Elementary Differential Topology. (AM-54), Volume 54 9781400882656

Table of contents :
CONTENTS
PREFACE
Chapter I. Differentiable Manifolds
§1. Introduction
§2. Submanifolds and Imbeddings
§3. Mappings and Approximations
§4. Smoothing of Maps and Manifolds
§5. Manifolds with Boundary
§6. Uniqueness of the Double of a Manifolds
Chapter II. Triangulations of Differentiable Manifolds
§7. Cell Complexes and Combinatorial Equivalence
§8. Immersions and Imbeddings of Complexes
§9. The Secant Map Induced by f
§10. Fitting Together Imbedded Complexes
REFERENCES
INDEX OF TERMS

Citation preview

Annals of Mathematics Studies Number 54

ELEMENTARY DIFFERENTIAL TOPOLOGY BY

James R. Munkres Lectures Given at Massachusetts Institute of Technology Fall, 1961

REVISED

EDITION

P R IN C E T O N , N E W JER SEY P R I N C E T O N U N I V E R S I T Y PR ESS

1966

Copyright © 1963, 1966, by Princeton University Press A L L RIGHTS RESERVED

Printed in the United States of America Revised edn., 1966 Second Printing, with corrections, 1968 Third Printing, 1973

To the memory o f Sumner B. Myers

PREFACE

Differential topology may be defined, as the study of those proper­ ties of differentiable manifolds which are invariant under differentiable homeomorphisms.

Problems in this field arise from the interplay between the

topological, combinatorial, and differentiable structures of a manifold. They do not, however, involve such notions as connections, geodesics, curva­ ture, and the like; in this way the subject may be distinguished from dif­ ferential geometry.

One particular flowering of the subject took place in the 1930’s, with work of H. Whitney, S. S. Cairns, and J. H. C. Whitehead.

A second

flowering has come more recently, with the exciting work of J. Milnor, R. Thom, S. Smale, M. Kervaire, and others.

The later work depends on the

earlier, of course, but differs from it in many ways, most particularly in the extent to which it uses the results and methods of algebraic topology, The earlier work is more exclusively geometric in nature, and is thus in some sense more elementary. One may make an analogy with the discipline of Number Theory, in which a theorem is called elementary if its proof involves no use of the theory of functions of a complex variable — otherwise the proof is said to be non-elementary. As one is well aware, the terminology does not reflect the difficulty of the proof in question, the elementary proofs often being harder than the others. It is in a similar sense that we speak of the elementary part of differential topology.

This is the subject of the present set of notes.

Since our theorems and proofs (with one small exception) will in­ volve no algebraic topology, the background we expect of the reader consists of a working knowledge of:

the calculus of functions of several variables

and the associated linear algebra, point-set topology, and, for Chapter II, the geometry (not the algebra) of simplicial complexes.

Apart from these

topics, the present notes endeavor to be self-contained. The reader will not find them especially elegant, however. vii

We are

not hoping to write anything like the definitive work, even on the most elementary aspects of the subject.

Rather our hope is to provide a set of

notes from which the student may acquire a feeling for differential topology, at least in its geometric aspects.

For this purpose, it is necessary that

the student work diligently through the exercises and problems scattered throughout the notes; they were chosen with this object in mind. The word problem is used to label an exercise for which either the result itself, or the proof, is of particular interest or difficulty. Even the best student will find some challenges in the set of problems. Those problems and exercises which are not essential to the logical continui­ ty of the subject are marked with an asterisk. A second object of these notes is to provide, in more accessible form than heretofore, proofs of a few of the basic often-used-but-seldomproved facts about differentiable manifolds.

Treated in the first chapter

are the body of theorems which state, roughly speaking, that any result which holds for manifolds and maps which are infinitely differentiable holds also if lesser degrees of differentiability are assumed.

Proofs of these

theorems have been part of the "folk-literature" for some time; only recent­ ly has anyone written them down.

([8 ] and [9].)

(The stronger theorems of

Whitney, concerning analytic manifolds, require quite different proofs, which appear in his classical paper [15].) In a sense these results are negative, for they declare that nothing really interesting occurs between manifolds of class C°°.

C1

and those of class

However, they are still.worth proving, at least partly for the tech­

niques involved. The second chapter is devoted to proving the existence and unique­ ness of a smooth triangulation of a differentiable manifold. follow J. H. C. Whitehead [1 4 ], with some modifications.

In this, we

The result itself

is one of the most useful tools of differential topology, while the tech­ niques involved are essential to anyone studying both combinatorial and differentiable structures on a manifold.

The reader whose primary interest

is in triangulations may omit §4 , §5, and §6 with little loss of continuity. We have made a conscious effort to avoid any more overlap with the lectures on differential topology [4 ] given by Milnor at Princeton in 1958 viii

them was necessary.

It is for this reason that we omit a proof of Whitney’s

imbedding theorem, contenting ourselves with a weaker one.

We hope the

reader will find our notes and Milnor’s to be useful supplements to each other.

Remarks on the revised edition Besides correcting a number of errors of the first edition, we have also simplified a few of the proofs. In addition, in Section 2 we now prove rather than merely quote the requisite theorem from dimension theory, since few students have it as part of their background.

The reference to Hurewicz and Wallman's book was

inadequate anyway, since they dealt only with finite coverings. Finally, we have added at the end some additional problems which exploit the Whitehead triangulation techniques.

The results they state have

already proved useful in the study of combinatorial and differentiable struc­ tures on manifolds.

CONTENTS P REFACE.............................................................. vii Chapter I . Differentiable Manifolds

§1

.

Introduction.....................................................

3

§2.

Submanifolds and Imbeddings.....................................17

§3 .

Mappings and Approximations ..................................... 25

§4 .

Smoothing of Maps and Manifolds.................................39

§5.

Manifolds with Boundary........................................ 47

§6.

Uniqueness of the Double of a Manifold..........................

Chapter II.

63

Triangulations of Differentiable Manifolds

§7 .

Cell Complexes and Combinatorial Equivalence.....................69

§8.

Immersions and Imbeddings of Complexes........................... 79

§9.

The Secant Map Induced by

f .....................................90

§10. Fitting Together Imbedded Complexes .............................

97

REFERENCES..........................................................

109

INDEX OF TERMS......................................................

111

xi

ELEMENTARY DIFFERENTIAL TOPOLOGY

CHAPTER I.

DIFFERENTIABLE MANIFOLDS

§1 .

Introduction.

This section is devoted to defining such basic concepts as those of differentiable manifold, differentiable map, immersion, imbedding, and diffeomorphism, and to proving the implicit function theorem. We consider the euclidean space sequences of real numbers, i > m; Then

||x|| =

1;

|x|.

is the set with of all

Bm ,

|x| < r.

m-tuples

Sm “1

the set with

x

m

by

||x|| < 1 ;

an open subset If of

x

h

that each point of M of

rf11 or of

:U

-*■ H111 (or Rm ) is

the pair

Rm

is non-bounded.

ly used only when

Bd M

with Cm (r)

as simply the space

M

is a Hausdorff spa

There is an in­

M,

(U,h)

is open in

is called a boundary point of

points is called the boundary of M

Rm

a homeomorphism of the neighborhood

on M. If h(U)

we say

and

Rm .

coordinate neighborhood x

||x||,

has a neighborhood homeomorphic with

in H211 or Rm ,

then

xm > 0.

where no confusion will arise.

with an open set

Rm_1 ,

for

and the r-cube

Definition. A (topological) manifold

such

= o

for which

with, a countable basis, satisfying the following condition: teger

i

is the subset of

Often, we also consider

(x1,...,xm ),

1.1

Rm

J((x1)2 + ... + (x“ )2)

The unit sphere

the unit ball

such that

if1 is the subset of

Rm_1 C if11 C Rm . We denote by

as the space of a^l infinite

x = (x , x , ...),

euclidean half-space

max |x^|,

Rm

1 2

M,

denoted by

if1

U

is often called a and

h(x)

lies in

and the set of all such Bd M.

(In the literature, the word

If

Bd M

is empty,

manifold is common­

is empty; the more inclusive term then is manifold-

I . DIFFERENTIABLE MANIFOLDS with-boundary.) denoted by also use

The set

Int M. Int A

(If

to mean

M - Bd M A

is called the interior of

is a subset of the topological space

X - C£(X-A),

in map

and is X,

we

but this should cause no confusion.)

To justify these definitions, we must note that if and

M,

^

: U-, -► if1

h2h2:: UU22-► -► if1 if1 are are homeomorphisms homeomorphisms of of neighborhoods neighborhoods of of xx with with open open sets sets if1 if1 , ,and and if if h-jh"1

h1 h1(x) (x) lies lies in in Rm_1 Rm_1,

so , does so does hg(x) :hg(x) For : otherwise, Forotherwise, the

would give a homeomorphism of an open set in

borhood of the point ly not open in

p = h1 (x)

Rm ,

in

if1.

Rm

with a neigh­

The latter neighborhood is certain­

contradicting the Brouwer theorem on invariance of

domain [3, p. 95]. One may also also verify verify that that the the number number

mm

is is uniquely uniquely determined determined by by

M;it it is is called called the the dimension dimension of of M, M,and and M M isiscalled calledananm-manifold. m-manifold. This may be done either by using the Brouwer theorem on invariance of domain, or by applying the theorem of dimension theory which states that the topo­ logical dimension of

M

is

m

[3 , p. k6] .

latter theorem we need to know that

M

Strictly speaking, to apply the

is a separable metrizable space; but

this follows from a standard metrization theorem of point-set topology [2 , p. 75]. Because

M

is locally compact, separable, and metric,

paracompact [2, p. 7 9 ]. open covering covering (1) ment of (2 )

M

Q

of

M

is

We remind the reader that this means that for

M,

there is another such collection

(& of open

any any sets sets

such that

The collection

(B Is is a refinement of the first, i.e., every ele­

(B is contained in an element of The collection

G ,.

(B is locally-finite, i.e., every point of

a neighborhood intersecting only finitely many elements of In passing, let us note that because finite open covering of

|

(a)

Exercise.

bounded

m-1 manifold or

M

M

M

has

(B .

has a countable basis, any locally-

must be countable.

If

M

is an m-manif old, show that

Bd M is aa non­ non­

is empty.

(b) Exercise. Exercise. Let Let MM be be an anm-manif m-manifold oldwith withnon-empty non-emptyboundary. boundary. Let

Mq = M x 0

and

M1 = M x 1

be be two two copies copies of of M. M.

The The double double of of M, M,

th

5

§ 1 . INTRODUCTION

denoted by D(M) , is the topological space obtained from MQ U M1 by identi­ fying

(x,0)

with

(x,1)

for each x

in Bd M.

Prove that D(X) is a non­

bounded manifold of dimension m. (c) n,

Exercise.

If M

and

respectively, then M x N

N

are manifolds of dimensions

is a manifold of dimension m + n,

Bc'l(M x N) = ((Bd M) x K) U (M x (Bd N)) .

11.2 .2Definition.

functions classCr

f1,...,fn

for all If

of class borhood

A

Cr Ux

< r < oo)

of x

if for each point x

such that Cr

Now

ff :: UU --► *•Rn Rn

f|(A n U^)

If If ff

isisofof

C°°.

Rn Rnis is differentiable differentiable

of A, A,

there thereis is

may be extended to

aa neigh­ neigh­

a function

on U .

f : A -► Rn

a ^ = df'Vdx^.

this matrix.

are continuous continuous on on U. U.

r, it is said to be of class C°°.

is differentiable, and and xx is isin in A, A,

to denote the Jacobian matrix of is

r

is any subset of Rm, then f : AA (1

and

if the partial derivatives of of the the component

through order

finite

which is of class If

Cr

and

_____________ I

If U is an open subset of ofRm, Rm, then then

is differentiable of class

m

f at

we we use Df(x) Df(x)

x — the matrix whose general entry

We also use the notation

df1,...,fn/cKx1,...,xm) for

f must be extended to a neighborhood of

x before these

partials are defined; in the cases of interest, the partials are independent of the choice of extension (see Exercise (b)). We recall here the chain rule rule for for derivatives, derivatives, which which states states that that D(fg) = =Df Df ••Dg, Dg, where where fg fgisisthe thecomposite compositefunction, function, and and the the dot dot indicates indicates matrix multiplication.

| (a ( )E a xe ) rc Exercise. ise.

Check that differentiability is well-defined; i.e.,

that the differentiability of ing space" (b) n

A —► R

Rm

for

A

f :A

Rn

does not depend on which "contain­

is chosen.

Exercise. 1

Let

U be open in Rm Rm;; let let UU CC AA CC U. U. IfIf

is is of of class class CC ,,and and x x is is in in A, A, show show that that Df(x) Df(x)

f : f :

is indepen­ is indepen­

dent of the the extension extension of of ff to to aa neighborhood neighborhood of of xx in in Rm Rm which is which is chosen.

_______ |

I . DIFFERENTIABLE MANIFOLDS

6

A

Remark. Our definition of the differentiability of a map f : is essentially a local one. We now obtain an equivalent global

Rn

formulation of differentiability, which is given in Theorem 1.5.

1 .3 on

C(l/2),

Lemma. There is a

Proof. Let Then

f

is a

C°°

function

f(t) = e"1^

for

cp : Rm -► C(1),

function which is positive for

g(t) = 0

for t < 0,

g ’(t) > 0

for

R1

which equals

and is zero outside

t > 0, and f(t)

g(t) = f(t)/(f(t) + f (1 —t)) . Then

Let that

C°°

is positive on the interior of

=0

for

1

C(1).

t
0. g is a

0 < t < 1,

C°° function such and

g(t) = 1

for

t > 1. Let that

h(t) = g(2t+2) g(-2t + 2).

h(t) = 0

for |t| > 1,

h(t) > 0

Then for

h

is a

C°°

|t|

< 1,

and

function such h(t) = 1

for

|t| < 1/2.

|

Let

cp(x1 , ...,y^) = h(x1) • h(x2) ••• hCx111) .

(a)

Exercise.

Generalize the preceding lemma as follows:

be an open subset of Rm ; let C be a compact subset of r m C real-valued function Y defined on R such that ¥ and is zero in a neighborhood of the complement of

U.

U.

Let

U

There is a

is positive on

C

7

§1 . INTRODUCTION Remark. Whenever an indexed collection {C^} of subsets said to be locally-finite, we shall mean by this that every point a neighborhood intersecting C^ for at most finitely many values This convention is convenient; it implies that only the empty set C, for more than finitely many i.

i.4 Lemma. space

X.

V. C U.

(i)

for each

i.

(2) Let

l

Proof.

{V^}

of

X

be a locally-finite open covering of there Is a closed covering (C^) of

i.

choose anIndex

asthe union of those

j(B)

such that j(B) = j.

Bfor which

B

(2)

V\

We construct the coverings

open set containing V I. Let

for only finitely many values of

X - (V2 U

U ...),

{C^}

{U^} . For

C Uj^j . De­

Each point has

neighborhood intersecting only finitely many elements of (B; borhood intersects

such that

(B be a locally-finite refinement of

(l) Let

element B of (B, V\

{V^}

for each

i

X is X has i. equal

be an open covering of the paracompact

X, where i = 1, 2,

X such that C. C V.

fine

{U^}

There is a locally-finite open covering

the normal space

each

Let

of of of can

a

this neigh­

j.

by induction.

Let

W 1 be an

whose closure is contained in

C1 = W 1 . Suppose

W 1 U ... U W^_1 U V. U V . +1 U ...

= X.

Let

W.

be

an open set containing X - (W1 u ... u W j_1 u v j+1 u ...) whoseclosure is

contained in

V\.

