Elementary Differential Topology. (AM-54) (Annals of Mathematics Studies)

ELEMENTARY DIFFERENTIAL TOPOLOGY BY

James R. Munkres Lectures

Given at Massachusetts Institute of Technology Fall, x g6 i

REVISED EDITION

Annals of Mathematics Studies Number

54.

ELEMENTARY DIFFERENTIAL TOPOLOGY BY

James R. Munkres Lectures

Given at Massachusetts Institute of Technology

Fall, 1961

REVISED EDITION

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1966

Copyright © 1963, 1966, by Princeton University Press ALL RIGHTS RESERVED

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

To the memory of Sumner B. Myers

PREFACE

Differential topology may be defined as the study of those properties of differentiable manifolyds which are invariant under differentiable homecmorphisms.

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, curvature, and the like; in this way the subject may be distinguished from differential 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. Than, 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 invoive 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 continuity 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 recently has anyone written them down.

([8) and [91.)

(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 uniqueness of a smooth triangulation of a differentiable manifold. follow J. H. C. Whitehead [14], with some modifications.

In this, we

The result itself

is one of the most useful tools of differential topology, while the techniques 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

than was necessary.

It is for this reason that we snit 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 Waliman'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 structures on manifolds.

ix

CONTENTS

PREFACE . .

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Chapter I. Differentiable Manifolds 0. .

Introduction .

§2.

Submanifolds and Imbeddings .

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§3.

Mappings and Approximations .

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§4. §5.

Smoothing of Maps and Manifolds . Manifolds with Boundary . . . . .

§6.

Uniqueness of the Double of a Manifold .

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47

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63

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Triangulations of

Chapter II.

Differentiable Manifolds §7.

Cell Complexes and Combinatorial Equivalence.

§8.

Immersions and Imbeddings of Complexes .

59.

The Secant Map Induced by f .

§10.

Fitting Together Imbedded Complexes .

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INDEX OF TERMS .

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REFERENCES . .

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111

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90

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.

ELENEVTARY DIFFERENTIAL TOPOLOGY

CHAPTER I.

DIFVERENTIABLE MANIFOIDS

Introduction.

§1.

This section is devoted to defining such basic concepts as those of differentiable manifold, differentiable map, immersion, imbedding, and diffeanorphism, and to proving the implicit function theorem.

We consider the euclidean space Rm as the space of all infinite x - (x1, x2, ...),

sequences of real numbers,

euclidean half-space

i > m;

Then Rm-1 C Hm C Rm. max Ixi'j, lixUU

- 1;

by

1xI.

is the set with

We denote

of all m-tuples

((x1)2 + ... + (xIl1)2)

The unit sphere

jxj < r.

Sm-1

the set with

by

lixti,

and

is the subset of Rm with 1ixII < 1;

and the r-cube

Cm(r)

Often, we also consider Rm as simply the space where no confusion will arise.

(x1,...,xm),

1.1 Definition.

for

is the subset of Rm for which xm > o.

Fin

the unit ball Bm,

such that xi = 0

A (topological) manifold M is a Hausdorff space

with a countable basis, satisfying the following condition:

There is an in-

teger m such that each point of M has a neighborhood haneomorphic with an open subset of Hm

If h :U

or of

Rm.

fl (or Rm)

is a homeomorphism of the neighborhood U

of x with an open set in Hm or Rm, 6OOa'dinate neighborhood on M.

0-1 ,

If

h(U)

the pair

we say M is non-bounded.

is often called a

is open in IF and

then x is called a boundary point of

points is called the boundary of M,

(U,h)

M,

h(x)

lies in

and the set of all such

denoted by Bd M.

If Bd M is empty,

(In the literature, the word manifold is common-

ly used only when Bd M is empty; the more inclusive term than is manifold3

I.

4 with-boundary.)

The set M - Bd M is called the interigr of

denoted by Int M. also use

Int A

DIFFERENTIABLE MANIFOLDS

(If A is a subset of the topological space

in map

h2 IF,

and is X,

we

but this should cause no confusion.)

to mean X - CI(X-A),

To justify these definitions, we must note that if

and

M,

h1

:

U1 -. if

: U2 -+ IF are ho¢neomorphisms of neighborhoods of x with open sets and if

lies in RP-1,

h1(x)

so does

h2(x):

For otherwise, the

would give a homeomorphism of an open set in Rm with a neigh-

h1h21

borhood of the point p - h1(x) ly not open in Rm,

in

IF.

The latter neighborhood is certain-

contradicting the Brouwer theorem on invariance of

domain 13, p. 951.

One may also verify that the number m M;

it is called the dimension of

M,

is uniquely determined by

and M is called an 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-

is m

logical dimension of M

[3, P. 46].

Strictly speaking, to apply the

latter theorem we need to know that M is a separable metrizable space; but this follows from a standard metrization theorem of point-set topology 12, p. 751.

Because M

is locally compact, separable, and metric,

paracompact [2, p. 791.

open covering Q of covering M (1)

ment of (2)

M,

is

We remind the reader that this means that for any there is another such collection

(B

of open sets

such that

The collection (A

M

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

is contained in an element of

The collection

(

a.

is locally-finite, i.e., every point of M has

a neighborhood intersecting only finitely many elements of

(g

.

In passing, let us note that because M has a countable basis, any locallyfinite open covering of M must be countable.

I

bounded

(a)

Exercise.

If M is an m-manifold, show that Bd M is a non-

m-1 manifold or is empty. (b)

Exercise.

Let MO . M x 0

and

Let M be an m-manifold with non-empty boundary.

