y each n , ( n )

C

... C n l ( n ) C

i s a compact s e t .

n a t u r a l t o define n 0 ( n )

t o be

O(n)

.

anti-commutes w i t h

J1,...,Jk-l

To complete t h e d e f i n i t i o n i t i s

=

-I

then the

=

I

.

-A

=

J.

J exp A

anti-commutes w i t h t h e complex s t r u c t u r e s

i f and only i f

commutes with

A

J1,...,Jk-l.

The proof i s similar and straightforward.

O(n)

Note that Assertions 1 and 2 b o t h put l i n e a r conditions on

**

Thus a neighborhood of

LEMMA 2 4 . 4 . Each n k ( n ) i s a smooth, t o t a l l y geodesic submanifold of O ( n ) . The space of minimal geodesics from Jf t o -Jf i n O,(n) i s homeomorphic t o n f + l ( n ) , f o r

A

J

i n %(n)

c o n s i s t s of a l l p o i n t s

A.

J exp A where

ranges over a l l small m a t r i c e s i n a l i n e a r subspace of the Lie algebra

T h i s c l e a r l y implies t h a t n k ( n )

O < f < k .

g.

i s a t o t a l l y geodesic submanifold of

O(n).

It follows that each component of

For t h e isometric r e f l e c t i o n of O ( n ) cally carry n k ( n )

A

ASSERTION 2 . C

( J exp A ) 2

i s s m a l l , this implies that 1

.

Thus we have n,(n)

Conversely i f

exp(J-lA J ) exp A

Jl,...,Jk-l.

Let n,(n)

-I.

=

=

above computation shows that

0

Suppose that t h e r e e x i s t s a t l e a s t one other complex s t r u c t u r e

DEFINITION.

n,(n)

i s a s y m e t r i c space.

i n a p o i n t of n k ( n ) w i l l automati-

to itself.

Now choose a s p e c i f i c p o i n t

2

exists a complex s t r u c t u r e J J = J@

w e see e a s i l y t h a t

cmutes with

*

These s t r u c t u r e s make Rn i n t o a module over a s u i t a b l e C l i f f o r d algebra. However, t h e C l i f f o r d a l g e b r a s w i l l be suppressed i n t h e following presentation.

**

J.

i s d i v i s i b l e by a PROOF:

which anti-commutes w i t h

on R n

n

A

2.

JrJs

for

Assume that

i s a complex s t r u c t u r e i f and only i f

J exp A

1.

A submanifold of a Riemannian manifold i s c a l l e d t o t a l l y geodesic if each geodesic i n t h e submanifold i s a l s o a geodesic i n l a r g e r manifold.

Jk.

Jk

E

nk(n),

and assume that t h e r e

which anti-commutes w i t h

...,Jk.

Setting

i s a l s o a complex s t r u c t u r e which anti-

A

However,

J1,

A

comutes w i t h

J1

,..., J k - l .

Hence t h e

formula

t d e f i n e s a geodesic f r m Jk

+

to

J k e x p ( n t A)

-Jk

i n llk(n).

Since t h i s geodesic i s

minimal i n o ( n ) , it i s c e r t a i n l y m i n i m a l i n n k ( n ) .

$24. THE ORTHOGONAL GROUP

IV. APPLICATIONS

136

a,(n).

Conversely, let 7 be any minimal geodesic from Jk to -Jk in

Cn/, as being a vector space gl4 over the quaternions H .

Setting y(t)

be the group of isometries of this vector space onto itself. Then n,(n)

=

Jk exp(nt A),

it follows from

a complex structure, and from Assertions

... ,Jk-l

Jl,

REMMRK.

geodesic

y

1,2

and anti-commutes with Jk.

that A

that A is

24.1

Before going further it will be convenient to set n

It follows easily that Jip

The point Jip

E

y($)

of the geodesic.