Let

To prove that the collection x

lies in only finitely many sets

V. U V j+ 1 U ... . As a result,

x

{W^}

C^ = W \ . covers

X,

Vj. Hence for some must belong to

note that any point j,

x

is not in

W 1 U ... U W ^

, by

the induction hypothesis. Transfinite induction may be used to prove (2) for an arbitrary index set. 1 .5 Theorem. Let of class

Cr . Then

in a neighborhood of

f A.

A

be a subset of

Rm ;

let

f : A

Rn

be

may be extended to a function which is of class Gr

8

I. DIFFERENTIABLE MANIFOLDS Proof. By hypothesis, for each point

borhood

Ux

of

x

such that

f|A n Ux

is of class

Cr

on

of the sets

Ux ; it is an open subset of

Ux . We choose

U

x

of

A, there is a neigh­

may be extended to a function which to be compact. Rm . Let

ing of

CC V\ for foreach each i. i. Let Let

function defined on on Rm which is positive on on

Let

be the union

be a locally-

{Ux}

by closed sets sets such such that that

M.

{V^}

M

finite open refinement of the covering M

of

Let

{C^}

be a cover­ be aa CC°°°°

and equals zero in in aa neigh­ neigh­

borhood of the complement of

V^.

E E

M, since it equals a finite sum in some neigh­

is is aa C°° C°° function on

Here we use Exercise (a) of 1.3.

borhood ofofany anygiven givenpoint pointof of M. M. Define Define cp^(x) cp^(x) == Y^(x)/E Y^(x)/EYj(x); ^^(x);

Then

then then

E . (x) = = 11.. E ccpp±(x)

For each if

i,

let f^ f^

denote denote aa Cr Cr extension extension of of f|A f|A nn V\ V\

A n V. V.is is empty, empty, letletf^ f^ be be the the zero zero function. function. Then Then Cr

tended to be of class

on

M

to to V^; V^;

may maybebe ex­ ex­

by letting it equal zero outside

V^.

Define f(x) = E1 cp±(x) f±(x)

.

This is a finite sum sumin in some some neighborhood neighborhood of of any any point point xx of of M, M, and and hence hence is of class for every

Cr i,

Furthermore, Furthermore, if if xx is is in in A, A, then then

on on M. M. so that

f(x) = E cp±(x) f(x) = f(x) . Hence in

f fis is thethe required requiredCr Cr extension extensionofof f f totothe theneighborhood neighborhood M M ofof A A

Rm .

1.6 m-manifold

M

Definition. Definition. A differentiable A differentiable m-manifoldm-manifold of class of Cr class is an Cr and a differentiable structure

differentiable structure of class coordinate neighborhoods

(U,h)

Cr on

on

M,

The coordinate neighborhoods in

(2)

If

and

on on M. M.

A

in in turn, turn, Is Isacollection

of

satisfying three conditions:

(1) (1)

(U1,h1)

M,

3D3D of of class classCr Cr

3D 3)cover cover M.M.

(U2,h2) belong to 3D,3D,then then

t^h'1 : hgd^n U £) - Rm is differentiable of class (3)

Cr .

The collection 3D is maximal with respect to property (2); i.e.,

if any coordinate neighborhood not in

3D

is adjoined to the collection

3D ,

is an

§ 1.

INTRODUCTION

9

then property (2) fails. The elements of 3D are often called coordinate systems on the differentiable manifold

|

M.

(a) (a)

on on

M M

ture ture

Let Let33 DD ''

Exercise. Exercise.

satisfying satisfying (1) (1) and and (2). (2 ). 3D 3D

of of class class

be be a a collection collection of of coordinate coordinate neighborhoods neighborhoods Prove Prove there there is is a a unique unique differentiable differentiable struc­ struc­

containing containing 3D'. 3D'.

Cr Cr

(We (We call call

3D’ 3D’ a a basis basis for for

3D, 3D,

by by

analogy w analogy with i t h the the relation relation between between a a basis basis for for aa topology topology and and the the topology.) topology.) Hint: Hint: M M

Let Let

3D consist 3D consist of of all all coordinate coordinate neighborhoods neighborhoods

(U,h) (U,h)

onon

which w h i c h overlap overlap every every elementelement of 3DT ofdifferentiably 3DT differentiably with class w i t h *Cr class ; this*Cr ;this

means that means that for for each each element element

of of 3Dr, 3Dr,

( (U U^ ^ hh ^ ^

l ^ h -1 : l^h-1 : hh(U (U n n U U 1) 1) -► -► RRm m

and and

h h"1 h,(U U 1)) — h"1 : : h 1 (U n n U -*■ Rm Rm

are differentiable of class class Cr Cr . . To show 2) 2) is is a differentiable structure, structure, the the are differentiable of To show a differentiable local formulation local formulation of of differentiability differentiability Is Is needed. needed. (b) (b)

Exercise. Exercise.

Let Let

M M bebea adifferentiable differentiablemanifold manifold ofof class class

(we often (we often suppress suppress mention mention of of the the differentiable differentiable structure structure confusion will arise). wi l l a r i sThen e). confusion

3D, 3D,

where where no no

Then may also be considered a differenti­ M mayM also be considered to betoa be differenti­

able manifold able manifold of of class class

Cr _ 1 , in in a natural way; one merely merely takes takes Cr_1, a natural way; one r r — — 11 basis for basis for aa differentiable differentiable structure structure 3D1 3D1 of of class class C' C' on on M. M. that the that the inclusion inclusion 5) 5) C C 3)^ 3)^

CrCr

is is proper. proper.

3D 3D as as aa

Verify Verify

This This proves proves that that the the class class

Cr Cr

of of a a

differentiable manifold differentiable manifold is is 'uniquely 'uniquely determined. determined. We see We see in in this this way way that that the the class class of of a differentiable differentiable manifold manifold

M M

may be may be lowered lowered as as far far as as one one likes likes merely merely by by adding adding new new coordinate coordinate systems systems to the to the differentiable differentiable structure. structure.

The The reverse reverse is is also also true, true, but but it it will will r re­ e­

quire much quire much w work o r k to to prove. prove. *

(c) (c)

Exercise Exercise ..

If If

M M

is is a a differentiable differentiable manifold, manifold, what what are are the the

difficulties involved difficulties involved in in putting putting a a differentiable differentiable structure structure on on

D(M)? D(M)?

(D(M)

was defined was defined in in Exercise Exercise (b) (b) of of 1.1.) 1.1.)

1.7 dimsnsions a subset of

Definition. Let

m M

and

n,

and let

M

_____

and

N

be differentiable manifolds, of

respectively, and of class at least f : A -► N;

then

f

Cr . Let

is said to be _of class

A

be

Cr

if

10

I.

for e v e r y pair

(U,h)

pectively, for which

DIFFERENTIABLE MANIFOLDS

and (V,k)

of coordinate systems of

M and

N,

res­

f(A n U) C V, the composite

kfh-1 : h(A n u) — Rn -p

is of class

C .

(Note that a map of class

2 C

though a manifold of class

C

2

is also of class

1 C

is not one of class

1

C ,

al-

until the differ­

entiable structure is changed.) The rank of at

h(p),

f (p) ,

f

at the point p

where (U, h)

respectively.

open subset

W

of

and

(V, k)

of

M

is the rank ofD(kfh~1)

are coordinate systems about

¥ C A C W,

M such that

for if

(U1, h 1)

are both differentiable, so that

1 D(kk1 )as its inverse. D(kfh~1)

|

D(k1k_1)

1 D(hh1 ) —

Similarly,

and

is non-singular, having

is non-singular, so

Exercise.

The standard

—

i : Rm -*■ Rm .

Rm

Similar­

Check that the definitions of differentiability given in 1.2

and i .7 agree.

______ |

1.8 let

M

and

Definition. Let N

at each point

have dimensions p

of

M,

f

m

and

f :M n,

N

be differentiable of class

respectively.

is said to be an immersion.

If rank If

f

morphism (into) and is an immersion, it is called an imbedding. homeomorphism of phism; of course,

| Rm-1

(a) ^m

M

onto

m = n

Exercise.

N

imbedding.

on

is a homeoIf

f

is a

and is an immersion, it is called diffeomor-

Note that

entiable manifold of class Cr

f = m

in this case.

is an imbedding.

ture of class

1

D(k1fhl )

C°° differentiable structure on

is that having as basis the single coordinate system if1.

^ k ”1

have the same rank.

(a)

ly for

(V1, k 1)

.

The requirements for a differentiable structure assure that

—

and

systems, we would have

D(k1fh"1) = D ( k 1k~1) • D(kfh"1) • D(hh^)

and

and

This number is well-defined, providedthere is an

were other such coordinate

kkij’1

p

Bd M

Bd if1 = Rm ~1

and the inclusion

Generalize this as follows: If Cr ,

M

is a differ­

then there is a unique differentiable struc­

such that the inclusion

Bd M -► M

is a

Cr

Cr ;

§ 1. (b)

INTRODUCTION

Exercise.

Show that the composition of two immersions is an

Exercise.

Let

immersion. (c) bounded.

Construct a

Cr

natural Inclusions of require

M

and

N

have class

CP ; let

differentiable structure on and

N

into

M x N

M x N

M

be non­

such that the

are imbeddings.

Why do we

to be non-bounded?

(d) H 1 x H1

M

M

Exercise*.

Construct a

so that the inclusion

C“ differentiable structure on

I : H 1 x H1 -*■ R2

is differentiable.

What

are; the difficulties involved in. putting a differentiable structure on M >: N

in general? (e)

carries

S1

immersion of (f) Cr

Exercise*.

Construct a

into a figure eight. B

2

into

C°° immersion of

S1 into

R^

which

Can such an immersion be extended to an

2

R ?

Exercise . Prove that two differentiable structures of class

on the topological manifold

M

are the same if and only if the identity

map from each differentiable manifold to the other Is a Cr diffeomorphism. (g) of class

Exercise*.

Construct two different differentiable structures

C°° on the manifold

R1,

such that the resulting differentiable

manifolds are diffeomorphic.

1.9 R

]

■)£

Problem.

Prove that any two differentiable structures on

give manifolds which are diffeomorphic. (It was long a classical problem to determine whether two differ­

ent differentiable structures on the same topological m-manifold always gave rise to diffeomorphic differentiable manifolds. turned out that the answer is yes if [5 ].The answer is

m
1.

The inclusion mapping zero vector at

x,

nate systems on

i : M -► T(M)

which carries

and the projection mapping

j ( a ) E x e r c i s e .

with

denote

M.

k(U n V) x Rm .

Let

Show that

(U,h)

and

k h"1

(V,k)

it,

x

into the

are maps of class

C

r—1

be two overlapping coordi­

is a homeomorphism of

Conclude that the topology on

h(U n V) x Rm

T(M) is well-defined.

§3 . MAPPINGS AND APPROXIMATIONS

27

Show also that it is separable and Hausdorff. (b) x

cf

M,

Exercise*.

Show that for some neighborhood

there is a diffeomorphism

is the natural projection of the tangent space at

x

g : ^_1(V) -► V x Rm

V x Rm -*• V,

onto

g

T(N)

is a fibrebundle, with Rm

Rm

as

as structural ______

Definition.

let g : M -*■ N ofclass

a vector at

be a

Cr_1,

assigning the matrix

fine

* g 1

(See [12].)

3 .3 CP ;

such that

is a linear isomorphism of

fibre and the non-singular linear transformations of group.

of the point

x xRm .

n : T(M) -► M

This shows that

and

V

g(xQ)

dg(v) = w

Let

Cr map.

M

and

a to the coordinate system

p

be manifolds of class at least

There is an induced map

defined as follows: Let

assigning

N

v

dg :T(M) -►

be a vector at (U, h), and

let

xQ w

to the coordinate system (V, k) .

be

We de­

if p = D(kgh“1)• a

Note that

dg

is a linear map on each tangent space;

differential of

I

(a)

it Is called the

g.

Exercise.

Check that

dg

Is well-defined, independently of

coordinate system. (b) "v

Exercise. Show that if

as velocity vector, then

dg(v)

f : [a,b]

M

is a

C1

curve having

is the velocity vector of the curve

gf : [a,b] — N. (c)

Exercise.

If

geometrically, noting that

f : [a,b] -► M f

is a

The preceding exercise then merely

ize

this formula to the case where the domain off

If

g

states that

Definition. Let

[a,b]

df into

d(gf) = dg• df.

is a manifold of

General

dimen­

is an immersion, an imbedding, or a diffeo­

morphism, how is this fact reflected by the map

3 .^

interpret

maps the 1-dimensional manifold

M.

sion greater than 1. (d)\ Exercise ^ .

C 1 map,

M

be a

Cr

dg ?

manifold.

Suppose we have an

28

I.

inner product

[v, w]

DIFFERENTIABIiE MANIFOLDS

defined on each tangent space of

ferentiable of class into

R

which carries

n/[v,

which is dif­

o < q < r, in the sense that the map of v

into

[v, v]

is of class

product is .called a Riemannian metric on note

M,

C^.

M. As usual,

T(M)

Such an inner

|(v|| is used to de­

v ].

Let us note that it follows from the identity [ v , w ] = ir |(v + w ||2 - t | f v — w ||2 that

the realfunction

T(M)

consisting of all pairs

|

(a)

fold

[v, w] is continuous on the subspace of

Exercise.

(v, w)

such that

*(v) =

T(M) x

tt(w ) .

There is a standard Riemannian metric on the mani­

Rn ; wesimply take the components

a1 and p1

of ~v and "w relative

to

and define [v,w] as their dot product,

the coordinate system i : Rn -► Rn E aV.

Let M If

v and

w

aretangent to

dot product in metric on

M. (b)

(U, h)

be a Cr manifold; at

Why do we need Exercise.*

f

Let

w

be a

C^+1 immersion.

[v,w]

df("w) .Check that this is a Remannian

[v,w] M,

be a Riemannian metric on and let a

and $

be the

under this coordinate system.

C^- matrix function

G(x)

to equal the

to be an immersion?

be a coordinate system on and

f : M —► Rn

x, let us define

T(Rn) of df(~v) and

corresponding to v unique

M

let

defined on

U

M;

let

matrices

Show there is a

which Is symmetric

and po­

sitive definite, such that Let

G(x)

be

["v,"w] = • G(x) the matrix function corresponding

to the coordinatesystem

(V, k) ; show that G(x)

= D(hk_1)tr • G(x) • D(hk“1)

Any correspondence assigning to each coordinate system a matrix

G,

with this equation relating

G

and G,

. (U, h)

is

about

xQ,

called classically

a covariant tensor of second order.

3.5 a1

Definition. If

be the components of ~v

v

is a tangent vector to

relative to the coordinate system

Rn

at

i : Rn -^Rn .

x,

let

§ 3 . MAPPINGS AND APPROXIMATIONS The correspondence Rn x Rn

(x, 01)

v

29

gives a homeomorphism between

which is an isomorphism of the tangent space at

We often identify (x,

ot) . If

x,

we define

T(Rn )

f : M

with

Rn

dQf(v)

Consider

Rn x Rn

onto

and

x x Rn .

for convenience, and we write

is aC1 map and

"v

is a tangent

"v =

vector to M

at

df(v) = (f(x), dQf ( v ) ) .

by the equation

F 1 (M, Rn ) ,

x

T(Rn)

the space of all

C1 maps of

M

into

Rn .

We topologize this space as follows: Choose a Riemannian metric on

M.

Given the C1

and the positive continuous function s(x) C 1 maps

set of all

g : M -► Rn

g

x

in M

denote

the

- g(x) | < 8 (x)

and each v

tangent to M at x. If

is called a 5-approximation to

approximation to

f.

fine C 1 topology on

The sets

W(f,

f.

5)

in some

Rn .

g

is in W(f, 5), C1

It is also called astrong

form a basis for what is called

the

F 1 (M, Rn) .