M1 - M x 1

be two copies of

M.

The double of

M,

INTRODUCTION

§1.

denoted by D(M), fying

(x,o)

is the topological space obtained from M0 U M1

with

for each x 'in Bd M.

(x,1)

bounded manifold of dimension (c)

n,

5

Prove that D(M) is a non-

m.

M

If

Exercise.

by identi-

respectively, then M x'N

and

N are manifolds of dimensions m and

is a manifold of dimension m + n,

and

Bd(M x N) = ((Bd M) x N) U (M x (Bd N)).

1.2

If U is an open subset of

Definition.

is differentiable of class

fl,...,fn

functions class

If A Cr

of class

borhood

Ux

is any subset of (1 < r < oo)

x

of

which is of class If

Rm,

then

f

if for each point

U.

: U

If

f

is of

: A - Rn is differentiable

x

of

there is a neigh-

A,

a function

on U.

Cr

f : A - Rn

is differentiable, and f

x

is in

A,

we use

this matrix.

Now

for

must be extended to a neighborhood of

f

Df(x)

at x - the matrix whose general entry

We also use the notation

aij -

Rn

C°°.

may be extended to

fl(A n Ux)

such that

to denote the Jacobian matrix of is

are continuous on

r

it is said to be of class

r,

f

if the partial derivatives of the component

Cr

through order

for all finite

Cr

then

Rm,

x before these

partials are defined; in the cases of interest, -.he partials arc independenn.

of the choice of extension (see Exercise (b)).

We recall here the chain rule for derivatives, which states that D(fg) - Df

Dg,

where

fg

is the composite function, and the dot indicates

matrix multiplication.

I

(a)

Exercise.

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

that the differentiability of ing space" (b)

A - Rn

Rm

for A

Exercise.

is of class

C1,

dent of the extension of chosen.

f : A -+ Rn

does not depend on which "contain-

is chosen.

Let U be open in apd f

x

is in

A,

Rm;

let U C A C U.

show that

to a neighborhood of

x

Df(x) in

If

f

is indepen-

Rm which is

ENTLAM MANIFOIDS

D

I.

6

Our definition of the differentiability of a map f: A -Rn is essentially a local one. We now obtain an equivalent global formulation of differentiability, which is given in Theorem 1.5. Remark.

There is a

1.3 Lemma. on

C(1/2),

f

is a Let

g(t) - 0

that

function c

is positive on the interior of f(t) . e-1/t

Let

Proof.

Then

C00

Cxx

for

: Rm -. R1

C(1),

and is zero outside

t > 0, and

f(t) - 0

function which is positive for

for

g'(t) > 0

t < 0,

for

for

I

C(1),

t < 0.

0.

Then g is a

g(t) - f(t)/(f(t) + f(1-t)).

which equals

C00

and

0 < t < 1,

function such g(t) . 1

for

t> 1. Let

h(t) .0 for

that

Then h is a

h(t) - g(2t+2) g(-2t + 2). h(t) > o

(t j > 1,

for

Iti 1

ft = fo

e > o,

for

t < E

and

ft a ft

- E. is a

C1 map for each

(2)

ft

(3)

dft: T(M) x R - T(N)

The homotopy

ft

It is a differentiable If

fo

is continuous.

is said to be a differentiable

N is of class

ft: MX R

t.

and

Cl.;

this is stronger than conditions (2) and (3).

Cr hanotopy if it is of class f1

ft

is an immersion for each

a regular (or differentiable) homotopy is an imbedding for each

Cr.

are both immersions, we make the (standard) conven-

tion that a "regular (or differentiable) homotopy means that

C1 homotopy if the map

t.

ft

If

ft"

fo

between them always and

f1

are imbeddings,

is said to be an iso o y if

ft

t.

The two notions of regular and differentiable homotopies in fact differ only slightly.

It will appear later as a problem, to prove that the

existence of a regular hamotopy between of a differentiable one.

fo

and

f1

implies the existence

It is not clear, however, which notion should be

given preference as being more natural.

A regular homotopy is natural in

MAPPINGS AND APPROXIMATIONS

43.

that it is equivalent to the existence of a path the maps

and

fo

: R - F1(M,N)

0

between

C1 topology is used for the func-

where the coarse

ft,

33

However, the homotopies one constructs in practice are usually

tion space.

differentiable rather than merely regular.

This is the case with us, so we

shall have little occasion to use the weaker notion. (a)

Let

Exercise.

ft be a homotopy between

gt be a homotopy between go by the equations

and

let

g1;

ht(x) = f2t(x) ht(x) = g2t-1(x)

Show that

ht

f1 - g0.

for

t < 1/2

for

t > 1/2

and

f0

Define

ht: Mx R - N

C1 homotopy if -ft

is a regular (or differentiable)

let

f1j

and

gt

are. (b)

R

C1 homotopy which is not a

C1 homotopy.

differentiable (c)

Construct a regular

Exercise*.

Let

Exercise*.

ft

be a regular

C1 hanotopy.

by the equation

F1(M,N)

0(t)(x) = ft(x)

Show that

0

Define

is continuous, if the coarse

-

C1 topology is used for F1(M, N).

State and prove the converse.

3.10

I

f : M-* N be a

Let

Theorem.

sion or imbedding, there is a fine

C1 neighborhood of

of immersions or imbeddings, respectively.

there is a fine

C1 neighborhood

neighborhood and

Proof.

g

Let

[CI)

f

Int C1 C1

also cover and

is an immersion.

matrices of rank r by M(p,q;r); M(m,n;m)

f

f

is an immer-

consisting only

is a diffeomorphism,

then

M,

f(C1),

g is in this g

is a diffeomorphism.

and

(U1,h1)

and

(V1,k1)

respectively.