PROOF: Any complex structure J3 e n 3 ( 1 6 r )

In order to pass to a stable situation, note that nk(n)

of

can be

. . .,Ji

structure J f3 21; =

1,

.

on Rn’

Then each J

E

nk(n)

@

=

R’6r into two mutually orthogonal subspaces V, and V,

as fol-

are

1.

b t V, C R16r be the subspace on which JlJ2J3 equals + I; and let V,

JA for

...,k - 1 .

be the orthogonal subspace on which it equals -I. Then clearly

DEFINITION. Let SFaces a,(n),

H4r

determines a splitting

J,J2J3J,J2J3 equal to + I. Hence the eigenvalues of J,J,J3

determines a complex

on Rn @Rn’ which anti-commutes with J,

16r.

lows. Note that JlJ2J3 is an orthogonal transformation with square

imbedded in nk(n+nt) as follows. Choose fixed anti-commuting complex structures J;,

=

LEMMA 24.6 - (3). The space a,( 1 6 r ) can be identified with the quaternionic Grassmam manifold consisting of quaternionic subspaces of

flk+l(n) which corresponds to a given

has a very simple interpretation: it is the midpoint

Let Sp(n/4)

can be identified with the quotient space U(n/2)/ Sp(n/‘c).

commutes with

This completes the proof of 24.4.

belongs to Qk+l(n).

CY

139

nk

denote the direct limit as n-r

m

=

of the

with the direct limit topology.

(I.e., the fine topology.) The space 0 =nois called the infinite orthogonal group.

V, f3 V,.

Since JlJ,J3

that both V, and V,

.

commutes with J, and J,

it is clear

are closed under the action of J, and J,.

Conversely, given the splitting H4k

=

V,

@

orthogonal quaternionic subspaces, we can define J3

V, E

into mutually

n3(l6r) by the

identities It is not difficult to see that the inclusions give rise, in the limit, to inclusions

nk+l ank .

-r

0

I

n,(n)

-r

-

THEOREM 24.5. For each k 2 0 this limit map fik+l 0 ak is a homotopy equivalence. Thus we have isomorphisms nh 0

E

nhw1

n, z

nh-,

n,

g

...

=

J31V2

=

-J1J,IV1 JlJ21V2

.

T h i s proves Lemma ?4.6 - ( 3 ) .

The space “lnh-l

J3P1=

*

3

is awkward in that it contains components of

(16r)

varying dimension. It is convenient to restrict attention to the component

The proof will be given presently.

of largest dimension: namely the space of 2r-dimensional quaternionic sub-

Next we will give individual descriptions of the manifolds for

k

= 0,1,2,

ak(n)

...,8. n,(n)

this way, so that dimHV1

is the orthogonal group.

nl(n) is the set of all complex structures on Rn

.

Given a fixed complex structure J, we may think of Rn as being a vector space Cn/2 over the complex numbers. n,(n)

spaces of H4r. Henceforth, we will assume that J a

can be described as the set of “quaternionic structures” on

the complex vector space Cn’2.

Given a fixed J2 en2(n)

We m y think of

=

dimHV2

3

has been chosen in

= 217.

LEMMA 24.6 - ( 4 ) . The space n4(16r) can be identified with the set of all quaternionic isometries from Vl to V,. Thus n4(16r) is diffeomorphic to the symplectic group S p ( 2 ~ ) .

PROOF: Given J,, E n4( 16r) Commutes with JlJ,J3

.

Hence J3J4

note that the product J3J4 antimaps V,

to V2 (and V,

to V 1 )

$24. THE ORTHOGONAL GROUP

IV. APPLICATIONS

140

W i n t o two mutually orthogonal subspaces.

s p l i t t i n g of Since

J3J4 commutes w i t h

and

J,

we see that

J,

i t i s e a s i l y seen that

-V,

Conversely, given

identities:

his proves 24.6 - ( 4 )

-

REMARK.

Jil T

=

J41V2

= -T-’

-

phisms of

PROOF:

Given

commutes with

J,J,J3

Let

Since

J2

W C V,

( 5)

=

X C W, i t i s not hard t o see that

J 6 i s unique-

If O(2r) C U(2r)

keeping

W

X

denotes t h e p o u p of complex automor-

f i x e d , then t h e q u o t i e n t space

U( 2 r ) / O ( 2r)

can

a,(

J5

E

a,( 1 6 r )

note that t h e transformation

J1J4J5

+ I . Thus J,J4J5 maps V,

and has square

V,

J,J4J5 coincides with

J,J4J5, i t follows that

J2W

J,J4Jg equals

c

V,

Given

J,J6J7

commutes with

square

+ I . Thus

J1J6J7

+ I.