Given a differentiable manifold The fine C 1 topology on

Riemannian metric on

M

N, let us imbed it differentiably

F 1 (M, N) is defined to be that induced

from the fine C 1 topology on F 1 (M, Rn ) .

(b) of

Rn ,

||d0f(v^ - d0g("v) | < s(x) |[v||

for each then

let W(f, 5)

f: M

such that

Ilf(X) and

on M,

map

This topology is independent of the

and the choice of the imbedding of

N (see Exercise

3 .6 ). To obtain the coarse C 1 topology, one

quire that the inequalities hold only for takes sets

W(f, 5, A) as a basis.

We

x

alters the definition to re­

in some compact set

A; one

shall make no essential use of the

coarse C 1 topology. |

(a)

Exercise.

Check that the sets W(f,s) form a basis for a

topology. (b)

Exercise.

The fine C° topology on F°(X,Y) (the space of al

continuous maps of the separable metric space space

Y) is obtained as follows:

ous function

s

on

Given

X, let a neighborhood of

such that

p(f(x), g(x))
o,

let

N

(V, k)

be a submanifold of

Rn ;

M;

0

5 >

(*)

C

be a compact

N

about

f(C).

such that if g : M -*• N and

g(C) c V , Ikfh-1 (y) - kgh_1 (y) I
n

for

^— 0 0 < t < 1/2

ft(x)

=h(x)

for 1 /2


0.

f^

be a

Cr ;

let

C1 differentiable

Prove there is a

Cr

I.

44 differentiable homotopy 5 -approximation to

F^.

f^.

f1

Let

beC1 maps of

tween them.

M

Outline: be a

Rn ;

C1 map

suppose

f^

a compact subset

fQ

and

f1

such that

is a

M

______ |

and N

be manifolds of class

N; let

ft

be a

Prove there is a

C1 ;

let

C1 differentiable

homotopy F^

f^,

for each

t.

Consider first the following problem:

For each

t,

of the open subset is a of

U

of

H01

or

Rm

C1 regular homotopy between

U; let

Y

be as in 4.1;

Define

fQ

C1 regular homotopy be­

is a 5 -approximation to

F^

F^

t.

into

5 (x) > 0.

Let

between them such that

ft

between

for each

4. 6 Problem*. and

DIFFERENTIABLE MANIFOLDS

into Hn

fQ

let

let

or into

and f^ .

Let

g^(x) = ¥(x)

A be

• f^(x) .

+e

ht (x) = ^ tp(s) gt + s (x) ds , -€ for

suitably

small

and

y

1 . Let

cp =

differentiable and equals

WC V

f :U

f^ outside some neighborhood of

A.

Lemma. Let

and

Cr map such that Cr imbedding

h

equals

(2)

h

is a 5 -approximation to

(3)

h(W)

(4)

If

Rn ,

then

of

Rn

subsets of onto

such

F^

A,

Rm , Suppose

Given

that

outside V. f.

is an open subset of

htl^) is also a

of

Rm ;

Rn . U

and

C°° submanifold

Therestriction of

is of class

Rm .

*f : U -*■ Rm is an imbedding. h : U -► Rn

outside,

h^(x) . Then

let

f(U1) of

is a

Cr ,

C°°submanifold

Rn .

* to f(U)

is a homeomorphism of

f(U)

g : 0 -► Rn

be the inverse ofthis

map.

g ( x \ . . . , x m) = ( x \ . . . , x m, ^ + 1( x ) , . . . , g n (x )) g

0

some neighborhood of

W be bounded open

is a C°° submanifold of U1

onto an open subset0 Then

f

and

jt be the projection

(1)

Proof.

and

U, V,

VC U.Let

is a

f

and

(1 - Y(x)) + on

s(x) > 0,there is a

of

= f^.(x) •

homotopy between

and

Rn

^( x )

f1

4. 7 with

cp is positive on ( - € , e)

e, where

since it equals

f(rtf)"1.

,

is a

§*+.

Further, if of

g

to

on

f(U1)

SMOOTHING OF MAPS AND MANIFOLDS

fCU^)

is a

C°° submanifold of Rn ,

(The converse is easy (2 .2 ).) For impose

JtfCU^ is of class C°°. its induced

C°° structure, and note that

relative to thisstructure. C°°,followed by

The map

g

the inclusion of

Consider

gQ : 0

Rn_m,

C°°

^.1, choose

on

irf(W)

f(U1)

into

-►

tion to

gQ

C°° map

this, which

is

C°°.

defined by the equation

outside

.

Rnbedefined by the equation

gQ

which is of class

nf(V) . We further require that

on any open subset of 0

where

gQ

is

C°°.

Let

g(x) =(x, gQ (x)); it is an

g :

e-approxima­

g. Define

h : U -*■ Rn by the equation

is properly chosen,

1+.8

h

will

Theorem. Let

M

be a non-bounded

h : M -►Rn

which is a s-approximation to f ,

it of

Cr manifold There is a

such that

f = girf.

(i < r) . CP

e

Let

imbedding

h(M)

is a

C°°

Rn .

Proof. Let an imbedding.

Let

s(x) > 0 .

be a

submanifold of

Cr imbedding.

h(x) = g*f(x). By 3.7, if a 5 -approximation to

be

f : M -*■Rn

jection

is a

Rn , which is

to be an e-approximation to

and equals

be of class C°° 0

gQ (x)

7t|f (U 1)

is the inverse of

g0 (x) » (gm+ 1 (x),...,gn (x)) Using

the restriction

5

For each Rn

be small enough that any &-approximation to x

in

M,

df(x)

has rank

m.

onto a coordinate m-plane, the map

f

is

Then for some pro­ d(*f)(x)

has rank m,

k6

I. DIFFERENTIABLE MANIFOLDS

so that

itf

is a Cr diffeomorphism of a neighborhood of

x

onto an open

set in the coordinate m-plane, by the inverse function theorem. covering of about

C^,

cover

M,

m-plane,

M

by sets

h^(C^)

C^

equals the unit m-ball, such that the sets

is an imbedding.

approximation to 5 (x) < 5^ then

such that for some coordinate system

and such that for some projection jr^f |C^

for

Let

in

C^.

Then if

Let

W^

be an open covering of

and

V± C Int C± . fQ = f.

M,

Int C^

onto a coordinate

s(x)

be small enough that

is a 5-approximation to

so that

jt^f!|C^

with

W^ contained in the open set

Rn ,

approximation to for

k < j.

f,

is an imbedding.

As an induction hypothesis, suppose

is a C°° submanifold of

s^-

*^f,

is a Cr map which is a (1 - l/2",_1)5(x) fj_i(Wk)

Rn

(IL , h^)

be a number such that any

f ’: M R n

ir^f1 is a 5-approximation to

Let

of

is an imbedding; let

x

Choose a

V^,

M -► Rn f

such that

We then construct a map

fj. Apply the preceding lemma to the map : h.(IntC.)^Rn to obtain a nap equals

fj_1

which is a 5 /2 ^ approximation to

f^ : M -► Rn

outside

,

V\,

a ^

and such that

fj_i>

submanifold of

Condition (k) of the preceding lemma guarantees we can choose

Rn . that

is As before, we let

a C°° submanifold of h(x) = lim^

^ fj(x),

Rn

which

for

fj so

k < j .

and note that

h

satis­

fies the conditions of the theorem.

4.9

Corollary.

non-bounded manifold

|

(a)

a non-bounded

f,

Exercise.

Let

C°° manifold.

such that

M Let

f.

is a

Cr differentiable structure on the

C°° structure.

be a non-bounded s(x) > 0.

Cr imbedding

h(M)

entiably isotopic to

2D is a

M, 2D contains a

di-ng> prove there is a to

If

If

h : M —► N

C°° submanifold of

Cr manifold;

f : M -► N

is a

let

N

be

Cr imbed-

which is a 8-approximation N,

and

h

is

CP differ-

§ 5 • Manifolds with Boundary.

our two There are additional technicalities involved in proving oar main theorems ^.5 and

9,

^ .

for manifolds with boundary.

First, we need to prove the local retraction theorem (5,.5.), which states that a non-bounded

Cr submanifold

a neighborhood which is retractable onto

M

of euclidean space has

by a Cr retraction. If r > 1 , r_ i it is easy to to find find such such aa retraction retraction of of class class CC; ;roughly, roughly, one one takes takes the the

plane normal to tion of class

M C

at

xx

M

and and collapses collapses it it onto onto x. x.

requires more work;

mann manifolds (5.1 - 5 -)0 .

To To construct construct aa retracretrac-

one needs first to study the Grass-

One also needs a topological lemma (5.7).

From this, we can prove the product neighborhood theorem, which stages that

Bd M

has a neighborhood in

M

which is diffeomorphic with

This in inturn structure Bd M x [0, 11 )).. This turn is is used used to to construct construct aa differentiable differentiable structure on

D(M),

the the double double of of M The theorems for

M

the non-bounded manifold

5.1

(5.8-5.10). M (5.8-5.10).

D(M)

Definition.

M(p,q)

n

r.

the same element of nations of n xn

\(A). X ( A ) .

of

de­

M(n,n+p;n)

are

Rn+^, »so they determine an element of A

and

B

determine

Gp ,11 if and only if the rows of each are linear combii.e., if

is an open subset of

has a natural topology and

A = CB

for some non-singular

pn (n+P)

(see 3.10),

C°° differentiable structure.

the identification topology:

V

is open in

G

p ,n

so that it

Let

G be given p ,n if and only if A,-1 (V)

M(n, n+p; n ) .

We note that be open in

A

M(p,q;r)

C.

M(n,n+p;n)

is open in

is the set of all

matrices;

Further, two matrices

the rows of the other;

matrix

pxq

The rows of any matrix

independent vectors of

Gp ,ji which we denote by

G p ,n

Rn+P.

denote the set of all

notes those having rank a set of

(5.11 - 5 .13 ).

The Grassmann manifold

n-dimensional subspaces of Let

then reduce to the corresponding theorems for

X:

M(n, n+p; n) —► G

M(n, n+p; n) .

JJ j II

For any

C

^7

in

is is an an open open map. map. For For let let UU

M(n, n; n), the map

C : A —► CA

1+8

§5.

is a homeomorphism of M(n, n+p; n) .

MANIFOLDS WITH BOUNDARY

M(n, n+p; n)

x”1(x(U))

M(n, n; n ) % so

x(U)

with itself, so that

is the union of the sets

Theorem. G^ „ is a non-bounded * Pj X : M(n,n+p;n) -* Gp, II is of class C°°. Proof.

(i)

G^ p ,„ n

(P

If

there is some projection

of

(P

it is a linear isomorphism.

n-tuple of vectors

(v1,...,vn)

.

Let

U be the U

set of all planes on uniquely determines an

U.

Thus, each vector

columns. nxn

Let

U

of

Then

let

is .open in

since

V,

since

cp : U —► M(n,p)

be

cp is constant on each set

X([P2 Q2 ]) P~1Q2 .

means that

Hence

the inverse of home omorphi sm.

G p ,n

such vectors,

Q

[P Q]

where

P

of

ancj hence in

M(n,p)into thematrix

is a continuous map of only if

x[P Q] = x[I P”1Q] = Y0 (P defined by the

whence of

P^’1Q 1 = V

cpQ^0 (Q) = cp([ I Q]) = I"1Q

M(n,p)

[IQ.,] = C[I Q?] 1 ^ *

equation

x"1((p), since the equation

cpQ

is

.

X[I Q 1] = x[I Q2]

[P1 Q 1] = C[P2 Q2 ],

since

Rn (n+P)

Then

cp induces a continuous map yq,

com­

n

U is open in

= X¥.

1-1,

onto

n

p

suppose it consists ofthe first

the arbitrary matrix

M(n,p) Let

n;

Y

It is

has

Some

V = x(U)

map

v^

G^ n -

the set of matrices of the form

M(n,n+p);

Gp

it maps

Now

be

into

Conversely, each

be the general element of

Hence

Let [IQ]

x(A)

has rank

and non-singular.

M(n,n+p;n) .

into

A

since there are

*

R-^n .

More precisely, let submatrix of

dimpn; the

which lie in it and project under

ponents which may be chosen arbitrarily; is home omorphi c with

of

Rn+-^ onto a coordinate n-plane

Each plane of

such n-tuple determines a plane of

nxn

in

Let us explain first the

the natural unit basis vectors for the coordinate n-plane.

U

C

is an n-plane through the~'origin in

*

which is a linear isomorphism on which

C00 manifold

is locally euclidean.

geometric idea of the proof. Rn+-P,

for all

is open.

5.2

map

C(U)

C(U) is open in

cp( [PQ])= P~1Q. x([P

Q.,]) = (CP2)_1(CQ2)

into M(n,p) ;

cpQ

= Q.

=

is Hence

is

I.

(2 )

G

p,hn a s

DIFFERENTIABLE MANIFOIDS

a C“

d iffe r e n tia b le

that two coordinate neighborhoods class

QQ

C

overlap.

'"V rV-/ 'X*

Let

cp,¥,U,V

(V, cpQ)

stru c tu re .

Q

Hence

o n ly p ro v e

be the corresponding maps and open sets Then

?

takes the columns

and distributes them in a certain way among the columns of

the identity matrix is of class

We n e e d

of the type just constructed have

which determine another coordinate neighborhood. of the matrix

1*-9

C°°

0

so that any

rank at least

n+1.

€-neighborhood of neighborhoods of (4) (n+p)J/nip.T

Let

x(A)

e

be distinct sub-

Then

X(U)

and

^(B), respectively, since

n+i .

of this one has

denote the e-neighborhood of

(a)

* Exercise .

Show that

(b)

Exercise .

Show that there is a

5.3

Lemma.

A.f* = f.

(f*

Proof. Let such that

\ (B )

A, and V, the X(V) X

are disjoint

is an open map.

G has a countable basis. Indeed G„ „ is covered by i/*11 P^n coordinate systems of the type constructed above.

Let

there is a neighborhood that

and

matrix within

M(n, n+p; n) . and

A.(A)

[ g ] must have rank at least

2n x (n+p)

U

B, in

For let

f(x)

G p, n

is compact. C°° diffeomorphism of

f : M -► G_ _ be a Cr map. Given x in p ,n U of x and a Cr map f*: U -► M(n,n+p;n)

is called a lifting of

f

over

G^ n

M, such

U.)

(V, cp ) be a coordinate system on G , as in 5.2, 0 Pfn lies in V. Now cp0 : V -*■ M(n,p) is a C°° diffeomorphism.

50

§5.

MANIFOLDS WITH BOUNDARY

Simply define f*(x) = Y(cp0 (f (x))) = [I cpQf (x) ]

|

(a)

Exercise*.

a diffeomorphism

g

Let

of

(V)

the natural projection of Let

(V, q>0)

with

Show that there is

V x M(n, n; n)

V x M(n, n; n)

onto

such that

For each

x

in

Xg”1 is

V.

be another coordinate system on

corresponding diffeomorphism. M(n,n;n)

be as in 5 .2 .

(V, cpQ)

VHV,

Gp n ,

and

let a map

g the

h^., carrying

into itself, be defined by the equation

(x, h ^ P ) ) = g g-1 (x, P) Show

that

h^ is merely multiplication by a non-singular matrix

show

that the map

x -► Ax

\ : M(n,n+p;n) -► G^ „

M(n,n;n).

5.4 tangent to

at

If we identify 3.5.

As

v

(See [1 2 ].)

x,

T(Rn)

f :M

then df("v) with

Rn

be a

is a tangent

Rn x Rn , then

Cr immersion. vector to

If "v

Rn

at

f(x).

x,

dQf(v)

Rn .

will be the map

which assigns this m-plane to the point

(n

= m + p, Let

is

Hence

t

t

of the immersion

f

x.

of course.) (U, h)

be

given by the equation

evaluated at

Pjm

The tangent map

ranges over an

m-plane through the origin in G

is

df(v) = (f(x), dQf(*v)) , as in

ranges over the tangent space at

t :M

I _____ j

Definition. Let

M

This will show that

is a principal fibre bundle with fibre and structural

Pj

group

is continuous.