Then the matrix D(klfhj1)'(x)

rank m = dim M for all x in hi(Ci).

Now the set

If

be a locally-finite covering of M by compact

are coordinate systems about Suppose

f

such that if

f

of

If

carries Bd M into Bd N,

sets, such that the sets

(1)

C1 map.

has

We denote the set of p x q

it is considered as a subspace of Rp

is open in Rmn:

If one maps each matrix into the

sum of the squares of the determinants of its m x m submatricea, the set is

the inverse image of the open set R - (0)

under this continuous map.

34

I.

DIFFE NTIABIE MANIF'OIDS

where x

The matrices D(kifh.jl)(x), pact subset

M(m,n;m).

of

K1

C1 neighborhood of

the desired fine

Let

(2)

of

W1

f

C1 neighborhood W2

a fine

This choice of the

but

converges to

gn

gn(xn) - gn(yn)

f,

consisting of immersions.

specifies

si

Within this, there is

g which are 1-1 on each

consisting of maps

and

si-

as in 3.6.

f,

For otherwise, there would exist a number where

such that the

si

We have just proved that there is a fine

be an imbedding.

f

C1 neighborhood

Hence there is a

lies in M(m,n;m).

neighborhood of K1

is in hi(Ci), form a ccm-

i

dgn converges to

uniformly on

df,

for two distinct points xn and yn

of

f(xn) - f(x) , gn(xn) -, f(x)

similarly, gn(yn) -+ f(y) .

Ci,

By

C1.

passing to subsequences and renumbering, we may assume xn-+ x and

Since

yn - y.

Hence

f(x) - f(y); since f is 1-1, x - y. We refer

f

and g to coordinate systems:

xn - hi(xn)I

g - kighil,

and yn - hi(yn)

let

f - k1fhj1

zn i

for sane point

on the line segment Joining xn and

zn i - x,

(3)

Let

f be an imbedding.

neighborhood W2

with W.

o(x);

1.4).

f

Let

1-1;

on each

we shall denote this

We specify this neighborhood by a continuous function

be the distance in N from f(D1)

is a homeomorphism, this distance is positive.

function on M which is less than ei/2 Cc s-approximation to

1-1

C1

f whose intersection

C0 neighborhood of

p.

of M by compact sets, with Di C Int Ci ei

.

is non-singular.

thus we assume N is given a topological metric

covering Di

u

We have just proved there is a fine

consists of maps which are globally

intersection by W3.

By passing

,

of f consisting of immersions which are

Now we prove there is a fine

Ci.

u

contradicting the fact that Df(i)

i,

yn.

we have

0 - D?' (50 for each

,

(xn - yn) /Ilxn - 'in II

to a subsequence and renumbering, we may assume Then since

,

Then

0 - g(xn) - g(yn) - Dgn(zn,i) ' (i - yn)

on

f which lies in W2.

Ci

to

Let

(see

1

f(M - Int Ci); 6(x)

(see 2.6).

Suppose

There is a

for each

since

be a continuous

Let g be a

g(x) - g(y),

C1:

gn

and a sequence

where-

MAPPINGS AND APPROXIMATIONS

$3.

x

is in Di and y

is in D,,

and

35

Then

Ei < Ej.

p(f(x), f(y)) < Ei/2 + E,/2 < EJ. g

Since

is

on Cp x

1-1

is not in

so that

Cj,

p(f(x), f(y)) > Ej

by choice of E. Thus we have a contradiction. Finally, we prove there exists a fine

intersection with W3 phism,

L(f)n f(M)

consists of homeomorphisms.

is empty.

Let

g be a C0 s-approximation to

Let

First we show that

so that

g(xn) - y,

Hence

Let

(4)

fine

f(M)

L(g)n g(M)

so that

y if and only if

f(xn)

If x

is empty.

Note that

f be a diffeanorphism.

g(M)

is in

then

Ci,

which equals

L(g).

is a closed subset of N

Exercise.

There is a fine

if

g(Bd M) C Bd N,

Let

C1 neighborhood W4

The following lemma shows that there is a

such that if

f

and this neighborhood, and

(C1)

since

is.

Co neighborhood of

(a)

p(f(xn), g(xn)) -+ 0,

is not in L(f),

g(x)

consisting of imbeddings.

f

contains only finitely many

Ci

L(f) . L(g).

g is a homeaanorphism.

and only if

If xn is a sequence in M hav-

L(f) . L(g).

This means that

It follows that p(g(x), f(x)) < Ei,

Let

L(f).

for x in C

s(x) < Ei

if xn is in C. Hence

p(f(xn), g(xn)) < 1/i

be less than the

Ei

f which lies in W3.

ing no convergent subsequence, then each terms of the sequence.

whose

f

is a hameamor-

to the closed set

f(Ci)

be a continuous function on M such that

s(x)

f

Since

and let

Ei < 1/i

distance in N from the compact set

of

Co neighborhood of

g lies in the intersection of W. then

g

is a diffeomorphism.

f be a homeomorphism of X

into Y;

let

be a locally-finite covering of X by compact sets such that the sets

Int Ci cover

Prove there is a fine C0 neighborhood of

X.

g lies in this neighborhood and hcmeomorphism. (b)

regular

(X

is 1-1 on each

Exercise*.

then g is a

Let M and N be C1 manifolds; let

C1 hcmotopy between the maps t,

lar homotopy Ft between f0 t,

Ci,

and Y are separable metric, as always.)

is an immersion for each

for each

g

such that if

f

f0

and

f1

prove that for some and

f1

which is a

is also an immersion for each

t.

ft

of M into N. s(x) > 0,

any

be a If

ft

C1 regu-

s-approximation to ft

Prove similar theorems when

DIFFERENTIABLE MANIFCIDS

I.