equals

J7,

anti-commuting with

J,J2J3, w i t h

J,,

...,J6

J,J4J5, and with

J , J 6 J 7 determines a s p l i t t i n g of

orthogonal subspaces:

into

i n t o two mutually orthogonal sub-

be the subspace on which

anti-commutes w i t h

PROOF:

X,

(mere

J,J6J7 equals

- I ) . Conversely, given

X, C X

X

note that

J2J4J6; and has

i n t o two mutually

+I) and

X,

(where

it can be sharn that

J7

I s uniquely determined.

is

T h i s space a7(16r), l i k e n3(16r),has components of varying dimen-

-I. Clearly

sion.

W.

Again we w i l l r e s t r i c t a t t e n t i o n t o t h e component of l a r g e s t dimen-

sion, by assuming that Conversely, given t h e subspace

dim X ,

W, i t i s n o t d i f f i c u l t t o show t h a t

REMARK. morphisms of

V,

If

U ( 2 r ) C Sp(2r)

keeping W

ASSERTION.

denotes t h e group of quaternionic auto-

f i x e d , then t h e q u o t i e n t s w c e

can be i d e n t i f i e d with n 5 ( 1 6 r )

Sp( 2 r ) / U( 2 r )

commutes b o t h with Since

J6

.

E

n6(16r) note that the transformation

J,J2J3 and with

(J,J4J6)2

=

r.

=

I,

J 1 J 4 J g . Hence

it follows that

J,J4J6 maps

J2J4Jg

The l a r g e s t component of

n7(1 6 r )

i s diffeomorphic t o R2’.

a

UMNA 2 4 . 6 - ( 8 ) . The space a , ( 1 6 r ) can be i d e n t i f i e d w i t h t h e s e t of a l l r e a l i s o m e t r i e s from X, t o X,.

PROOF. c-utes

Given

dim X p

t h e Grassmann manifold c o n s i s t i n g of r- dimensional subspaces of

can be i d e n t i f i e d LEMMA 2 4 . 6 - (6). The space a6( 161’) with the set of a l l real subspaces X C W such that W s p l i t s as t h e orthogonal sum X fB J , X . PROOF.

=

Thus we obtain:

J 5 i s uniquely determined.

itself.

-I.

L F M 24.6 - ( 7 ) . The space i 6 r ) can be i d e n t i f i e d with the r e a l Grassrnann manifold c o n s i s t i n g of real subspaces of X s Rzr.

.

p r e c i s e l y the orthogonal subspace, on which JIW

w i l l be the orthogonal

+ I . Then J , X

I

i t s e l f ; and determines a s p l i t t i n g of spaces.

be t h e

be i d e n t i f i e d with a6(16r).

J3

The space n5( 1 6 r ) can be i d e n t i f i e d with t h e s e t of a l l v e c t o r spaces W C V, such that ( I ) W i s closed under J , ( i . e . , W i s a complex vector space) and (2) V1 s p l i t s a s t h e orthogonal sum W 8 J, W .

LEMMA 24.6

X C W

Let

l y determined.

J 4 i s uniquely determined by the

J41V,

J2J4J6 equals

subspace on which i t equals

Conversely, given any such isomorphism

is a quaternionic isomorphism. T : V,

subspace on which

-+v,

J 3 J 4 1 V , : V,

141

J,J4J6 W

determines a

into

&nce

with

JrJ8

determines

If

J8

E

a8(16r) then t h e orthogonal transformation

J,J2Jj, J,J4J5,

maps

X,

and

J2J4J6; b u t anti-commutes w i t h

isomorphically onto

X,.

J7J8 J,J6J7.

Clearly this isomorphism

J8 uniquely.

Thus w e see that

n8( 16r)

i s diffeomorphic t o t h e orthogonal

IV. APPLICATIONS

142

$24. THE ORTHOGONAL GROUP

group* O ( r ) .

DEFINITION.