Ax , and

h(x) ,

is of class

and

a coordinate system on

M.

t(x) = \(D(fh“1)), where X is the projection of

Then locally the map the matrix

M(m,m+p;m)

t

D(fh"1)

onto

is

Gp m .

Cr~1.

Similarly, one has the normal map of the immersion, n : M -+ G m, p which carries x into the orthogonal complement of t(x) . Now the matrix D(fh“1) is

mxm

is non-singular; and

= \( [R(x) I]),

suppose it is of the form

non-singular. Then locally where

R = - (P 1Q)"^P .

n

Hence

[P(x) Q(x)]

where

isgiven by the equation n

is also of class

P

n(x) Cr_1 .

51

I . DIFFERENTIABLE MANIFOLDS 5.5

a

Theorem

Cp imbedding;

(local retraction theorem).

M non-bounded.

There is a neighborhood

and a CP retraction r : W -*• f (M) . for

y

in

Let

W

f :M

of

(We recall this means that

Rn

f(M)

r(y) = y

f(M).)

Proof.

Consider the normal map

n : M -► G^ ^ and the tangent map m,p of the imbedding f (p = n m) . Both are of class Cp_1 .

t : M -► Gp ,m There is a fine

C° neighborhood of

lies in this neighborhood, Rn ,

n(x)

and

n :M

Gp,m of class

Theorem U.2.

(In the case

Cp

t(x)

n

which

are independent subspaces of

n(x);

t

is

Let that

v

r = 1,

n

E be the subset of

is only of class

Let

also a map of

lies in

Choose

which lies in this neighborhood, using

use Exercise (c) of that section.)

E

such that for any map

i.e., their intersection is the zero vector (see Exercise (a)).

a map

of

n

t(x)

class

C°,

and we need to

be the orthogonal complement Cp .

M x Rnconsisting of pairs

the subspace

n(x).

(Ifwe

(x, "v)such

used the map

ninsteadofn,

wouldbe what is called the normal bundle of the imbedding.)We prove

that

E

isa

CP submanifold of

Given

x

in

in a neighborhood of t*.(x) = B(x)

is an

M; x.

let

Define

n*

n*

Then

and

and

Let (U, h)

g : U x Rn -► Rm

x Rn ,v n ) )

t*

be

be U;

is a

pxn

A

(U x Rn , g)

lies in

E, then

is a v

matrix,

and B

and

let

t

and

jointly form a x

h(x) = (u1 , ..., um ) .

by the equation

= ( h ( x ) ,

[v 1 . ,.

Vn ] • I

")

lB(x ) I

Cp coordinate system on

'

M x Rn . Now if

is some linear combination of the rows of

(x, ~v)

A(x) , so

-V = [y 1 ... yp p 0 ... 0 ] • r A(x)i L B(x)J

thflt

for some choice of (x, v)lies in g(x, v)

n

a coordinate system about

v Then

n.

CP liftings of

the rowsof

are defined on

g (x , ( v 1 , . . .

dimension

n*(x) = A(x)

m x n matrix;

linearly independent set. such that

M x R n of

E.

lies in

tersection of

E

y 1,

y-^.

As a result, h(U)x R-^. with

Conversely, if "v (x, v)

lies in

is of this form, then E

Hence the coordinate map

U x Rn

if and only if g

carries the

onto the open subset h(U) x Rp

of

in­

Rm+^,

be

52

I.

Consider the f(x)

+ "v; v;

let

F

Cr map of

constructed above.

M x Rn

Rn

which carries

be the restriction of this map to

We prove that coordinate system

DIFFERENTIABLE MANIFOLDS

dF 6F

has rank

n

(x, ”v) into

E.

at each point of

MxO.

g(x, "v) = (u1 , . . . ,um, y 1 , . . . ,yp , 0 , . . . , 0 )

on

Use the E

which was

Then

Fg_ 1 (u, y) = f h " 1 (u) + [y1 . . .

yp ] * A ( h “ 1 (u))

,

so that

5(Fg-1)/Sy = A(h_1(u ))tr , and

cS(Fg-1)/3u k = a (f h '1)/S u k + ( [y1 ---yp ] • 9(Ah 1) /3uk) tr .

The last term vanishes at points of

g(M x o ) ,

since

y = 0

at such points.

Hence D(Pg_ 1 ) at points of g(M x 0 ) .

=

[D(fh-1) (u)

The columns of

A( h_ 1 ( u ) ) t r ]

D ( fh 1) span the tangent plane t ( x ) ;

§5. MANIFOLDS WITH BOUNDARY the rows of

A

span the plane

n(x);

53

by choice of

n

these spaces are

independent. Now, space F

F : E -► Rn

MxO;

is a homeomorphism when restricted to the sub­

furthermore, each point of

MxO

has a neighborhood

maps homeomorphically onto an open subset of

tion theorem). which

F

maps homeomorphically onto an open set in

V be the

MxO

in

Cr diffeomorphism of

tion of

W

I

onto

(a)

map

n

E

Rn

on which

V

onto

W,

it of

MxRn

F it F " 1 = r

MxO

in

E

(see Lemma 5.7). dF has rank

intersection of these neighborhoods and

There is a natural projection a

(by the inverse func­

It follows that there is a neighborhood of

There is also a neighborhood of let

Rn

which

n;

set W = F(V). onto

M.

Since

F

is

is the required retrac­

f (M) .

Exercise.

Construct the fine

C° neighborhood of the normal

required in the preceding proof. Hint:

hoods of

Use the fact that

n(xQ)

and

t(xQ)

5.6

X

is an open map to construct neighbor­

consisting of subspaces which are independent.

Corollary. Let

There is a neighborhood

W

of

f (M)

f : M -*■ N

in

N

be a Cr imbedding;

M

non-bound

and a Cr retraction r : W —►

f (M) . Proof. Let hood

U

of

Then

g“1r0g

I |

gf(M)

g:

N -► R^-

in R^

be a Cr imbedding.

and a Cr retraction

is the required retraction,

rQ

of

defined on

There is a neighbor­ U

onto

gf (M) .

W = g~1 (U) .

# (a)

M

and

N non-bounded.

simply when ric.

N

such that

normal to E

dffw)

v

f : M -► N

The normal bundle

E

M x Rp .

be a Cr immersion, with E

of

f

may be

be the subspace of

tangent to M

as asubset of

r > 2;

described most

having the derived Riemannian met­ M x R^

consisting of pairs

is a p-tuple representing a vector tangent to

for each "w

is described

manifold of

Let

is a submanifold of -Rp

In this case, let

(x, v)

case,

Exercise.

at

M x T(N) .)

N

and

x. (In the more general Then

E

is a C1*”1 sub­

Prove the tubular neighborhood theorem:

I. DIFFERENTIABLE MANIFOLDS

^

If of

M x 0

f :M

N C Rp

is a Crimbedding, there is a neighborhood U r—i in the normal bundle E and a C diffeomorphism h of U

with a neighborhood of Hint:

Let

f(M)

in

N

F : E -*• Rp

such that

be the map

be a retraction of a neighborhood of

5.7 space

X.

Lemma.

Let

f: X

pose each point

x of

N

neighborhood of

=

f.

F(x, v) = f(x) + "v;

onto

Let

A

N;

let

let

r

h = rF.

be a closed subset of the locally compact

Y

be a home omorphi sm

A

has a neighborhood

phically onto an open subset of Y.

h|M x o

Then

when restricted to Ux which

f

A onto an open subset of

f

A;

sup­

maps homeomor-

is a homeomorphism of some

Y.

(X and Y are separable met­

ric, as always.) Proof. Let

U

be the union of

locally 1-1 at each point such that f(x) ,

f|B

where

is xR

i-i, andx

of U.

Furthermore, if

then

f|B

are in

is a homeomorphism, (1)

If

xn

and

yn

of

Un

U.

If

C, f

C

is

1-1 on

we may assume

xn -► x

of

f(x)

since

Then

locally

1-1

at

(2)

=f(y) ;

Un

f

is

of

A - Un

1-1

no set

in U,

on no set

of

such that

and so that* UnUA,

f(xn )

f(xn ) -+

C

f is

i-i.

C,

we

Let

Un

U 1 is com­

By passing to subsequences

and

x

and on C,

U

is

e

y

will be

x = y.

? or large

and f

such that

x

on

xR

is f

is

i-i

points

But

f

1-1 on xR

U1 of

is

n. on

1-1

CUA,

on

VUA.

is small enough that

there are points

where

Since

there are points

where f

so it con­

is 1-1

f(xn ) =

= H y ^ • By

numbering, we may assumexR -► x,

U

1-1

C is a compact subset of

is compact and lies

and f

is small enough that

is

be the e/n-neighborhood of C,

If

U

x, so we could not have

If

Y,

U

x.

and yR-*■ y; f

we prove there is a neighborhood V Let

e

f(xR )= ? ( Y n ) •

such that

is open in

from some nonwards.

of

Then

is any subset of

on whose closure

where

renumbering, C.

f(Ux )

f (xn )

C is a compact subset

be the e/n-neighborhood of pact and lies in

Now

must converge to

prove there is a neighborhood of

B

U .

is a homeomorphism. For let

B.

tains all elements of the sequence f|U

the neighborhoods

U1

(using (i)). UR - A

and

yn

passingto subsequences and re­ Is apoint of

C.

Then

f(yn) "**

fis

55

§5. MAJHFOLDS WITH BOUNDARY f(x) .

Nov

1-1

at x,

so wecould not have

(3)

^(xn) =

We now prove the lemma.

ing sequence

AQC A^...

to be the empty set. such that

Vn

of compact sets, where

In general, suppose

is a compact subset

we may choose a neighborhood

Vn+1

Vn+1

U

is a compactsubset of Let

V

be

neighborhood

V

of

5.8

dg(x)

M

Cr function

g

be a on

M

{(U^, h^)} let

{U^} . Let

is zero for

{cp^}

f

and

set in = 0,

if1 about so that

in

h?(x)

at points of Let Bd M.

1-1

on

Define

VQ

An ,

VnUA. By (2 ),

on

VnUAn+1

such that

Vn+1UA.

V

.

Cr manifold.

Then f

1 - 1 on

is

the

There is a non-negative

such that for each

x

in

Bd M,

be a locally-finite covering of

For each

U^ n Bd M.

For k(x)

h^ k-1

i,

d(h? k~1)/duJ = 0

the

g(x) = 0

M

by

m^*1 coordinate function

Furthermore, if x, then

k(x) = (u1,...,um ) cKh? k“1)/dum

is

is a non-singular transformation of an open

onto an open set in

D(tLk-1) is non-singular, since

is empty.

f is

1-1

is

n.

be the union of the increas­

be a partition of unity dominated by the

dim M = m. x

k(x):

AQ

is locally

is a neighborhood of

is another coordinate sustem about the point positive at

lai,ge A

f

1.

has rank

coordinate systems;

h?(x)

^or Let

of the compact set

and

But

A.

Proof. Let

covering

of U

Vn

the union of the sets

Lemma. Let

real-valued and

Yn ~* x '

f | C U A is a homeomorphism, so that

if1,

at points of

^(h? k"1)/^um

is non-negative for all

and

Rm “1,

h? k~1(u1,...,um ~1,0 ) for

j < m.

Since

must be non-zero at points of

x,

Rm ~1;

this derivative must be positive

Rm “1. g(x) =

Furthermore, if

q>j_(x) • h^(x) . Then k(x) = (u1,...,um )

g(x) = 0

whenever

x

is in

is a coordinate system on

M,

then

Whenever

h® •

x

the first term vanishes, because

is in

Bd M,

second term is strictly positive. Bd M.

5(h“ k - 1 ) / 3 u m

S ( g k _ 1 ) / S u m = E±

cp1 -

Hence

dg(x)

has rank

.

h®(x) = 0; 1

if

x

the

is in

56 56

I . DIFFERENTIABLE MANIFOLDS 5 .9

phism

p

0)

(x,

Theorem.

L et

JVI be a

Cr m an ifold .

o f a neighborhood o f Bd M w ith whenever

x

There is ~ a

Bd Mx [0, 1)

Cr d iffeo m o r-

such th a t

p(x) =

i s in Bd M. Such a neighborhood i s c a lle d a product

neighborhood o f the boundary. P r o o f. Cr r e t r a c t io n g

on

By 55.6 . 6, ,

thth ere ere i si s a aneighborhood neighborhoodW Woof f Bd Bd MMin inM M andand aa

r : W-► BdM.M. By By 5 5. 8, .8 , ththere ere i si s a a nnon on-n-negative egative

MM such suchthth a ta ti if f xx iiss in in Bd Bd M, M,

D efin e

f : W- + Bd M x t o , 00)

i s in

Bd M,

then

g(x) g(x)

==00

by the equ ation

f( f ( xx)) = ((xx,, 0) 0 ) and

d f(x )

Cr Crfu n fucntio c tio n n

and and dg(x) dg(x)

has hasrank rank

ff(( y ) = ( r ( y ) , g ( y ) ) . i s n o n -sin g u la r:

k(x) == ( u \( .u. \. ,. u. . m) , u m)be be aa coord coordin inate ate system system about about x ; x; l e t

If

1 .1 . x

For l e t

h* == (k|Bd (k|Bd M M)) xxi .i .

I

D (hfk ’ )

=

One u ses Theorem 11.11 . 1 1 to prove th a t each Ux

which

Sin ce th a t

f

f

x

f

V

Bd M x 00

of

in

)

Bd M M x [0, 00 00)

morphism o f

l i e s in

Bd M M x [[0, 0 , 00)

(x, t/&(x)),

5.10

p

if

t
> 00

Bd M M x [0, 00) .

Bd M, i t fo llo w s from

i s a homeomorphism o f some neighborhood

(x , t) o f

and

Bd M has a neighborhood

i s a homeomorphism when r e s t r ic t e d to

2 .6 2. 6,, we may co n stru ct a Cr fu n c tio n

D(M),

in

c a r r ie s homeomorphically onto an open su bset o f

a neighborhood

Mx 1,

s(gk_1)/aum

0

whenever x

D(M)

a s fo llo w s :

p 11 : U U11 -► Bd M11 x( - 1 , 0] M11 . LLetet

P : U -► Bd M x ( - 1 ,

1 )be

i s in

and

M11 =

Bd M.

We

L et

be product neighborhoods

U be the union o f UQ and

U1 U1

in

the homeomorphism induced

by

pQ

CP d i f f e r e n t ia b l e s tru c tu re on

D(M)

i s w e ll-d e fin e d i f we

§5. MANIFOIDS WITH BOUNDARY require (1) and

M1

P

to be a

in D(M)

P C

2/3.

tt be the projection of

enough that

g(M)

lies in

is a diffeomorphism of imate

N.

2 / 3 ).

Proof, Let

k >

into

irf = f

M

Let

onto

k >|3f(t)| for all

NxR

N x(—1/3, N x 0.

onto

(By 3.7,

as closely as desired by taking

where

is of the form y

is in

function on
(y)| y )| < 1< / 31 ,/ 3 ,so so wwhheenn ss == cp(y), q>(y), P(is!) P(lsl) == 1.1. N x 0.