36

is an imbedding and M is compact, or when

ft

ft

is a diffeomorphism be-

tween non-bounded manifolds.

(1) and (2) suffice for the

Generalize the proof of 3.10.

Hint:

case of an immersion or an imbedding; for the case of a diffeomorphism, (3)

and (4) may be applied to the map F : M x R - N x R defined by the equation

F(x, t) - (ft(x), t). Exercise*.

(c)

Show the hypothesis that M is compact is needed

in the preceding exercise when

is an imbedding.

f be a hcmeomorphism of M onto

Let

Lemma.

3.11

ft

and N are topological manifolds. such that if

g

There is a fine

lies in this neighborhood and

carries M onto

coordinate system

about

(Vi,ki)

prove that

for all

g(Ci)

We prove that

i.

f( i).

z

Sm-1

contains

of radius

1

+ si

f(Di)

cover

N.

kig(x) - kif(x) II < ei

II

g is onto;

z,

when

in particular, we

Said differently, the composite map

= Bd Bm under h lies outside

and the B(1 - si);

B(1 - 8i).

lies in B(1 - 8i)

radial projection from

of Rm - z

but not in h(Bm). onto

Sm-1.

Let

7.

be the

Then Xh carries Bm

Sm-1.

into

On the other hand, consider to the identity; we merely define This homotopy carries

Sm-1.

and

B(1 + 8i)

is a well-defined map carrying Bm into Rm,

h(Bm)

Suppose

such that for some

(see Exercise (a)).

Then the sets

contains'

image of the unit sphere we prove that

then g

and so that the sets Di = (kif)-1(B(1 - si))

(see 1.4).

h - (kig)(kif)-1

f

equals the unit m-ball

kif(Ci)

f(Ci),

Let g : M -* N be a map such that Ci,

Ci

small enough that the ball

si

is contained in ki(Vi),

is in

g(Bd M) C Bd N,

M by sets

also cover M

Int Ci

and the sets Choose

x

Co neighborhood of

We consider first the case in which Bd M is empty.

Choose a locally-finite covering of

still cover M

where M

N.

Proof.

Bm,

N,

x,

so

h(x)

Ft(x). lies outside

hISm-1

Ft(x) - t

:

Sm-1 _ Rm.

h(x) + (1-t)

It is homotopic x

for x

along the straight line between B(1-8i).

h(x)

Hence the map XFt is well-

in

MAPPINGS AND APPROXIMATIONS

S3.

Sm-1

xhle-1

defined, and is a homotopy between

Sm-1 and the identity.

Consider the homology sequence of the pair homanorphism of it induced by

o

Hm_1(sm-i)

(xh Sm-1)*

(xh)*

0 - ;(B, ;_1 (sm-1)

0 Sm-1),

is the zero-homomorphism of the infinite cyclic group H(Bm,

(xh),

because

and the

(Bm, Sm-1)

xh:

0 - Hm(Bm, Sm-1) -p

Now

37

xh maps Bm

into

Sm-1.

phism of the infinite cyclic group

And I

(xh ISm-1)

_1(Sm-1)

is the identity hommnorxhISm-1

because

is homoto-

This contradicts the commutativity of this diagram.

pic to the identity map.

(This argument is the only place where we use a bit of algebraic topology.)

Now let us consider the case in which Bd M D(M)

be the double of

if

: D(M)

and

be the double of

D(N)

There is a positive continuous function D(N)

and

p(f(x), i(x)) < E(x),

Now D(M) - MOU M1; 6 o

let

be the homeomorphism induced from

: D(M) -« D(N)

on D(N).

as in 1.1;

M,

E1

on

M.

a

Similarly,

is non-empty.

N;

Choose a metric

f.

e(x)

then

Let

g

on

let p

such that

D(M)

is onto.

determines two positive continuous functions p

induces two metrics

po

and

p1

on N.

I.

38

Let

g : M

if

N;

we require

g

DIFFERENTIABLE MANIFOLDS

g(Bd M) C Bd N,

f relative to the metric

to be an ee-approximation to

and an e,-approximation relative to and hence is onto.

(a)

g induces a map g : D(M) - D(N).

Exercise.

pi,

If p0,

then g is an E-approximation to

Then g is necessarily onto as well.

Prove the existence of the sets

Ci

used in the

preceding proof. (b)

fold

M;

let

Exercise.

Let A be a closed subset of the non-bounded mani-

B be a closed set containing A in its Interior.

there is a positive continuous function

f : B - M is a

C

0

B(x)

defined on B

8-approximation to the identity, then

Prove

such that if

f(B)

contains

A.

Smoothing of Maps and Manifolds.

§4.

We now approach the two main goals of this chapter.

The first of

our theorems states that if M and N are C00 manifolds, and is a

C1 immersion, imbedding, or diffeomorphism, then

f : M - N

f may be approxi-

C0° immersion, imbedding, or diffeomorphism, respectively.

mated by a

proof of this appears in 4.2 - 4.5, except for the case where

The

is a dif-

f

feomorphism and M has a boundary, which is postponed to Section

5.

The second theorem states that every differentiable structure of class

on a manifold M contains a

C1

C0D structure.

case where M is non-bounded appears in 4.T - 4.9;

The proof in the

the other case is treat-

in Section 5.