Let us consider this diffeomorphism

r

the l i m i t a s

-+O(r),

and pass t o

( J , , . . . , Jk) -space i f no proper, non-

i s a minimal

V

J l ,...,

t r i v i a l subspace i s closed under t h e a c t i o n of

It follows that O8 i s homeomorphic t o t h e i n f i n i t e

-c m .

orthogonal group 0 .

a8( 16r)

143

Jk. Two such

and

minimal v e c t o r spaces a r e isomorphic i f t h e r e i s a n isometry between them

Combining t h i s f a c t with Theorem 24.5, we obtain the

J,,. . . , Jk.

which commutes with the a c t i o n of

following. LEMMA 2 4 . 8 ( B o t t and Shapiro) . F o r k f 3 (mod 4), any two minimal ( J 1 , . , . , J k ) vector spaces a r e isomorphic.

THEOREM 2 4 . 7 ( B o t t ) . The i n f i n i t e orthogonal group 0 has t h e same homotopy type as i t s own 8 - t h loop space. Hence the homotopy group n i 0 i s isomorphic t o ni+8 0 f o r i

Sp

If

= a 4denotes

argcunent a l s o shows t h a t 0 and t h a t

n R n R Sp,

Rnnn 0 .

Sp

%

20.

L

The proof of 24.8 follows t h a t of 2 4 . 6 .

complex numbers o r the quaternions.

has t h e homotopy type of t h e & - f o l d loop space

For

has t h e homotopy type of the & - f o l d loop space J3

1

0 1

"i SP

1

z2 0

3

Z

Z

4 5

0

Z2 Z2

6 7

T h i s g i v e s t w o non-isomorphic minimal

-J,J2.

H

to

=

5,6

complex s t r u c t u r e s spaces.

k

For

The dimension i s equal t o

HI.

and H

HI. @

HI, with

8.

we o b t a i n t h e same minimal vector space

H

@

HI. The

we again o b t a i n t h e same space, b u t t h e r e a r e two J 7 i s equal t o

p o s s i b i l i t i e s , according a s

Z

H

J5,J6 merely determine p r e f e r r e d complex or r e a l sub-

7

=

Call these

4.

a minimal space must be isomorphic t o

4

=

k

For

0

Z

However, t h e r e a r e two p o s s i b i l i t i e s , according as

+J1J2 or

k

J3J4 mapping

0

0 0

a

2

a minimal space i s s t i l l a 1-dimensional v e c t o r space

3

=

i s equal t o

For

0

2

or

= 0,1,

Clearly any two such a r e isomorphic.

spaces, both with dimension equal t o

0

z2

k

over t h e quaternions.

The a c t u a l homotopy groups can be tabulated as follows. n i 0

k

m i n i m a l space i s j u s t a 1-dimensional v e c t o r space over the reals, the

t h e i n f i n i t e symplectic group, then t h e above

i modulo 8

For

+J1J2J3J4JgJ6o r t o

-J1J2J3J4J5J6.Thus i n this case t h e r e a r e two non-isomorphic minimal

vector spaces;

The v e r i f i c a t i o n t h a t t h e s e groupsare c o r r e c t w i l l be l e f t t o the reader.

SP(1) i s a ?,-sphere, and that

(Note that

SO(3)

For

i s a p r o j e c t i v e 3-space.) with

The remainder of this s e c t i o n w i l l be concerned with the proof of

Consider a Euclidean v e c t o r space

*

For

k

Jl,

>

8

V

with anti-commuting complex

>

k

Periodically.

8

L

L'.

and

a minimal v e c t o r space must be isomorphic t o L

onto

L

L',

L'. The dimension i s equal t o 1 6 .

i t can be shown that the s i t u a t i o n r e p e a t s more o r l e s s

However, t h e cases

mk

Let

...,J k' i t can be shown that

i

8

=

mapping

For

It i s f i r s t necessary t o prove an a l g e b r a i c lemma.

Theorem 24.5.

structures

J7J8

c a l l these

k

k

5 8 w i l l s u f f i c e f o r our purposes.

denote t h e dimension of a m i n i m a l

...,J k ) - v e c t o r

(Jl,

space.

From t h e above d i s c u s s i o n we see t h a t : ak(l 6r)

I n f a c t any a d d i t i o n a l complex s t r u c t u r e s

i s diffeomorphic t o

ak-8(r).