5 5, , wewe mmaayy

tion to the zero-function as desired. g ( * £ ) ”1 (y, g(*£)~1(y,

0);

u s i n g 3.7, we see that if

f f “1 = identity.

q>(y) q>(y)

as as close close aann aapppprrooxxiimmaa­­

g

approximates

1

—

L et

(y, 2p(x)/3,

and

^(x,t)/St = (1 - a)e(x) + a + (1 - € (x)) ( 1 /p(x)) • a 1 > e ( x ) > 0 Let which

f1 = fg.

equals f

Then

near ¥ - ¥ .

f1

is a

Cr diffeomorphism of

Furthermore,

if we set

f.|(x,t)

¥

with

. f(¥)

= (X^T^),

then 0< ^T1(x,t)/^t < 1 Thisfollows that

from the fact that

for

t < p(x)/3

T 1(x,t) = T(x,€(x)t)

for

t < p(x)/3,

^T1(x,t)/^t = c(x) • dT(x,c(x) t)/dt, which is positive and less than

by choice of Step the equation

e. 2

. Now we define a diffeomorphism h

h(x,t) = (x, cp(x,t)),

of

¥

with itself by

where

cp(x,t) = a( 2t/p (x)) *t + [1 - a(2t/p(x))]T 1 (x,t) Again,

so

h

is a diffeomorphism because

cp(x,o) = 0 , and

cp(x,t) = t for

6k

I . DIFFERENTIABLE MANIFOLDS

t > p(x)/3,

anddcp(x,t)/dt > 0

for

t 0

and

,

T^x^tJ/t = dT.j(x,t*)/dt < 1,

0 < t* < t. Let

f2

= f1 h_1 . Then f2

f(W)

which equals

then

T2(x,t) = t

f

5

W - W.

If

M

Y

of

MxO

in

W

with

f (x,t) = (X2,T2),

M x R +.

is compact, the completion of the proof is easy.

small enough that

f^(x,t) = (Xj,Tj),

Cr diffeomorphism of

Furthermore, if we set

in a neighborhood

Step 3. Choose

near

is a

where

M x [0 ,5 ]

T^ = T2

is contained in

Y,

and define

and

X 5(x,t) = X 2(x, a(t/s)•t) f^

is a diffeomorphism because the map

M

for

tQ < 5 , If

and

T^(x,t) = t

x

X(x,tQ)

is a diffeomorphism of

t < 5.

for

M is not compact, more work is involved.

We

need the follow­

ing lemma:

6.2 ball.

Let

Let

Lemma . C

U

be a compact subset of

whose closure is

U; let V

contained in U. Let

f(Ux [0,a]) (1)

f1

C

[o,a] into

U x 0, such that f(x,t) = (X(x,t),t).

Cr diffeomorphism

f1 = (X^t)

of

Ux[o,a]

with

such that is the identity on

(2 )X 1(x,t) = (3)

is a

be a neighborhood of

fbe a Crimbedding of U x

Rmx [o,a] which equals theidentity on There is a

m R whose closure

be an open set in

If f

X(x,t)

UxO

outside

and on

C x[o,&],

5 > 0.

for some

Vxto,a/2].

is the identity on some set

xx[o,b],

then

f1

isthe

identity on this set also. Proof. Let and

equals

which equals Let

0outside 1

for

f*(x,t)

cp(x) be a C°° function on V.

Let

t < 1/3

r(t)

e a/2.

is in

X.,(x,t) = X(x,t(1 - rep)) = x.

in

Rm ,

on

"J

t.

so is

h^

for

x

f

is the

t(1 - yep),

so that

f

is a diffeomorphism, and for

h^-(x) = X 1(x,t)

be an imbedding of

By Exercise (b) of 3.10, there is

such that if

so condi­

Thus condition (3) holds.

suffice that themap

foreach

0

gt(x) = X(x,t),

Then there is a positive constant

|X.(x,t)

U

sQ

such

- X(x,t)| < 50

and |dX., (x,t)/dx - dX(x,t)/dx| < b Q for all

x

in

U.

By uniform continuity of such that if

x

X

|X(x,tQ) - X(x,t) | < &0

is in

U

and

and

|t - tQ | < eQ .

entries of

|dX(x,t)/dt *dcp(x)/dx|

minimum of

eQ

Now

and

X.,(x,t) = X(x,tQ),

there is a constant

The first term is within

M x

be the maximum value of the in

U.

where

SQ/2

with a ball in

asdesired.

=

Also,

- t r(t/e) [dX(x,tQ)/dt • dcp(x)/dx]

of

dX(x,t)/dx,

M.

U^,

and the second is within

f

f^

near

(1)

T± (x,t) =t

on

(2)

X^(x,t) =x

for x

c

C^

be a compact

For convenience, assume

which equals

Y,

so that

Cover

M

by a locally-

for which there is a diffeomorphism

Rm . Let

Induction hypothesis:

Choose

to be the

Hence our desired result follows.

that CC^} covers

withf(Y)

e

Hence 11 - tQ |

|X.,(x,t) - X(x,t) | < 5 Q ,

that

finite collection of open sets of fL

Choose

tQ = t( 1 - rep).

Completion of the proof of the theorem.

h^

eQ

|dX(x,tQ)/dx - dX(x,t)/dx| < 5Q /2

Let

for

h X ^ ( x , t ) / b x = dX(x,tQ)/dx

of zero.

dX/dx,

sQ /2M.

|t * r(t/e) • cp(x) | bj

is convex

in particular, it for which

It also contains a point L^(y) < b^,

Bd c

with

is a cell

(P for which

J / i,

and contains

d^

L^(x) = b^;

Li (x) > y

such

since otherwise

S

would lie entirely in the region Li(x) > b^, inequality the set Then (P^.

so that discarding the L^(x) > b^

c.

S n (J^ Since

from the set of inequalities for

By convexity,

S

is not empty; S n (P C d^,

contains a point since

d^

S

is open in

z

c

would not change

such that (P,

S n (P1

must be a cell of dimension

L^z)

= b± .

is open in

m - 1.

§7.

CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE

The uniqueness of the cells

d^

is easy:

Bd c

73

is contained in the

union of p planes of dimension m-1. Hence the intersection of any other m-'l plane with Bd c lies in the union of finitely many planes of dimension less than m-1, and hence has dimension less than m-1 .

7.4 into which d^

Definition. If

c

is an m-cell, each of the m - 1 cells

Bd c decomposes is called a face of

is called a

m - 2 face of

is also a face of

7.5

c,

as

c.

And so on.

is c

Lemma. Let

c;

each

d^

m - 2 face of a

By convention, the empty set

itself.

cbea cell given by a systemof linear equations

and Inequalities. Replacing some inequalities by equalities determines a face of

c, and conversely. Proof. We proceed by induction on

m =: 0, the lemma is trivial.

If

m,

m > 0, let

c; adjoin to the system determining

c

the dimension of

(P

a set of equations for (P.

dowithout loss of generality.

equalities for

c, so numbered that those for which

LetL^(x) >

Then let us replace some Inequality

J < V, this determines an m-1 face In the case

j > p,

dj of c,

1 < i< p

be the in­ form a

^j(x) > "bj ^ as

consider the subset

b^

of c

This

an equality.

we have just proved. satisfying

bj. If the hyperplane fa3e

c;

let

e

^j(x) = b-

contains

if it does not intersect be the intersection of

c,

bitrarily near each point Hence

e

q

of e

e

Otherwise,

it is a cell. . Now

is an m - 1

are points y

plane,

of CP for which

so that ar­ Lj(y) < "bj •

c.

is convex, It must lie in some m-1 faced^: otherwise,

be the smallest integer such that

be a point of q

e

this gives the trivial

c with this hyperplane;

lies on the boundary of Since

(P,

it gives the empty face.

the intersection of CP withthis hyperplane

d_. .

L.(x) = J

•

let

If

bethe m-plane containing

we clearly may

minimal set.

c.

not in

d1U...U d

e

.j;let

y

lies in

d1U...U d . Let

be a

point of '>

Then Li(x ) > b i

for

1 < ^

and

Lq(x ) =

Li(y) > b±

for

i < q

and

Lq(y) > bq .

e

x

not in

7^

II.

The point that

z

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

z = (x

+ y)/2 lies in

does not

e,

but

L^(z) > b^

adjoined, determine the

m - 1 cell

Furthermore,

e

an equality.

By the induction hypothesis,

d^

c,

which contains

e

is a face of

Rn

d^,

and of

K

c.

is a collection

such that

(1)

Each face of a cell in K

(2)

The intersection of two cells of K Each point of

many cells of

e.

Lj(x ) >

Definition. A (rectilinear) cell complex

of cells in

so

with the equation

is obtained from this system by replacing

7.6

(3)

i < q,

lie in d ^.-.U d , contrary to hypothesis.

The system of equations and inequalities for L^(x) = b^

for all

K.

|K|

is in

K. is a face of each of

them.

has a neighborhood intersecting only finitely

(Here |K|

denotes the union of the cells of K.)

In order that this definition be non-vacuous, we need to note that asingle cell

c,

along with its faces, constitutes a cell complex.

This follows

from the preceding lemma (see Exercise (a) below). A subdivision of a cell complex [K f| = |K|

and each cell of

K

is a cell complex

K' is contained

in one ofK.

of. K. is the dimension of the largest dimensional cell in ton of

K,

denoted by

dimension at most

[

p.

Kp ,

(a)Exercise.

faces, show that

K

If

K

Kp

such that

The dimension K.

is the collection of all cells of

One checks that

K'

is a subcomplex of

The p-skeleK

having

K.

consists of a single cell, along with its

is a cell complex.

(b)

Exercise.

Show that the simplex

a = vQ ... vm

is an m-cell,

and that the notion of "face" is the same, whether one uses the definition in 7.1 or the one in 7A.

Show that a simplicial complex is a special kind of

cell complex.

7.7 ]K2 \ .

Then

Lemma. Let K1

and

K2

K 1 and

K2

be cell complexes such that

have a common subdivision.

|K^ | =

§7.

CELL COMPLEXES ADD COMBINATORIAL EQUIVALENCE

Proof. cell.Let c1 and

It is clear that the intersection of two cells is again a

Lbe the collection of

is a cell

of

K1

and

Exercise.

e1 n e2, where (b)

c2 is a cell of

K2 .

e^

or

is a face of

c^;

the form Cj nc2, where Then

L is a cell complex,

Show that L

m = 1,

is an m-cell of

K

is a cell

K

complex.

K,

let

a., * c,...,a

* c.

I

has a simplicial subdivision.

is already a simplicial complex.

Choose an interior point

c.)

is of the form

We proceed by induction on the dimension

is a simplicial subdivision of

plices

c1 n c2

and conversely.

Lemma. Any cell complex

Proof. m=o

Show that any face of

Exercise.

7.8

and

all cells of

IK, | = |K2 | = |L|.

(a)

L

Km ~1 ,

m

c

adjoin to the collection

c;

(Recall that

* c

L

K ’ be canonically defined.

(a)

division of

Exercise.

lying in L

K.

If

c

as the centroid of K

Bd c.

the sim­

which is a subdivision of c

c

c,

the

in

is already simplicial, the K1

is called

K.

Check that

Kr

is a complex, and that it is a sub­

K.

7.9 |K., I = |K2 |,

7.10 in

If

K,

centroid is the same as the barycenter, and this subdivision the barycentric subdivision of

K.

of

It is often convenient to choose order that

If

denotes the join of

If we carry out this construction for each m-cell K'

K.

the m - 1 skeleton of

be the simplices of

of

of

In general, suppose

a 1,...,a

result will be a simplicial complex

I

75

Corollary. K1

and

If

K2

Theorem.

K1

and

K2

have a common simplicial subdivision.

Let

K1

and

K2

Rn . There are simplicial subdivisions

respectively, such that

are simplicial complexes such that

Kj U K 2

be two finite simplicial complexes Kj

and

K2

is a simplicial complex.

of

K1

and

K 2,

76

II.

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

Proof. A rectilinear triangulation of L

such that

plex of

Let J.j

of

Rn

L1

of

Rn :

be the simplices of

K1.

which contains

ct1 .

is to take any rectilinear triangulation affine transformation a1;

is a simplicial complex

|L| = Rn . We prove that some subdivision of

a rectilinear triangulation

angulation

Rn

the collection

h1

of

h 1(J)

Rn

K1

is a subcom­

Choose a rectilinear tri­

One way to construct such a

J

of

Rn

and find a non-singular

which carries one of its simplices onto

will be the required complex.

(An affine trans­

formation is a linear transformation composed with a translation.) let

be a rectilinear triangulation of

a common subdivision of

Rn

containing

(using 7.9)*

tope of a subcomplex of

L 1,and

Then

ck;

this subcomplex is a subdivision

ing some subdivision of

K 2 as a subcomplex.

Let

vision of

and let

be the subcomplexes

L 2,

having polytopes

7.11 and

K 2 are not finite.

of the sets

K]

|K2 |,

You

for

Definition.

ing situation:

K2

L

Rn

contain­

be a common

subdi­ of

Let

K1

L

respectively.

[0,1] xO

K1

in

R2

and

|K2 | is the union

n = 1,2, ... .

The proof of Lemma 7.8 generalizes to the follow­

be a subcomplex of the simplicial complex

be a simplicial subdivision of K.J

triangulation of

K1.

will need some further hypothesis to avoid

|K., | is the set

[o,1 ] xl/n

7.12

tending

and

be

of

Problem*. Generalize this theorem to the case in which

the case where

K.J

| and

L1

|^ | is the poly­

L2 be a rectilinear

and

Similarly,

let

Similarly, let

L1

J1

to a subdivision of

K;

let

K 1. We define a canonical way of ex­

K,

without subdividing any simplex outside

St(K1,K). We will call it the standard extension of

K.J

to a subdivision of

K: Every simplex of plex in

of K

A =

ces of

which is outside

St(K1,K) - | | K

belongs to

lie in

A.

St(K1,K).

are Each 1-simplex

K 1,

of course, as does every sim­

The simplices whose interiors lie subdivided step-by-step as follows:

a

of

K

whose interior lies in

A

No

§7.

CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE a

is subdivided into two 1-simplices, the barycenter of vertex.

In general, suppose the For any

Let

be its barycenter, and for each simplex

Bd c,

s * a

let

a,

its boundary has already been subdivided.

a

is

a

itself.)

(a)

Exercise.

Problem. Let K

quence of simplicial subdivisions of

K

This means that any simplex of

a

simplices integer

positive

as well. a

such that

Let

lim^

belongs to

I

such

Let K 1, K 2, ... that Kj_+1

^

for all

i

minimum of

in

5(x) ,

for

x

A^

is a

K.

be a simplicial complex; let |K|.There is a subdivision K*, the diameter of a.

outside

greater than some

Exercise.

a of

se­

denote the collection of those

(a)

such that for any simplex

be a

be a

which does not intersect

Show that this collection is a subdivision of

continuous function on

{A^}

equals

Na.

Let K

the

K.

be a simplicial complex; let of |K|.

Kj_+1

(Convention:

|K|.

locally-finite collection of subsets

simplex of

o.

Check that the collection of simplices obtained

this way is a complex whose polytope Is

A^.

of the subdivision of

After a finite number of

steps, we will have the required subdivision of

7.13

s

be a simplex of the subdivision of

join of the empty set with

being the extra

m - 1 simplices have already been subdi­

vided. a

m-simplex

77

d( x)

K1

be a of

K

a is less than the

|

§8.

Immersions and Imbeddings of Complexes

In this section, we define the notion of a Cr map where

K

is a simplicial complex and

M

f : K -► M,

is a differentiable manifold, and

we develop a theory of such maps analogous to that for maps of one manifold into another.

In particular, we define (8.2) the differential of such a map

and use this to define (8.3) the concepts of immersion, imbedding, and tri­ angulation (which is the analogue of diffeomorphism) . As before we define (8. cj) what is meant by a strong

C1 approximation to a map

f : K -*■ M,

and

prove the fundamental theorem which states that a sufficiently good strong C1 approximation

g

to an immersion or imbedding is also an immersion or

imbedding, respectively.