The fundamental tool needed for proving these theorems is the fol-

0

lowing "smoothing lemma."

where V CU.

a compact subset of the open set V, Cr map,

1 < r.

IF

Let U be an open subset of

Lemma.

4.1

Let

be a positive number.

8

Lot

or

Let A be

Rm.

f : U -' Rn be a

There is a map

f1: U -+ Rn

such that (1)

f1

is of class

(2)

f1

equals

CP,

for all

lDf1(x) - Df(x)l < 8,

on any open set on which

f

x.

is of class

There is a differentiable Cr homotopy ft between

such that Proof.

Rm,

V.

and

CP

A.

1 2/3

Define

(see 1.3).

ft(x) = a(t) fi(x) + (1 - a(t)) f(x)

Then ft f1 - f,

so that

Ift - fl - a(t) If1 - If

and

IDft - Df I

Finally,

ft

(a)

I

is of class

Exercise.

open in Rm and Iim

and

by

f1

- a(t)IDf1 - DfI < 8

on any open set where

then

and

f1

are.

Show that if U

is

Show that if U is open in

ft(U) C Fin.

ft(U) C Hn.

it need not be true that

f

so that this relation will hold.

(Hint:

Modify the con-

Replace

C(e)

throughout the proof.)

C(e) n FIm (b)

V,

n and

for

t < 0

ft(x) - F(x, tan (at))

for

o < t < 1/2

ft(x) - h(x)

for

1/2 < t

A.

Vi.

t < 1

ft(x) - f(x)

Exercise.

U of

x

on Mx R; and for any compact subset B of x

f

in

M, B.

.

Strengthen the preceding theorem as follows:

A be a closed subset of M and let borhood

we

h,

8/21 approximation to for

t,

F(x, t) - Fi+1(x, t-i)

is of class

and

f

Cr differentiable homotopy Fi(x, t)

F(x, t} - FVx, t)

Then. F

f.

Lemma 4.1 gives us a Cr differentiable homotopy between

From this we obtain a

gi.

is of

gi

Furthermore, if we require

is.)

g1_1

in U1

x

for

s(x)/21 approximation to

Define

of course.)

outside V1

on W1U...U Wi.

f i

can make

Cp

gi hi(x)

Also,

gi_1 that it carries

already be of class

Cp

Let

in a neigh-

Then we may add the following to the conclusion of the

theorem: (3) fi(x) - f(x)

and

ft(x) - f(x)

for each x

in A.

SMOOTHING OF MAPS AND MANIFOLDS

S4.

In the above proof, take

Hint:

covering

of

(U, M - A)

to be a refinement of the

(Ui)

Let B be a closed subset of the open set U of such that h is of class

map h : M - N

outside

f

equals

M.

Co 8-approximations to

Corollary.

4.3

f : M - N be a

Let M and N be manifolds of class

Cr differentiable homotopy

between

ft

and

f

fi.

We remind the

is required to be an immersion for

t.

4.4

Corollary.

f : M - N be a

f1: M

let

f

f1;

(a)

ft

There is a

(1 < r < p < co).

N be a

Cr diffecmorphism and a

let

Cp imbedding

and

f1.

(1 < r < p < oc).

There is a

Cr differentiable isotopy

3s a diffeomorphism for each

Exercise*.

f

Cp;

Let M and N be non-bounded manifolds of class

Cp diffeomorphism )f1: M -' N,

f and

N be manifolds of class

Cr differentiable isotopy between

Corollary. : M

Let M and

Cr imbedding

and a

N,

4.5 Cp;

ft

let

Cp;

Cp immersion

There is a

(1 < r < p < co).

reader that by our convention (3.9), each

are, of course,

ft

This follows from Theorems 3.10 and 4.2.

Proof.

r - 0.

f.

Cr immersion

and a

f1: M-+ N,

and

the preceding theorem hold.

Consider the preceding theorem in the case

Show that the results still hold, except that h and only

Prove there is a Cr

in a neighborhood of B

Cp

and (1) and (2) of

U,

Exercise.

(c)

if Ui C U.

gi = gi_1

Assume the hypotheses of the preceding theorem.

Exercise.

(b)

and choose

M,

43

ft between

t.

State and prove the stronger forms of these three

corollaries, obtained by applying the results given in Exercises

(a) and

(b) of 4.2. (b)

f0

and

f1

Exercise*.

be

Let M and N be manifolds of class

Cr maps of M into

homotopy between f0

and

f1.

Let

N;

let

8(x) > 0.

ft be a

Cr;

let

C1 differentiable

Prove there is a

Cr

44

DIFFERENTIABLE MANIFOLDS

I.

differentiable homotopy Ft between

C1 maps of M into

be

f1

Let

tween them.

suppose

Rn;

t.

N;

let

ft

let

C1;

C1 differentiable hanotopy Ft for each

t.

Consider first the following problem:

For each

t,

is a 8-approximation to

Rm into

C1 regular homotopy between

is a

a compact subset of

let

U;

be as in 4.1;

7

fo

C1 regular homotopy be-

be a

ft

C1 map of the open subset U of Hm or

be a

is a

ft,

between them such that Ft

ft

Ft

such that

f1

Prove there is a

8(x) > 0.

Outline:

and

0

Let M and N be manifolds of class

Problem*.

4.6

and

for each

ft

8-approximation to

f

let

and

fo

let

or into

Hn f1.

Let A be

gt(x) = Y(x) ft(x).