Jg,Jlo,.. . ,Jk on R16r

give J8Jk on

...,

r i s e t o anti-commuting complex s t r u c t u r e s J8Jg, J 8 J l 0 , J 8 J l l , and hence t o an element of ak-8(r).However, f o r our purposes i t Xl; w i l l be s u f f i c i e n t t o s t o p w i t h k = 8.

For

k

>

8

mo

= 1,

m4

=

ml

mg

=

= 2,

m6

REMARK.

These numbers

=

mk

i t can be shown that

mk

m2 = m3 m7 =

=

=

8, m8

4, =

16.

16m~-~.

a r e c l o s e l y connected with t h e problem

of c o n s t r u c t i n g l i n e a r l y independent v e c t o r f i e l d s on spheres. example t h a t

J,,

...,Jk

Suppose f o r a r e anti-commuting complex s t r u c t u r e s on a vector

F IV. APPLICATIONS

144

space

of dimension

V

u

f o r each u n i t vector

d i c u l a r t o each other and vector f i e l d s on an

the

V

E

r can be any p o s i t i v e i n t e g e r .

Here

rmk.

to

Thus w e obtain

u.

linearly

k

For example we o b t a i n

(rmk-l)-sphere.

f i e l d s on a

(4r-l)-sphere;

f i e l d s on a

( 1 6 r - I ) - s p h e r e ; and s o on.

and Radon.

..., Jku

J l u , J2u,

vectors

k

7 v e c t o r f i e l d s on an

Then

eigenspaces of

A;

are perpen-

eigenvalues of

AIMh;

independent 3

(8r-l)-sphere;

...,

we see that

Commentarii Math. Helv. V o l . 1 5 ( 1 g 4 3 ) , pp. 358-366.) J .

.'

F . Adam has r e c e n t l y proved that t h e s e estimates a r e b e s t p o s s i b l e . k f 2 (mod 4 ) .

PROOF of Theorem 24.5 for minimal geodesics from

of n k ( n )

at

T

c o n s i s t s of a l l matrices

J

i s skew

2)

A

anti-commutes with commutes with

t

corresponds t o a geodesic

J,,

-

T

-

T.

can compute

E

T

Since n k ( n ) KA

e x i s t s by

j u s t as before.

from J

(nL4)

A.

to

A given

A

E

T

=

-+

24.8.

-J i f and only i f

mutes w i t h

O(n),

we

(-A2B + PABA

-

BA2)/4

-,J

S p l i t t h e v e c t o r space Rn

eigenvalue

so as

J,

,...,J k - l , J

and

A.

a l l equal, except f o r s i p . *

*

for a

e x p ( n t A)

.

Then t h e eigenvalues of For otherwise

Mh

BIMj

B

Mh

as follows.

T

Let

to M

j

. . . Q M,

of

must be

would s p l i t as a sum of

BIMj

We are d e a l i n g with t h e complex eigenvalues of a r e a l , skew-symmetric transformation. Hence these eigenvalues a r e pure imagiinars; and occur i n conjugate p a i r s .

Bw)

J

Mh

B

B

on

T.

Since

w

M

E

I.

J.

BIMh

com-

It follows e a s i l y that

J1,...,Jk-l

and anti-

i s an eigenvector of

+ 1

v

E

KA Mh

corresponding t o the then

( - A ~ B + ~ A B A- m 2 ) v 2

2

( a Bv + 2a Ba v + Bah)v j j h

Ti 1 (aj

+ ah)2 BV v

E

M

;

j*

The number of minimal spaces

Mh C Rn

i s given

&n must be 2 3 .

T h i s proves t h e following

k f 2 (mod 4) ) :

ASSERTION-

-> ( 3 + 1 1 2 / 4

B(Mh.

T.

E

For otherwise we would have a minimal geodesic. (always f o r

Such an isomorphism

Mj.

For a t least one of t h e s e t h e i n t e g e r

= n/mk+l.

where the bar i n -

It i s a l s o c l e a r that

(ah + a j ) 2 / 4 . For example i f

Now l e t u s count.