(The theorem also holds for triangulations, with

the additional hypothesis

that

g

carries Bd|K|

From now on, we restrict ourselves divisions, unless otherwise specified.

into

Bd M.)

to simplicial complexes

The integer

r

(1

and sub­

0

and

so

D(dffe)(x) =

x = b. Since U

for any

the do­

of b such

that

"u.

such that any

Cr map;

K

non-degenerate

f on

U

is an

Cr

a angle-

U.

€ = min || df^(x)||/|| x — b || f

vector in

is a continuous

be a non-degenerate

for any

for all b

is non-degenerate,

be a non-degenerate

U.

be a simplex of

in

e > 0. Let

Cr map which is a

The angle between bn/e = a.

\\v\\ 29 /it;

"v

0

8.8 Theorem. K be finite.

let

df^(x)

b

and

and

Hr is less

6

than

|K| and all

5

= ccc/it;

5 -approximation to

,

is

a;

b

the same as that

where "u = (x

|| "v - "w

let

- b)/||x-b||.

|| jt/|[v||,

which is less

||"v - "w ||> \\v\\ > |(v|| 0/«.)

is not acute,

f : K -+ Rn

There is

lie in

||"v - ”w || > |(v|| sin 9 >

is acute, then

Let

x

dgb (x)

and "w = D(g|o)(b) •"u

and

(For if

and if

K f;

Gbetween

The angle

between "v = D(f|a)(b) •"u

let

- b),

is a

U. Let a

than

x

: K -► Rn

U,

^(b,K) ;since

g : K' -► Rn on

when

there is a

on

Proof. Let

(x

Df(b) • u

which is a 5-approximation to

approximation to

x ^ b

and

Df(x) •"u

there is some neighborhood

f

> 0,

Rn

df^x) = Df(b) •

Df(x) •u

a compact set,

Given

is a unit

and its value is zero when

angle is less than

finite.

and "u

we wish to make the angle between

Df(b). Now the angle between function of

o,

is a point of

K

a& > 0

be a

such

Cr immersion, or imbedding;

that any

s-approximation to

f

is an immersion, or imbedding, respectively. Proof.

We

first make two preliminary

definitions, andprove a

special case of the theorem. (1) Let face of each.

a1

and

a2

The angle from

be two simplices, with a1

to ct2

a = c^n a2

is defined as follows:

a proper Let

a

§8.

IMMERSIONS AND IMEEDDINGS OF COMPLEXES a.

be the barycenter of jo

ments.

and

za,

mum such angle

intersect

a.

different from

a2 - St (a) .

e(y,z)

lies in the face of

is positive:

a2 - St (a);

over

y

between the line seg­ a

o1

and

a

z

The mini­

to

So we would get the same minimum if z

and

ct2 .

If we extend the ray from

since y and

a

opposite

This angle is positive.

is called the angle from

e

e

Now

where

a2

is any point of

Consider the angle

85

z,

z

it will

ranged only

then-range over compact sets,

e

is

positive. 6

Now the angle of

a 1 anda2.

a

and

a2

If we

from

a

to

c?2is a function of the vertices

vary these vertices continuously

to become degenerate),

then

e

(without-allowing

varies continuously as well

(see Exercise (a)).

(2) projection of itself. c?.j

fined points px into

a

o

onto

Second, suppose

opposite

a

Let

a;

y

are parallel.

let of

a

s1

be a proper face of the simplex

as follows: p

First, it is the identity on

belongs to

a1 - a.

be the barycenter of and

x

of

The projection of

a . We define the

a

Let a.

s1

be the face of

There are uniquely de­

such that the line segments a1

onto

a

a

ya

is defined to carry

and p

x. Projection is a continuous map of

o

onto

a;

furthermore, it is

natural in the sense that it is invariant under a linear isomorphism of with another simplex (see Exercise (b)). The fact about projection which is useful for us is the following: Let

a = a^n a 2

be a proper face of both

and

a2 ;

let

p

belong to

II. TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

a

into

and

x,

q

then the angle between the line segments a

than the angle from

az

such that angle from

a1

(3)

to

to

is parallel to

xq,

F : L -► Rn be a 1 - 1

e1

a > o

tion to

itfollows that

For let a = it.) a.

G

is

is no less

Now no vector

Rn

onto

0

having diameter less than Lg

a.

t(a)

Cr map;

of

let

d(a) a

8

isthe mini­

e

of

is

and

such that for any simplex

a

r(cr)/d(a).

tQ s

be con­

and thickness greater than

is a 8-approximation to f|s. 90

a

The diameter

The thickness

f : a -► Rn

There is an

Bd

of the simplex

tQ ,

§9. Proof. Then

Given

b

f(x) - Fb (x)= h(x,

11x - b ||

0.

(See

THE SECANT MAP INDUCED BY f a,

in

b) ,

where

(2) of 1 .10.)

It follows that there is an approximation to f less than

€

;

center

of

s.

of a simplex

Df(x) - DFb (x) = b

of

||h(x, b) ||
‘**>vp

be the general point of L s(x) =

Z

Pb (x)

E a±

and -

s.

a. L g(v±) =

Since

Z

Lg

of

andFfe are

s; linear

a. f(v±)

Fb (v.)

,

so that II L s (x )

Now 5

||h(x,b) ||


K. Given

and thickness greater than

C(o, o, o)

.

(To illustrate, the accompanying illus­

intersected with

R(o), R(l), and R(2).) Take

as the collection of all such cells and their faces.

The cell complex

§9. THE SECANT MAP INDUCED BY

J

has two properties which are important for us: (1)

Any cell of

unit cube

J

C(o,...,o),

(2)

is the image of one of the cells contained in the under a translation of

The simplex

subcomplex of

J,

spanned by

J

is the polytope of a

m.

into a simplicial complex

step process described in Lemma 7.8; c

Rp .

0,m€1,...,m€

for each positive integer

Now we subdivide

of

93

f

L

by the step-by-

at each step we use the centroid

as the interior point of the cell

c

which determines the subdivision.

It follows that conditions (1) and (2) hold for the complex a result, the simplices of diameter

L

c

have a minimal thickness

L

tQ > 0,

as well.

As

and maximal

d. The similarity transformation on

Rp

which carries

x

into

x/m

does not change the thickness of any simplex, and it multiplies the diameter by

1/m.

Therefore the image of

rectilinear triangulation of most

d/m. Since

(a) Let

K1

m

Z1

under this transformation will be a of thickness at least

*

Exercise .

K 1,

etc.

and diameter at

is arbitrary, the theorem is proved.

Let

K

consist of a 2-simplex and its faces.

be the barycentric subdivision of

vision of

tQ

K,

K2

Show that the thickness of

the barycentric subdi­ approaches zero as

n -► so.

9.5

plexes

K.

Unsolved problem*. Generalize this theorem to non-finite com One would expect

tQ

and

e

to be continuous functions on

but it is not entirely clear even what the proper conjecture should be.

|K|,

9*+

II. TRIAMGUIATIOMS OP DIFFERENTIABLE MANIFOLDS 9-6

5 > 0,

The orem.Let

f : K —► Rn

there is a subdivision

L^-, : K ’ -► Rn

induced by

f

K*

be a

of

is a

K

Cr map;

K

finite.

Given

such that the secant map

5-approximation to

f.

We now obtain the following generalization of this theorem, which will follow immediately from the lemma which succeeds it:

9.7

Theorem.Let

subcomplex of to

f

K.

Given

> 0,

Cr map;

there is a

(1)

g equals the secant map induced by

(2)

g equals

Here K1

f K

outside

let

K1 be a finite

&-approximation

g : KT

Rn

outside

9.8

K

subcomplex of

K.

Given

mationg : K.J

Rn

to

:K* -► Rnto h

equals Proof.

is defined on to extend

h

K1

inducedby St(K1,K)

f : K -► Rn

e

> o,

f |K1

f.

Here

f

outside

on

K.J.

St(K1 ,K).

outside

Lemma. Let

f

St(K., ,K) .

is the subdivision of

that every simplex of

and

5

be a

such that

(3)K ’ equals !

h

f : K -► Rn

K ’;

and condition (3) means

appears in

be a

Cr map.

there is a

5

> 0

Let

K1 be a finite

such that any

can be extended to an

K’

K ’.

5 -approxi­

e -approximation

is the standard extension of

K.J

(see

7.12),

St(K1,K).

The subdivision |K1 | and on

K' of

K

is specified, and the map

h

|K| - St(K1 ,K). The only problem remaining is

to each simplex of

K'

and to see how good an approximation

whose interior lies in h

St(K1,K) - |K1 |,

is.

We carry out this extension

step-by-step, first to the 1-simplices, then

the ?-simplices, and so on.

For this reason, it will suffice to prove our lemma for the special case of a single simplex: Special case. Let L its faces;

5-approximation tion.

h: L T

be a

L1 = Bd a .Given

let

g: L^ -► Rn to

e

complex consisting of a simplex > 0,

f|L1 may be

Rn to f|L , where

there is

5

> 0

a and

such that any

extended to an e-approxima­

L’ is the standard extension of L^

§9. THE SECANT MAP INDUCED BY

f

95

Proof of special case. An obvious extension of one mapping the line segment a

where

ya

is the barycenter of

a.

g(y) f(a ),

But there are difficulties here,

definition, "tapering off" between

a.

g(Bd

and

a)

We modify this

f(a)

by a smooth function

rather than linearly. Let

for

would be the

linearly onto the line segment

for the resulting map will not be differentiable at

a(t),

g

a(t),

1/3 < t,

as usual, be a monotonic

and equals

1

ty + (1-t) a,

where

mined, and

is unique if

y

y

for

is in

t > 2 /3 .

Bd a

x / a.

and

C°° function which equals If

x

is in

0 < t < 1;

Furthermore, if

a,

then

o

x =

t

is uniquely deter­

L.J

is any subdivision

y of

Bd a,

and

functions

of Let

L ? is the standard extension, then y x g

and

tare

oneach simplex of L T, except at the point r ’ bea C map of a subdivision L., of L 1 into

C°° a.

x = n R .

Define h(x) = f(x) + a(t(x)) •[g(y(x)) - f(y(x))] Since

a(.t(x)) = 0

each simplex of to

f

If

g

for

in a neighborhood of

L ’. We show that

h

a,

h

is of class

f

as

on

f|Bd a.

|| h(x) - f(x) || < || g(y) - f(y)||,

C° approximation to

Cr

is automatically a good approximation

Is a good approximation to

First note that as good a

x

g

Second, we

compute d(h-f)/dx for

a ’(t)[g(y)

- f(y)J - b t / h x

so

h will be

is. t > 1/3.

+ a(t)[dg/dy -

It equals /by]

•b j / b x

,

96

II. TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

where

and

f

are written as column vectors, as usual.

a(t) * b j / b x

and L.J

g

and the map

a'(t) • b t / b x

Now

are bounded independent of the choice of the subdivision g.

Hence we may make

simply by requiring

|g(y) - f(y)|

d(h-f)/dx

as small as desired,

|b g / b j - df/dy|

and

to be sufficiently

small.

'

(a)

Exercise.

l^g/^y - df/dy| than

Compute exactly how small

need to be in order that

|g(y) - f(y)|

|d(h - f)/dx|

and

should be less

5.

(b)

Exercise.

Complete the proof of the lemma using the result

for the special case proved above. (c)

Exercise.

Suppose thatwhenever then

g

carries

a simplex

a

(d) planes in

of

a

K

aof

also carried

by g

P

carries a simplex

into into

CP Let

CP ,

a1of

K1

, as well. Prove then

CP 1 ,

h

carries

(P

^

into the plane

CP ,

that whenever f carries a

into

(P

,

as well.

be a finite number of the

Prove Theorem 9-7, with the additional conclusion that

each simplex

plane

f

Exercise.

Rn .

(e)

Show that the following addendum to the lemma holds:

K

Exercise.

which fcarries into into

oneof the planes

CP

CP ^ is ______________ j

Show that (c) does not hold if you replace the

throughout by a half-plane, for Instance

Hm .

§10.

Fitting Together Imbedded Complexes.

Suppose now we have 3ntiable manifold

M,

CP imbeddings of two complexes into the differ-

whose images in

M

overlap.

We would like to prove

that by altering these imbeddings slightly, we may make their images fit to­ gether nicely, i.e., intersect in a subcomplex, after suitable subdivisions. If

M

is euclidean space, this is not too difficult; the construc­

tion is carried out in

..

10 2

The basic idea is to alter the imbeddings so

that they are linear near the overlap (using the results of §9), for we know from §7 that two rectilinear complexes always If M on

M,

intersect nicely.

is not euclidean space, we must use the coordinate systems

which look like euclidean space, and carry out this alteration step-

by-step.

The process is not basically more difficult, but it is somewhat

more complicated (10.3 - 10.4).

After this, however, the two main theorems

of Chapter II fall out almost trivially (10.5 - 10.6).

10.1

Definition.

phisms;

let

f^lK^)

(K.|,f.|)

and

(K2,f2)

imag;es of of

K1

with of

f1

f 1 : K, -► Rn

f 2 ( |K 2 1)

and

f2 : K 2

Rn

be homeomor-

be closed subsets of their union.

are said to intersect in a sub comp lex if the inverse

( |K-, |) n f 2( |K2 1)

and K 2,

L 2.

and

Let

are polytopesof subcomplexes

respectively, and

f^^i

a li-near isomorphism of

They intersect in a full subcomplex if

whose vertices belong to

such a case, there Is a complex

L^, K

L 1 and

either for

L^

L1

contains every simplex

i = 1

and a homeomorphism

L2

or for

i = 2.

f : K -* Rn

In

such

that the following diagram is commutative:

Here

i1

and

i2

are linear isomorphisms of

97

K1

and

K2

with sub complexes

98 of

II. K

TRIANQULATIONS OF DIFFERENTIABLE MANIFOLDS

whose union equals

morphism;

K.

The pair

it is called the union of

the subcomplex

(K,f)

(K^f.,)

is unique up to a linear iso­ and

(K2,f2) .

Let

L

denote

i^K-j) n i2(K2) .

We will be interested in this situation in the case where f1 : K 1 -*• Rn

and

f : K -► Rn

f2 : K 2 -*■ Rn

will be a

are, in addition,

Cr imbeddings.

The map

Cr non-degenerate homeomorphism in this case, but it

may not be an imbedding, because the immersion condition may fail. for some point is.

b

of

|L|,

the map

(See the example of 8.1.)

on

13£(L.,Kj)

for

j = 1 and

df^

However, if 2,

f^

1-1,

even though

b

f

should happen to be linear

then the union is necessarily an

ding, for in this case dffo = f |St b for 1-1.

may not be

That is,

in |L|;

and

This fact will be of importance in what follows.

f

imbed­

is given to be

Another case in

which the union is an imbedding is given in Exercise (c).

I

(a)

Exercise.

Let

f p : Kp -►

f . : Kj -*■ Rn

be a homeomorphism,

j = 1,2.

If

f1:

K 1 -► Rn .and

f1 :

K.J -► Rn and f?:K p -*• Rnintersect in a full subcomplex, where K.J ! K2 are the barycentric subdivisions of K 1 and K p, respectively.

and

(b) union

Exercise.

Prove existence and essential uniqueness of the

(K,f). Outline: where v J

fj(vJ), simplex

Rnintersectin a subcomplex, prove that

of

Form an abstract complex Is any vertex

K.,let the collection

J

of

K.

K

whose vertices are the points

(j = 1,2).

If vq

{f .(vjb ,...,f.(v^)}

J

u

Then take a geometric realization of this abstract complex, and define To show

f

isa

be a simplex of

y

j

••• vp

K.

f.

is a homeomorphism, you will need to consider the limit set of

f. (c)

Exercise.