Define ht(x) =

for suitably small

where

c,

cp

S p(s) gt+s(x) ds is positive on

(-e, e)

and

0

outside,

E

and S

Let

- 1.

Ft(x) = ft(x) .

differentiable hanotopy between and equals

Let

Lemma.

with WC V and V C U. Rn

: U

8(x) > 0,

of

fo

Then Ft

is a

Let

it

(2)

h is a s-approximation to

(3)

h(W)

(4)

If

is a U1

C

f

outside

h(U1)

Rn

onto

Rm.

is an imbedding.

Rm,

Suppose

Given

such that

V. f.

C00 submanifold of R.

is also a

0 of

Rm;

f(U1)

is a

let

a

to g : 0

f(U)

is a haneanorphism of

g(x...... xm) = (x1..... xm, gin+1(x),...,gn(x))

since it equals

f(U)

Rn be the inverse of this map.

Then

Cr,

C°° submanifold

C°° submanifold of R.

The restriction of

g is of class

A,

A.

is an open subset of U and

onto an open subset

and

is a

on sane neighborhood of

of : U _ Rm

Cr imbedding h : U -

h equals

Proof.

f1

be the projection of Rn

Cr map such that

there is a

then

and

and W be bounded open subsets of

U, V,

(1)

Rn,

- T(x)) + ht(x).

outside some neighborhood of

ft

4.7

f

(1

E

f(xf)-1.

Further, if of

g

on

f(U,)

f(U1)

is a

C00 submanifold of

Co structure, and note that

its induced

relative to this structure.

The map g

followed by the inclusion of

C00,

go: 0

Consider

Co

4.1, choose

on

be of class

C0D

go

af(V).

on any open subset of 0

.

which is of class

g0

We further require that go

where

go

is C. Let g :

g(x) - (x, g)(x)); it is an e-approxima-

h : U _ Rn by the equation

is properly chosen,

4.8

Let M be a non-bounded

Theorem.

Rn be a Cr imbedding.

submanifold of Proof.

an imbedding. n

of

h(x) - gaf(x).

h will be a B-approximation to

Let

s(x) > 0.

h : M - Rn which is a s-approximation to

jection

which is C.

g.

Define

f : M

Rn,

1 (x),...,gn(x))

outside

0 - Rn be defined by the equation tion to

into

to be an e-approximation to

and equals

nf(W)

Co map

defined by the equation

Rn-m,

fo(x)

is a

alf(Ui)

is the inverse of this, which is

f(U1)

go(x) _ (g

Using

the restriction

Rn,

(The converse is easy (2.2).) For impose

nf(U1) is of class Co.

to

45

SMOOTHING OF MAPS AND MANIFOLDS

S4.

f,

By 3.7, if

e

(1 < r).

Let

f - gnf.

Cr manifold

There is a such that

Cr imbedding

h(M)

is a

C0D

Rn.

Let

s

be small enough that any 6-approximation to

For each x in Rn

M,

df(x)

has rank

m.

f

is

Then for some pro-

onto a coordinate m-plane, the map d(af)(x)

has rank in,

46

MANIPDIDS

I.

is a Cr diffecmorphism of a neighborhood of x onto an open

xf

co that

Choose a

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

covering of M by sets about

Ci,

cover

M,

xiflCi

approximation to

xifICi

for x in

8(x)< bi

is an imbedding; let

is a 8-approximation to

xif'

Bi-

be small enough that

a (x)

f': M - Rn is a 8-approximation to

Then if

Ci.

of Rn onto a coordinate

x1

si be a number such that any

Let

is an imbedding.

(Ui, hi)

Int Ci

equals the unit m-ball, such that the sets

hi(Ci)

and such that for some projection

m-plane,

then

such that for some coordinate system

Ci

Let Wi be an open covering of M,

xif,

xif'ICi

so that

f,

is an imbedding.

with Wi contained in the open set Vi,

and Vi C Int Ci. Let

is a Cr map which is a (i - 1/23-1)8(x) f,_1(Wk)

fj_,: M - Rn

As an induction hypothesis, suppose

f0 - f.

is a CO* submanifold of

Rn,

approximation to for

f

such that

We then construct a map

k < J.

Apply the preceding lemma to the map

f3.

f3_1h1 1

: h3(Int C3)

Rn

to obtain a nap fM Rn which is a a/23 approximation to equals

f3_1

outside V3,

f(W3)

and such that

f3_,, which

is a C0° submanifold of

Condition (4) of the preceding lemma guarantees we can choose

Rn.

that

f3 (Wk)

is

a CO* submanifold of Rn for k < j

As before, we let

h(x) - lim1

00

f3 so

.

and note that h satis-

f3(x),

fies the conditions of the theorem.

4.9

Corollary.

non-bounded manifold

(a)

If 1) is a

M, 1) contains a C° structure.

Exercise.

Let M be a non-bounded

a non-bounded C°° manifold. ding, prove there is a to

f,

such that

Let s(x) > 0.

Cr manifold;

If f

let N be

: M-+ N is a Cr imbed-

Cr imbedding h : M-+ N which is a 8-approximation

h(M)

entiably isotopic to

Cr differentiable structure on the

f.

is a

Cc* submanifold of

N,

and h is

Cr differ-

Manifolds with Boundary.

§5.

There are additional technicalities involved in proving our two main theorems 4.5 and 4.9,

for manifolds with boundary.

First, we need to prove the local retraction theorem (5,.5),

which states that a non-bounded- Cr submanifold

M

a neighborhood which is retractable onto M by a it is easy to find such a retraction of class

plane normal to M at tion of class

Cr requires more work;

mann manifolds (5.1 - 5.4).