Mj;

to

for v c Mh,

and a similar computation a p p l i e s f o r

s

=

a l s o commutes with

=

by

...,k-1; +JIB .

and anti-commutes w i t h

=

on

-

be the negative a d j o i n t of

i s skew-symmetric.

Thus

J.

&j

(ah - a j -

2)

for

.

KA. 1

be

Then

.

(ah - aj)2B/4

we o b t a i n a n eigenvalue

l i n e a r transformation, and has a w e l l defined complex determinant which w i l l be denoted by

of t h e l i n e a r transforma-

an easy computation shows that

Jk-iv

The condition

J , , J , ,..., J k - l

Choose a base p o i n t composition

J,J2

=

B

minimal it follows from 24.8 t h a t t h e r e e x i s t s an isornet-y

c c

defining

for

-+

Can be constructed much as b e f o r e .

rT

iv

an eigenvector

be t h e f i x e d anti-commuting complex s t r u c -

i n t o an

Make Rn

-

we c o n s t r u c t a map

n&(n)

i ( a l +...+a r ) m k / 2 .

tion

i n g subspace n k + , ( n ) has only f i n i t e l y many.

f

h # j

Now f o r each

has i n f i n i t e l y many components, while the approximat-

To describe t h e fundamental group

i s equal t o

Therefore t h e t r a c e of A

has an i n f i n i t e c y c l i c fundamental

be ascribed t o the f a c t . t h a t n,(n) group.

The d i f f i c u l t y i n t h i s case may

IV. APPLICATIONS

148

149

Now l e t u s r e s t r i c t a t t e n t i o n t o some f i x e d component of T h a t i s l e t us l o o k only a t matrices c

such that

A

trace A

=

nClk(n).

icmk/2

where

i s some constant i n t e g e r . Thus t h e i n t e g e r s 1) 2)

3)

al,

= a2 I ... a1 +...+ a = Max Iahl 2 3 a1

h

ar c,

satisfy E 1

(mod 2 ) , ( s i n c e

=

The o b j e c t of this appendix w i l l be t o give a n a l t e r n a t i v e v e r s i o n

-I),

and

f o r t h e f i n a l s t e p i n t h e proof of Theorem 1 7 . 3 ( t h e fundamental theorem

ah i s equal t o

Let

- 3.

-9 the sum of t h e negative

ah and

exp(nA)

( f o r a non-minimal g e o d e s i c ) .

Suppose f o r example that some the positive

APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION

...ar

ah.

p

of Morse t h e o r y ) .

a

space

be t h e sum of

n(M;pJq),

=

a

Given t h e s u b s e t s

nal

C

R

C

na2

C

...

and given t h e information that each

of t h e p a t h

nai

has t h e

homotopy type of a c e r t a i n CW-complex, we wish t o prove that t h e union

Thus

R

a l s o has t h e homotopy type of a c e r t a i n CW-complex. P - 9

hence

2r

2p 2~

2

+ c.

=

c,

4X

1 %>

2 2p 2 r

+

q

>

r,

More g e n e r a l l y consider a t o p o l o g i c a l space

Now

(ah

X, C X1 C X2 C

-

aj

-

2)

+ c;

1

>

aj

hence

~

ah where

follows that t h e component of

r

=

>

n/mk

n %(n)

X (ah - ( - 3 )

- 3)

P

,

with

n.

=

( n ) , as

n

x,

It

-

m

=

xo

XIOJll

" xlx

[1,21

Xi?

x2 x [ 2 , 3 1

u

u

... .

X x R.

T h i s i s t o be topologized as a subset of

. DEFINITION.

Passing t o t h e d i r e c t l i m i t , w e o b t a i n a homotopy equivalence on each component.

To w h a t e x t e n t i s t h e homotopy type of

I t i s convenient t o consider t h e i n f i n i t e union

i s approximated up t o higher and

higher dimensions by t h e corresponding component of

of subspaces.

determined by t h e homotopy types of t h e

0

tends t o i n f l n i t y

...

and a sequence

X

the sequence

T h i s completes t h e proof of 24.5.

p(x,.r)

=

[Xi)

X

i s t h e homotopy d i r e c t l i m i t of

i f t h e p r o j e c t i o n map

p : X,

+

X,

defined by

i s a homotopy equivalence.

x,

EXAMPLE some Xi,

We w i l l say that

1.

and that

Suppose that each p o i n t of X

i s paracompact.