Let

f. :

-► Rn

whose intersection Is a full subcomplex. and let

1.

Int i1 ( |K1 |)

be the inclusion of K. and

10.2 subcomplexes of

Int ip(]K2 |),

in

then

Lemma. Let P

whose union is

whose images are closed In

Rm ;

be Let K.

Cr imbeddings f : K -* Rn If

f : K —► Rn

f : P -► Rm P;

let

f|Q

let A be finite.

|K|

be their union,

is the union of

is a

be a and

(j = 1,2)

Cr Imbedding.

Cr map. f|A

Given

be

Let

I

Q

and

Cr imbeddings

A

bo

§1 0 . FITTING TOGETHER IMBEDDED COMPLEXES s > 3, and

5 -approximation

there is a

g : P 1 -► Rm

f

such that

g|A! intersect in a full subcomplex, and their union is a Furthermore,

g

equals

f,

St5(f~1f (A) ,P) = St("S^CS^(f~1f (A)))) \ Proof. Let

PQ

and

P1

equals

faces, so it contains f(P) - f(PQ)

to

carry a point of

A.

f(A),

We assume so that any

P - PQ

P1

P.

Td £( f-1 f (A)) ; it

along with their

is less than half the distance from

&-approximation

into a point of

g

to

f

cannot

g(A). We also assume that

5 -approximation

be the sub comp Lex of

Theorem 9.7, there is a

equals

P

P

5-approximation

the; secant map induced by P’

CP imbedding.

g : P1

R

to

f,

&

is

g |Q,1 and

CP imbeddings.

Let

and

5

f(A),

g|Q1

outside

be the complex whose polytope is

small enough that for any g|A1 are

P,

all stars being taken in

consists of all simplices whose images intersect

f

to

99

f

outside

whose poLytope is g : P’

on the subcomplex St(P1,P).

Rm P.J;

to

f

further,

T5£(P0 ,P) , By which equals g

equals

II.

100

Let Now

g|Q1

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

Q1

and

be the part of

P.,

that lies in

Q,

so that

P1 = Q 1 U A.

g|Af are linear imbeddings, so the images are finite recti­

linear complexes in euclidean space.

By Theorem 7.10, one may choose sub­

divisions of these complexes whose union is a complex. These subdivisions -1 " i I " i M induce, via g , a subdivision P^ of P1 such that g|Q1 and g|A I T t

intersect in a subcomplex. so that 10.1). Pn '

glQ^'and

We take the barycentric subdivision

P'

in

Am .

P1 ,

P.J

in the standard way to a subdivision

(see 7.12) .

We now claim that For suppose

of

g|Am intersect in a full subcomplex (see Exercise (a) of

Extend this subdivision of of

II

P1

g(cr1)

Then

g|Q,n

intersects

a1

and

g|A,M

intersect in a subcomplex.

g(a2),

where

a1

must lie in

so that in particular, sect in a subcomplex.

cr.,

|P0 I,

lies in

Hence

Q,,M

and

a2

hy our original assumption on

|Q1 |.

g|Q,n

is in

and

But

g|Q.JM

g|Am

and

is

8,

g|AM *

inter­

intersect in a subcomplex.

A similar remark shows this subcomplex is full. It follows that the union of ding;

g|Q,M

and

we apply the remark at the end of 10.1.

b1

in

|Q,|

5.

But

g

union of

and

b2

in

|A|,

then

is linear on

P" ’,

which contains

g|Q11T and

10.3

Corollary.

euclidean space. throughout, system

g|A,M

is a

Let

M

b1

For if

must lie in

is a

Cr imbed­

g(bj) = g(b2),

choice of

|P0 1*

TTE( |PQ |,PIM) .

Hence the

be a non-bounded

providing we assume that

f(A)

Rm

Cr submanifold of some is replaced by

M.

f-1f(A)

carries

P?

of

P

^ ^ ( f -1f(A),P')

lemma applies.

and

f~1f(A)

in the standard way. into

which

to

f"1 (M - U) . Extend to

The result is that

f

U.

We then take the subcomplex TfFE^f'~1f (A) ,P1)

P

and subdivide them finely enough that each has diameter

less than one-fourth the distance from a subdivision

M

is contained in some coordinate

Proof. First take the collection of those simplices of intersect

for

Cr imbedding.

The preceding lemma holds if

(U,h) on

g|A,M

consider the

J

of

Cr map

There is an approximation

P*

whose polytope is

hf : J -*■ Rm . G : J’

Rm

The preceding

to

hf;

G

equals

§10. FITTING TOGETHER IMBEDDED COMPLEXES hf

and

J f equals

g := h”1G

on

J

outside

St^(f”1f(A),Pf).

S t \ f “1f (A) ,P ’),

obtain the desired map;

and

g = f

no simplex of

Pf

101

Hence we may define

outside

^ 5(f“1f (A) ,P')

outside

J

to

needs to be sub­

divided at all.

sure

g

The preceding lemma guarantees that

G

is a homeomorphism.

is, we need to have

f

closely enough that

g

approximate

To be

g(P - S t \ f -1 f (A) ,P ' ) ) are disjoint. Similarly, r —1 the lemma guarantees that G is a C imbedding; the fact that h G g ( ^ 5(f”1f (A) ,P'))

and

is an imbedding follows from Exercise (a) of 8.3. cise (c) of 10.1 to prove that

(a)

dinate neighborhood

10.4

M.

and

g» : L ’ -► M

to

f

Let

g

disjoint union of

M.

{’7^}

(*) and

g1

to

and

respectively.

Cr imbeddings whose images are closed in

5-approximations

g

(8

f': K ’ -*■ M

and

is to be continuous on

carries each simplex of

(See Exercise (a) of 7.13.)

Choose a neighborhood

Assume f

and

8

is small enough

g,

the

L

into a coordinate

Order the simplices of

of

g(A^)

in

M,

for each

respectively,

M.

that for any 8-approximations g ’(A^)

i,

is contained in

f'

and is

f'f_1(M - V^). fQ = f

and

C1 imbeddings, where L;

Rn .

|L|.)

gQ = g.

Induction hypothesis.

of

be

Cr submanifold of

is a locally-finite collection of compact subsets of

disjoint from Let

f(A).

in such a way that each simplex is preceded in the ordering

by all its faces. so that

lies outside the coor­

be a non-bounded

Cr imbedding.

and

Assume

neighborhood on L : A 1,A2,...

|K|

f(x)

respectively, which intersect in a full subcomplex,

such that their union is a

Proof.

M

there are

and

if

chosen containing

g : L -► M

s(x) > 0,

Note that the following additional conclusion hol

g(x) = f(x)

(U,h)

Theorem.

f : K -*■ M Given

is an imbedding.

Exercise.

in the preceding corollary:

Let

g

Finally, one uses Exer­

f^

and

Suppose

f^: -+ M

is a subdivision of g^

are

Furthermore, if

K

and

and L^

g^: L^

M

are

is a subdivision

(1 - 1/21 ) • &(x) approximations to denotes the subcomplex of

L^

f

and

whose

g,

II.

102 polytope is

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

A-jU.-.U A^,

suppose that

f^

-► M

and

intersect in

a full subcomplex, and their union is an imbedding. Let

h : Q -► M

and

Consider

Kill

.

Now ;

Bd A^+1

A^+1

Cr imbedding which is the union of

and

as subcomplexes of

let

P

in order that

P

should be a complex;

Extend

h

to

P

are

nate system on to

h

M.

Cr imbeddings, and

Given

such that

L^,

e > 0,

h T[Q T and

J^.

h(A^+1)

h 1 |K^

and

g^

h : P

on

M

point

.

is a

is contained

CP map;

in a coordi­

€-approximation

h ’: P 1 -* M

CP imbeddings which intersect in

a full subcomplex such that their union is an imbedding. same as the union of

each

We may have to subdivide

The map

are

Q, =

we assume this done without

there is an

h'|A^+1

(K^,f^)

which we also

by letting it equal

Now apply the preceding corollary. andh|A^+1

so that

be thecomplex obtained by identifying

with the corresponding point of

change of notation.

h|Q

Q,

is the polytope of a subcomplex of

denote by of

be. the

h'|J.JUA^+1,

This union is the

so the union of the latter

pair is also an imbedding. e

If mation to to

f,

h,

is small enough, so that

f^+1 = h* |K_!

-*■ M is an

Cr map

e

is small enough,

g^+1 : L^+1 -► M

satisfied, so that

f^

and

g^

similarly for

g ’ and

L 1.

information to enable us to and

f* = lim. 1 — ► 00 1

f.

and

K ’2.= lim K .;and

The induction hypothesis does not contain enough

gj_+*| more closely. Q 1 equals

St5(h-1h(Ai+l),P). Hence f^+1 3 —1 for simplices outside St (f^ §j_(Ai+1)

equals

i.

do this, so let us examine the definitions of

outside

carried

may be extended

Thus the induction hypothesis is

are defined for all

Corollary 10.3 tells us that

on

UA^+1 — M

5 (x)/2 l+1 approximation to

which is a

We would like to define

(*)

(1 - 1/21+1)5 (x) approximation

h r:

Here we use Lemma 9 . 8 again.

g^: L^ -► M.

f^+1

5 (x)/2 l+1 approxi­

will be a

as desired. Further, if

to a

h 1:

5,

x K^,

carries a point into

V^.

outside

lection of subsets of

x

of

equals

Q f^,

and

h*

equals

andK^+1

h,

equals

K^,

by the original assumption K

into

Thus it is certain that

only if f^+1

equals

f^,

** and

Kj_+1

5 —1 St (f (V^),K) . These sets are a locally-finite col­ K,

so that

lim f^

and

lim

do exist (see 7.13).

103

§1 0 . FITTING TOGETHER IMBEDDED COMPLEXES Similarly,

g^+1

equals

g^,

st3(g^

and

L-j_+1

=

exist in this case as well. approximations to

f

and

equals

L^,

St5(A1+1, L±)

C

outside

St5(Ai+1, L) . Hence

The limiting maps will automatically be g, respectively.

s(x)-

That their union is a Cr im­

bedding is easily checked. (a)

Exercise.

By examining the proof of this theorem more closely,

draw the following additional conclusion: g(|L|), equals

then we may so choose f,

outside

tostrengthen

f*

L,

f’

and

K ’ equals

K,

and

be

f1

Apply Exercise (a) of 10.3 and Exercise (d) of 9 . 8 10.4 as follows:

Assume f

and

g

Cr submanifold

N

carry subcomplexes of

g 1 may be chosen so as to carry these subcomplexes into

g : L -► M

Theorem. Let

M

be a

C.r triangulations of

Cr manifold;

M.

let

M.

Then

N.

f : K —► M

There are subdivisions of

I

and K

and

which are linearly isomorphic. In fact, given 5(x) >

and

g 1: L' -►

M

Proof.

0,

there are

L 1 with

f

and

g

-*■ D(M)

carries Bd |K|

f'(Bd |K |) = Bd M, f 1( IK|)

= M.

10.6 triangulation.

D(M),

Theorem. If

M

If M

g'

&(x) > 0

so that any

In the other case, we consider

and apply Exercise (b) of 10.4 to assure that into

Bd M.

so thatf 1( |K |) C M.

Similarly, if 5

(f’)~1

are necessarily onto maps, using Lemma 3.11.

Our result then follows from Theorem 10.4. as a submanifold of

such that

f f: K f -► M

K ’.

In the non-bounded case, we choose

5 --approximations to

f 1 : K'

5 -approximations

which are Cr triangulations of M,

is a linear isomorphism of

M

is any neighborhood of

that

respectively, into the non-bounded

10.5

L

and

V

St5(f-1 (V) ,K) .

(b) Exercise.

and

K'

If

If

Again,

5

if5

is small enough, is small enough,

is small enough, g T(|L|) = M.

is a non-bounded Cr manifold, M has a

is a manifold having a boundary, any

o f the boundary may be extended to a

Cr triangulation of

M.

Cr

CP triangulation (If

104

II.

f : J -► Bd M

is a

triangulation of

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS CP triangulation of

g : L -*■ M

of

J with a subcomplex Proof. Let

systems

(U^,h^),

M of

C^j

covers

i= 0,...,m,

M.

and containing a Then

chosen

M

Uj ^ ± y

a neighborhood of

Cover M

by

contains

IL

is a

M

lies in

coordinate

is the union

if

5^(x) > 0

Rm (see 2.8). (Cjj)

contained in be the

com­

be a

5^(x) -approximation to

(K q ,gQ),..., (K^, g^) intersect

in

This union will be a

is chosen small enough that

We triangulate the space

g^(|K^|)

let

f : J -*■ Bd M be

|J| x[o,1)

of J,

a

as follows:

CP

tri­

Suppose

J

and each positive integer

cells

a x [1 - 1/n, 1- 1/n+1]

contained in

Rn+1.

The collection of all such cells is a cell complex |J| x [0,1);

simplicial complex without kind.

and

x (1 - 1/n),

Kq

be the subcomplex of

P:BdMx[o,1)-*M

which are

this cell complex may be subdivided into a

subdividing any cell ax (1 - 1/n)

We denote the resulting simplicial complex by Let

let

a

n,

consider the

whose polytope is

Cr

3.11).

M has a boundary;

a

Rm

let

C^.(see Exercise (b) of

Rn . For each simplex

of a

Cr imbedding whose image contains

-► M

chosen so that

Now suppose angulation.

+1

such that the collection

a full subcomplex and their union is an imbedding. triangulation of

m

so that h^(U^)

Sj_:

i = 0,...,m,

Cr

L.)

neighborhood of

hT1 :

is a

is a linear isomorphism

be a finite rectilinear complex in

Uj_1 % j •

f

of disjoint bounded open sets in

Let

hT1 , for

an extension of

g_1 f

be a closed set contained in

hj_(V^j), plex

such that

M be non-bounded.

countable collection Let

Bd M,

of the

second

K.

K whose polytope

is

|J|x[0 ,5 /6 ];

be a product neighborhood of the boundary.

The

composite

J when restricted to closed in

M

Let Int M. L

KQ ,

is a

and contains h : L -► M

By subdividing

L

L

Cr imbedding

JL->

g :KQ -*■ M

M, whose image is

P(Bd M x [0 ,4 /5 ]) in its interior.

be a

which intersects the set

P(Bd M x [0,3/4]), of

x [ 0,1) BdM x [0,1)

Cr triangulation of the non-bounded manifold

if necessary, we may assume that any simplex of P(Bd Mx(4/5))

is disjoint from the set

using Exercise (a) of 7.13.

Let

consisting of simplices whose images under

LQ h

be the subcomplex

intersect

§10.

FITTING TOGETHER IMBEDDED COMPLEXES

M - P(Bd M x [0,4/5)), and their faces.

Then

whose image is closed in

M - P(Bd M x [0 ,4 /5 ])

and is contained in

M,

contains

M - P(Bd M x [0,3/4]).

h : LQ -► M

105

is a

Cr imbedding in its interior,

By Theorem 10.4, there are

Bd

1

approximations

g 1: KQ -*■ M

1

and

h 1: LQ -► M

to

g

and

h,

respectively,

whose intersection is a full subcomplex such that the union is an imbedding. According to Exercise (a) of equals

KQ ,

|j|x [0 ,1 /2 ].

on

thus be the required h 1(|LQ |)

cover Now

h( ILq I)

10.4, we may assume The union of

Cr triangulation of

M,

g*

(K^,gr)

g(|KQ |)

contains

P(Bd M x [0 ,1 /2 ]).

K -+ M

interior. then

and

in its interior, and

g ’ and

Exercise (b) of

h 1 will contain these two and

Thus in this case

g 1( |KQ |) g T( |KQ |)

automati­

and

h 1( |LQ |)

M.