Cr retraction.

Cr-1;

and collapses it onto

x

of euclidean space has If

r > 1,

roughly, one takes the To construct a retrac-

x.

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

states that Bd M has a neighborhood in M which is diffeomorphic with This in turn is used to construct a differentiable structure

Bd M x [0, 1).

on D(M),

the double of M The theorems for

(5.8 - 5.10).

M

then reduce to the corresponding theorems for

the non-bounded manifold D(M)

5.1

The Grassmann manifold Gp,n

Definition.

n-dimensional subspaces of Let

M(p,q)

(5.11 - 5.13).

denote the set of all

notes those having rank

r.

the same element of Gp,n nations of

n x n matrix

p x q

matrices;

The rows of any matrix A

a set of n independent vectors of Gp,n which we denote by

is the set of all

Rn+p.

)..(A).

Rn+p,

of

M(p,q;r)

de-

M(n,n+p;n)

are

so they determine an element of

Further, two matrices

A

and B determine

if and only if the rows of each are linear combi-

the rows of the other;

i.e., if A - CB

for some non-singular

C.

M(n,n+p;n)

is an open subset of

has a natural topology and

Rn(n+p)

(see 3.10),

C0D differentiable structure.

Let

so that it

Gp,n be given

the identification topology: V is open in Gp,n if and only if

>v-1

(V)

is open in M(n, n+p; n). We note that a.: M(n, n+p; n)

be open in

M(n, n+p; n).

For any

Gp,n

C

47

is an open map.

in M(n, n; n), the map

For let C

U

: A- CA

48

MANIFOLDS WITH BOUNDARY

53.

x-1(x(U))

M(n, n+p; n).

is open.

Theorem:

Gp,n

5.2

map

is the union of the sets

X(U)

M(n, n; n), so

Proof.

(1)

is a non-bounded

is of class

: M(n,n+p;n) - Gp,n

x

Gp,n

(P

n-tuple of vectors

the

dim pn;

Let us explain first the

onto a coordinate n-plane

Let U be the set of all planes on

.

which lie in it and project under

(v1,...;vn)

in

Each plane of U uniquely determines an

is a linear isomorphism.

it

C

COO.

of Rn+P

it

which is a linear isomorphism on

which

for all

l9 is an n-plane through the--origin in

If

there is some projection

Rn+p,

C(U)

C°° manifold of

is locally euclidean.

geometric idea of the proof.

C(U) is open in

with itself, so that

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

U.

Thus, each vector vi has p

ponents which may be chosen arbitrarily;

into

Conversely, each

the natural unit basis vectors for the coordinate n-plane. such n-tuple determines a plane of

s

since there are n

com-

such vectors,

U is homeomorphic with Rpn. More precisely, let

submatrix of A has rank

n x n

M(n,n+p;n).

Hence V . X(U)

Let [I Q]

of

M(n,n+p);

into

Gp,n.

It is

Now

q,

m

:

To . XT.

since

U--* M(n,p)

means that

Hence

the inverse of homeomorphism.

W

Then

To,

.(I

only if

P-1Q] =

TO(P-1

whence vo

M(n,p) 011 Q21;

[I Q,] Q).

P-1

q,([P Q])

since the equation

[P1 Q1] = C[P2 Q2],

since

is

into the matrix

is a continuous map of

be defined by the equation

induces a continuous map

P

and hence in

M(n,p)

of

To

x[P Q] -

x-1((?),

where

[P Q]

GpIn.

since MI Q1] = x[I Q2]

1-1,

is constant on each set

x([P2 Q2]) P21Q2.

let

onto V,

M(n,p)

Let

is open in

map the arbitrary matrix Q

Y

Some

Gp,n.

suppose it consists of the first n

Than U is open in Rn(n+p)

and non-singular.

it maps

n;

Let U be the set of matrices of the form

columns. n x n

be the general element of

x(A)

x([P1 Q1])

PT1Q1 . (CP2)-1(C%) _

of V into

q,o,o(Q) - q,((I Q]) . I-1Q = Q.

M(n,p);

Hence

T.

q,o

is

is a

49

DIFFERE 'IAB1E MANIFOLDS

I.

M(n,p)

U C M(n,n+p;n) m

V C Gp'n

(2)

We need only prove

Gpn has a C°° differentiable structure.

that two coordinate neighborhoods

of the type just constructed have

(V, (Pe)

cpN,NT,U,VN

class

Let

CO overlap.

be the corresponding maps and open sets

Then Y

which determine another coordinate neighborhood.

takes the columns

of the matrix Q and distributes them in a certain way among the columns of the identity matrix is of class Hence

C°°

(poT0 I = cp

so that Y is certainly of class C. Likewise R

I,

on the open set U of M(n,n+p;n),

-

is of class

C°°

since

9((P Q)) = P-'Q.

(on the open set (U n U)

where it is

defined).

Gp,n is Hausdorff.

(3)

so that any

e > 0

Choose

Then the matrix

Gp,n.

spaces of

rank at least

matrix within

B, in M(n, n+p; n).

e-neighborhood of neighborhoods of

X(B)

be distinct sub-

must have rank at least

( B I

2n x (n+p)

and

)L(A)

n+t.

of this one has

a

Let U denote the e-neighborhood of A, and V, the

n+1.

and

%(A)

Gp,n

(4)

For let

Then

and

X(U)

).(V)

a(B), respectively, since

has a countable basis.

.

Gp,n

Indeed,

are disjoint

is an open map.

is covered by

coordinate systems of the type constructed above.