X

l i e s i n t h e i n t e r i o r of

Then using a p a r t i t i o n of u n i t y one

can c o n s t r u c t a map f :X+R tr

6*

SO that

pondence

f(x)

2

i+l

for

x q! XiJ

x + ( x , f ( x ) ) maps

and

X

20

for a l l

x.

Now t h e corres-

homeomorphically onto a subset of

X

i s c l e a r l y a deformation r e t r a c t . and

f(x)

Therefore

p

X,

which

i s a homotopy equivalence;

i s a homotopy d i r e c t l i m i t . EXAMPLE 2.

w i t h union

X.

Let

Since

X

be a CW-complex, and l e t t h e Xi be subcomplexes

P : Xz

-

X

induces isomorphisms of homotopy groups

i n a l l dimensions, i t f o l l o w s f r m W t e h e a d ' s theorem that X direct l i m i t .

i s a homotopy

150

1,

APPENDIX

151

HOMOTOPY CF A MONOTONE U N I O N

I

The u n i t i n t e r v a l

EXAMPL3 3.

[0,11

l i m i t of the sequence of closed subsets

is

LO1

not

the homotopy d i r e c t

a s follows (where it i s always t o be understood that

[l/i,ll.

n

0

and

5 t 5 1,

,...) .

= 0,1,2

The main r e s u l t of this appendix i s the following. THEOREM A . Suppose that X i s t h e homotopy d i r e c t l i m i t of (Xi] and Y i s t h e homotopy d i r e c t l i m i t of ( Y i ) . Let f : X + Y be a map which c a r r i e s each Xi i n t o Yi by a homotopy equivalence. Then f i t s e l f i s a homotopy equivalence.

f

Taking u

to Assuming Theorem A, t h e a l t e r n a t i v e proof of Theorem 17.3 can be given as follows.

of homotopy equivalences.

C K,

Since

n

C K2 C

K

U nai

=

Define

Suppose that

(obtained by r e s t r i c t i n g f ) f

and

=

Yi

fz

-

+

K

=

U Ki

a r e homotopy

homotopy inverse

g : X,

t

Y,

by

f,(x,t)

=

fi : Xi+

i s homotopic t o t h e i d e n t i t y .

(x,n+2t)

hl(x,n+t)

E

X,+l

x [n+ll

=

fn,

h:

=

identity.

o < t < $

for

35t 5

1

.

In fact a

$ can be defined by the formula o < t < +

(x,n+2t)

hLt"

=

hl(x,n+l)

Proof that the composition

Yi Of

X,.

.

hlg

=

(x,n+l)

.

i s homotopic t o the i d e n t i t y map

Note that

o < t < t

t < t < h

c P

h l ( x , n +3 ? - t) Define a homotopy

$ : %'Xn

for

i s a homotopy equivalence.

hl

=

However counter-

let

%:%-%

on t h e other hand has the following

=

hl(x,n++)

We must prove

examples can be given.

Define the hornotopy

which i s c l e a r l y homotopic

(x,n+4t)

f c must a c t u a l l y be homotopic t o the i d e n t i t y .

be a one-parameter family of mappings, w i t h

X,

+

This i s w e l l defined since

(f(x),t).

i s a homotopy equivalence.

and that each map

-

f

g(x,n+t)

Under these conditions i t would be n a t u r a l t o conjecture

For each n

X,

%

:

hl( x , n + t )

c i s a homotopy equivalence. FEWARK.

that

Xi

:

f,

It i s c l e a r l y s u f f i c i e n t t o prove t h a t

tht

hl : X,

We w i l l show that any such map

i s a l s o a homotopy equivalence.

PROOF of Theorem A.

CASE 1 .

The mapping

hO

proper t i e s :

...

d i r e c t l i m i t s (compare Examples 1 and 2 above), it follows t h a t t h e l i m i t

-

t h i s d e f i n e s a map

R e c a l l that we had constructed a commutative diagram

KO

mapping

f,.

= 0

Q : Xc

+

%

as follows.

.

h < t < 1

For

0

5u