* 10.7 Problem . Let f:

will

g 1(|KQ |)

in its interior.

closed sets, if the approximation is good enough;

Let

and

and (Lq ,h ')

providing

P(Bd M x M / 2 ,4 /5 ])

M - P(Bd M x [0,3/4])

3.11 applies to show that the images of

do cover

g,

M.

contains

cally contains

equals

f|KQ

If

A

be a closed subset of the manifold

be a Cr imbedding such that KQ

is a subcomplex of

K

f( |K|)

contains

such that

may be extended to a triangulation of

f|KQ M.

A

M.

in its

triangulates

A,

1 06

II. TRIANGUIATIONS OF DIFFERENTIABLE MANIFOLDS * Problem . Let

10.8 of

Rn . Let

Rn

onto

M

K

in

plex

r:W -*■ M

Rn

be the retraction of a neighborhood

lying in If

M

W

is linear on

such that

sufficiently close to Given

approximation

f

If

in

M.

is of class

Cr

f,

g

TO be a C map.

there exists to

compact,

finite

subcom­

and use the fact that if

f: K -+ M

K' -► M

M is not

as the union of

is isotopic to

e(x) > 0, g:

M

is a Cr triangulation of

then the secant map induced by

* 10.9 Problem . Let

lowing:

r|K

f: K-+M.

Instead writeK

|K^| C Int |K^+1 |,

K£,

of

is compact, one may take the secant map induced by

this willnot suffice. with

W

Prove there is a rectilinear com­

chosen smooth triangulation

plexes

T* C submanifold

be a non-bounded

which was defined in 5.5.

Hint: asuitably

M

f.

g: K r -*• Rn

equals g

on

K^.

Prove that any map

Specifically, prove the fol­

8(x) > 0

such that for any 5-

there is a map

f^:

|K| x

I -»■

that (1)

f^

on the cell

a x I,

for each simplex

a

of

K 1, (2)

f^ : K' -► M

(3)

fQ = f

is an

and

e-approximation to

f,

for each

t,

and

f1 = g.

Hint: Use the techniques of Problem 5.15.

10.10 Definition.

Let

of the non-bounded n-manifold if for each

x

in the

M.

N 1, ...,

be non-bounded submanifolds

They are said to intersect transversally

intersection of any collection N.

of these 1

submanifolds, there is

a coordinate system

h: U -► Rn about

such that that of

h(U n N. ) is contained in a plane in xk N. , for each k. xk

complex of

K,

L

then we

and

f_1g

is a linear isomorphism of

may require that

(fM^g1

in

M

Rn whose dimension is

10.11 Problem. Generalize Theorem 10.k as follows: is a subcomplex of

x

i

= f_1g

(1) H on

If

H

onto a sub­ |H| .

M su

§10.

(2 )

Let

N1 , . ..,

tr a .a s v e r s a lly . and

L,

and

g'

FITTING TOGETHER IMBEDDED COMPLEXES

N

b e n o n -b o u n d e d s u b m a n ifo ld s o f

If

f

and

g

r e s p e c t iv e ly , in t o to

c a r r y th e s e

H in t:

c a r r y s u b c o m p le x e s

K ( j_ ) a n d

r e a d i l y fr o m E x e r c is e

( 1 ) , p ro c e e d a s f o llo w s : A ssum e

p le x .

...

A1, A2,

d e n o te th e

in d u c t io n h y p o th e s is i n and

g ^ : L_^

M

f

th e

s u b c o m p le x o f

f

and

^

g,

-► M

and

and

(1

-

f 7_L 1gg.1 = f 1 ,

w h o s e p o ly to p e I s in te r s e c t in

is

o fL

t h e p r o o f o f 10.1+ a s f o l l o w s :

r e s p e c t iv e ly , L^

H

s im p lic e s

a r e Cr I m b e d d in g s w h i c h a r e

to

K f ’

DL .

(2 ) w i l l f o llo w

To o b ta in c o n d it io n Let

w h ic h i n t e r s e c t

t h e s u b m a n i f o l d s I L ,t h e n w e m a y r e q u i r e

s u b c o m p le x e s I n t o

C o n d itio n

M

107

1

o f 9.8.

a f u l l subcom ­

not

in

S uppose

|H | .

H.

f^

/ 21 ) s ( x )

on

H U Aj

(d )

A l t e r th e

-*• M

a p p r o x im a t io n s

Let

U ... U A . ;

J.

d e n o te

suppose th a t

a f u l l s u b c o m p le x a n d t h e i r u n i o n i s

a Cr i m b e d d i n g .

T h e o re m .

10.12

n o n -b o u n d e d m a n if o ld c lo s e d i n g

such

M.

G iv e n

th a t

f _1g ' :

(1 )

( 2)

M.

g:

5(x) > 0, L 1 —>- K

_1

If

Let

Let

fg

Is

f:

L

K

M

5-a

p p r o x im a tio n

g!

o f th e

g (L )

: L T -► M

lo

lin e a r .

lin e a r on th e g’ = g

s u b c o m p le x

r e q u ir e

th a t

If

c a r r i e s s u b c o m p le x e s

g

b e a s m o o th t r i a n g u l a t i o n

b e a Cr i m b e d d i n g , w i t h

th e re is a is

M

on

H

of L,

we

m ay

|H|. L., , . . . ,

L^

o f L in to

th e t r a n s v e r s a l l y i n t e r s e c t i n g n o n -b o u n d e d s u b m a n ifo ld s N1,

of

M,

a n e ig h b o r h o o d o f m ay r e q u ir e P ro o f.

We a p p l y 1 0 . 1 1

L * —►M ,

w h e re

( f ’ ) ~ 1g n : -1

.1

g (L ± )

th a t

¥ e m ay assum e th a t

w i t h a s u b c o m p le x s io n .

r e s p e c t iv e ly , a n d i f

e

L ’ -► K

K , s in c e

in C

f ~ 1g

N±

f

fo r each

t r ia n g u la t e s I,

th e n we

. i s a l i n e a r is o m o r p h is m o f

t h i s m a y b e o b ta in e d b y a p r e lim in a r y

t o o b t a in € - a p p r o x im a tio n s f ' : i s s m a ll e n o u g h t h a t f ’ I s l i n e a r , a n d e q u a ls

is

K ' -► M

and

a tr ia n g u la t io n .

f -1 g

on

|H | .

We l e t

H

s u b d iv i­ g ": Then g' =

II. TRIANGULATIONS OP DIFFERENTIABLE MANIFOLDS

108

To obtain (2), choose a neighborhood in tain

g !f(LjL)

g«(Li)

C

N±

of

g(L±) .

Corollary. Let

g 1: L ’ -+M

is linear.

If

g1

of

K

to

g

f-1g

If

M

M.

K -+M

Given

and

g: L

s(x) > 0, M,

M

and

L

which triangulate f

g*

1 0.H Problem. (M C Int N) .

a triangulation of Hint:

Bd M

and

g,

a neighborhood

of

M.

N±) .

Then

there is a 5 -approxima­ such that H

of

V

f"1g ’: L 1-*- K

L, then

we may

1 0 .1 2 .

Let

the subcomplexes

to be full subcomplexes, we may obtaining triangulations

f

N

Bd M

to be the submanifold

will be the desired triangulation of

M

be a Cr submanifold of

N

Show there is a Cr triangulation of

and

g of

M.

which is closed

N

which induces

M.

Suppose that both

the hardest case.)

D(N)

will con­

be smooth tri­

other case, we form the manifoldD(M) . As-suming

of

f'CK^)

on |H].

D(M). We now apply 10.12, taking

N

and contains

is non-bounded, this followsat once from

D(M). A restriction of

in

f:

which triangulates

double the triangulations of

N^

is small enough,

is linear on the subcomplex

equal to g

Proof. In the

e

lies in

N± .

angulations of the manifold

choose

If

f(K^)

(see Exercise (b) of 3.11; by 10.11, both lie in

10.13

tion

so that

M

and

N

Extend the imbedding of

M

in

D(M)

into

i:

M -*■ N

Int N.

which induces a trianguiation of Apply 10.11, letting

have boundaries.

Bd N;

(This is

to an imbedding

Let let

N 1 = i(V), N2 = Bd M, and

f

i of

be a trianguiation g

be a trianguiation

N 3 = Bd N.

REFERENCES [1]

R. H. Bing,

Locally tame sets are tame,

Ann. of Math., 59 (195*0,

11+5-1 58. [2]

J. G. Hocking and G. S. Young,

[3l

W. Hurewicz and H. Wallman,

Topology, Addison-Wesley, Reading,

Mass., 1961. Dimension Theory, Princeton Univ. Press,

Princeton, N. J., [1+]

J. Milnor,

Differential Topology (mimeographed), Princeton Univ., 1958.

[5 ]

J. Milnor,

On manifolds home omorphi c to the 7-sphere, Ann. of Math.,

[6]

J. Milnor,

61+ (1956), 399-^05. Two complexes which are homeomorphic but combinatorially

distinct, Ann. of Math., 71+ (19 6 1 ), 5 7 5 -5 9 0 . [7 ]

E. E. Moise, Affine structures in 3-manifolds, V. The triangulation theorem and Hauptvermutung, Ann. of Math., 56 (1 9 5 2 ), .9 6 —1 1U .

[8]

D. A. Moran, Raising the differentiability class of a manifold in euclidean space (thesis), Univ. of 111., 1 9 6 2 .

[5 ]

M. Morse, On elevating manifold differentiability, Jour. Indian Math. Soc., XXIV (1 9 6 0 ), 3 7 9 -^0 0 .

[1 0 ]

J . R . Munkres,

Obstructions to the smoothing of piecewise-differenti-

able homeomorphisms, Ann. of Math., 72 (1 9 6 0 ), 521 -551+. [11]

C. D. Papakyriakopoulos,

A new proof of the invariance of the homology

groups of a complex, Bull. Soc. Math. Gr£ce, 22 (191 *3 ), 1-1 5*+. [1 2 ] N. Steenrod,

The topology of fibre bundles, Princeton Univ. Press,

Princeton, N. J., 1 9 5 1 . [131

J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math., 73 (1 9 6 1 ), 151 +-2 1 2 .

[*\b 3 J. H. C. Whitehead, On C1-complexes, Ann. of Math., 1+1 (191 +0 ), 809 -821+.

[1 5 ] H. Whitney,

Differentiable manifolds, Ann. of Math., 37 (1 935), 61+5-

680. [i6]

J . Stallings, The piecewise-linear structure of euclidean space, Proc. Cambridge Philos. Soc., 58 (1 9 6 2 ), 1+8 1 -1+8 8 .

[17]

M. Kervaire, A manifold which does not admit any differentiable struc­ ture , Comment. Math. Helv., 3^ (1 9 6 0 ), 2 5 7 -2 7 0 .

109

INDEX OF TERMS angle-appr oximati on

8. 5

approximation of a map, for manifolds for complexes ball,

Bm

3.5 8.5

7.1

barycentric coordinates

7.1

barycentric subdivision

7.1

basis for a differentiable structure

1 .6

boundary, Bd M

1.1

C°

6 -approximation

of a manifold, df of a complex, df^

euclidean space

cell complex

7.6

euclidean half-space

3 .5 , 3 . 8

combinatorial manifold

8.4

1 .1 7.2 7.6

double of a manifold, D(M)

7.2

7. 1

3.3 8.2

dimension, of a manifold of a cell of a cell complex

3.5

combinatorial equivalence

8. 1 1.6

differential of a map,

cell

coarse topology

1 .7

differentiable structure

1.0

barycenter

1.3

differentiable map, in euclidean space of a manifold of a complex

1.1,5.10

Rm

1.0 if1

1 .0

F 1 (M,N)

3.5

face, 7.1

of a simplex of a cell

7.4

complex

7.1

coordinate neighborhood

1.1

fine

coordinate system

1.6

full subcomplex

cube,

1.0

Grassmann manifold,

curve

3. 1

H*1

Df(x)

1.3

imbedding, immersion,

for manifolds for complexes

3.5

of manifolds of complexes

8. 5

interior of a manifold, Int M

diameter of a simplex

9.2

"intersect in a subcomplex"

Diff M

6.3

diffeomorphism

1.8

differentiable homotopy

3.9

differentiable isotopy

3.9

differentiable manifold

1 .6

Cm

5 -appr oxima ti on,

3 .5 ,3. 8

Cr topology,

10.1 5.1

Gp n

1.0 1 .8 8.3

10. 1 5.3

lifting of a map limit set of a function,

1.1

L(f)

2.9

linear map of a complex

7.1

linear isomorphism of a complex

7.1

111

INDEX OF TERMS

locally contractible locally-finite

5.16 1 .1 , 1 .2

Riemannian metric

3 .J+ 9. 1

secant map

5.1

simplex

7.1

manifold

1 .1

skeleton

7.6

manifold-with-boundary

1 .1

sphere,

non-bounded manifold

1.1

standard extension of a subdivision

non-degenerate map

8. 1

normal bundle

5 .5

normal map

5.4

open simplex

7. 1

subcomplex

orientable manifold

5.9

subdivision,

paracompact

1 .1

M(p,q),

M(p,q;r)

partition of unity

2 .5 ,2 . 6

star,

Sm_1

1 .0

St(x, K)

7.12

7.1

strong

C 1 approximation

strong

C° approximation

3 .5 ,8 . 5

3.5 7.1

of a complex of a cell complex

7 .1 7.6 2. 1

submanifold

piecewise-linear

7. 1

polytope

7.1

tangent bundle,

product neighborhood

5 .9

tangent map

radius of a simplex

9.2

rank

1 .7

rectilinear trianguiation

7.10

refinement

1 .1

regular homotopy

3.9

union of complexes

retraction

5.5

velocity vector

T(M)

tangent space

3.2 5 .4 3 .1

tangent vector

3. 1

thickness of a simplex

9* 2

trianguiation

8. 3 10. 1 3. 1

PRINCETON MATHEMATICAL SERIES E d ito rs : W u -ch u n g H siang, John M iln o r , and E lia s M. S te in

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The Classical Groups, Their Invariants and Representations, by HERMANN WEYL An Introduction to Differential Geometry, by LUTHER PFAHLER EISENHART Dimension Theory, by WITOLD HUREWICZ and HENRY WALLMAN The Laplace Transform, by DAVID VERNON WIDDER Integration, by EDWARD J. McSHANE Theory of Lie Groups: I, by CLAUDE CHEVALLEY Mathematical Methods of Statistics, bv HARALD CRAMER Introduction to Topology, by S. LEFSCHETZ The Topology of Fibre Bundles, by NORMAN STEENROD Foundations of Algebraic Topology, by SAM UEL EILENBERG and NORMAN STEEN ­ ROD Functionals of Finite Riemann Surfaces, by MENAHEM SCHIFFER and DONALD C. SPENCER Introduction to Mathematical Logic, Vol. I, by ALONZO CHURCH Homological Algebra, by H. CARTAN and S. EILENBERG Geometric Integration Theory, by HASSLER WHITNEY Qualitative Theory of Differential Equations, by V. V. NEMICKII and V. V. STEPANOV Topological Analysis, revised ed itio n , by GORDON T. WHYBURN Continuous Geometry, by JOHN VON NEUMANN Riemann Surfaces, by L. AHLFORS and L. SARIO Differential and Combinatorial Topology, edited by S. S. CAIRNS Convex Analysis, by R. T. ROCKAFELLAR Globe Analysis, edited by D. C. SPENCER and S. IYANAGA Singular Integrals and Differentiability Properties of Functions, by E. M. STEIN Problems in Analysis, edited by R. C. GUNNING Introduction to Fourier Analysis on Euclidean Spaces, by E. M. STEIN and G. WEISS

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