(n+p):/n!pi

(a)

Exercise*.

Show that Gp'n is compact.

(b)

Exercise*.

Show that there is a

5.3

Lemma.

C°° diffeomorphism of

Gp,n

onto G

Let

f : M - Gp,n be a

there is a neighborhood U of x that

).f* = f.

Proof.

such that

f(x)

(f*

Let

and a

Cr map

is called a lifting of (V, qt0)

lies in V.

Given x

Cr map.

f

f*: U - M(n,n+p;n) over

cpo: V - M(n,p)

M,

such

U.)

be a coordinate system on

Now

in

is a

Gp,n,

as in 5.2,

Cm diffeemorphism.

MANIFOLDS WITH BOUNDARY

45.

50

Simply define

f*(x) - Y(co(f(x))) - [I 1pof(x)I (a)

Let

Exercise*.

onto V. and

be another coordinate system on Gp,n,

:p0)

xg-1 is

such that

with V x M(n, n; n)

the natural projection of V x M(n, n; n)

Let

Show that there is

be as in 5.2.

(V, 4V 0)

,-1(V)

of

a diffeomorphisn g

.

g

the

ti

let a map hx, carrying

For each x in V n V,

corresponding diffecmorphisn.

into itself, be defined by the equation

M(n,n;n)

(x, hx(P)) - a g _'(x, P)

Show that

and

is merely multiplication by a non-singular matrix Ax,

hx

show that the map x - Ax

is continuous.

This will show that

: M(n,n+p;n) - Gpn is a principal fibre bundle with fibre and structural group

(See [12].)

M(n,n;n).

5.k

tangent to M at If we identify 3.5.

Let

Definition.

then

x,

Rn be a

f : M

df(v)

is a tangent vector to Rn at

with Rn x Rn, then

T(Rn)

f(x).

df(v) . Mx), do fnv ), as in

As v ranges over the tangent space at

m-plane through the origin in Rn.

If v is

Cr immersion.

x,

d0f(v)

The tangent map

t

ranges over an

of the immersion

will be the map t : M - Gp,m which assigns this m-plane to the point (n . m + p,

be a coordinate system on

(U, h)

is given by the equation evaluated at t

x.

of course.)

Let

Hence

f

t(x) . 7.(D(fh1)),

and

h(x),

is of class

where the matrix D(fh

is the projection of

X

Then locally the map

M.

M(m,m+p;m)

onto

1)

t

is

Gp,m.

Cr-1.

Similarly, one has the normal map of the immersion, n : M - Gm'p, which carries D(fh-1)

x

into the orthogonal complement of

is non-singular;

is m x m and non-singular. 7.([R(x) I]),

where

suppose it is of the form

t(x).

Now the matrix

[P(x) Q(x)]

where

Then locally n is given by the equation

R - - (P-Q) 1

.

Hence n is also of class

P n(x)

Cr-1 .

a

(local retraction theorem).

Theorem

5.5

and a Cr retraction r : W -+ f(M).

y in

for

(We recall this means that

r(y) . y

Consider the normal map n : M - Gm,p and the tangent map

of the imbedding

Gp'm

There is a fine

and

i(x)

Cr-1.

Both are of class

(p - n - m).

f

such that for any map $ which

neighborhood of n

Cc

lies in this neighborhood,

are independent subspaces of

t(x)

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

Rn,

a map

of class

: M -. Gp..m

Pi

(In the case

Theorem 4.2.

n is only of class

r - 1,

Let

t is also a map of class

?1(x);

Co,

and we need to

be the orthogonal complement

t(x)

Cr.

Let E be the subset of M x Rn consisting of pairs that v lies in the subspace

(x, v)

that E is a Cr submanifold of Mx Rn of

in a neighborhood of t*(x)

be

t*

linearly independent set. n*

and

T*

Let

(U, h)

We prove

Cr liftings of

1'I

is a p x n matrix,

and

t

and

jointly form a

be a coordinate system about x

are defined on U;

let

h(x)

(u1,

..., um).

g : U x Rn,.. Rm x Rn by the equation

Define

rA(x)

g(x,(v1,...,vn))

Than

$,

n.

is an m x n matrix; the rows of A and B

- B(x)

such that

let k and

dimension

Then Z 1 (x) - A(x)

x.

such

(If we used the map n instead of

P1(x).

E would be what is called the normal bundle of the imbedding.)

Given x in M;

Choose

Cr which lies in this neighborhood, using

use Exercise (c) of that section.) of

f(M)

f(M).)

Proof. t : M

f : M - Rn be

Let

There is a neighborhood W of

M non-bounded.

Cr imbedding;

51

DIFFF.EENTIABLE MANIFOLDS

I.

(U x Rn, g)

_

l-1

(h(x), [v1... vn]. L B(x) J

is a Cr coordinate system on M x Rn.

Now if

(x, v)

lies in E. then v is some linear combination of the rows of A(x), so that

v . [y' ...

for some choice of yt, ..., yp. (x, v)

g(x, v)

lies in E.

As a result,

yp o

... o

A(x)

B(x)

Conversely, if v is of this form, then (x, v)

lies in E

if and only if

lies in h(U) x R. Hence the coordinate map g carries the in-

tersection of E with U x Rn onto the open subset

so that E

is a Cr submanifold of M x Rn.

h(U) x Rp

of 0+1),

52

I.

DIFFERENTIABLE MANIFOLDS

z

y

M OF

f(c)

Consider the f(x) + v;

Cr map of M x Rn -. Rn which carries

(x, v) into

let F be the restriction of this map to E.

We prove that coordinate system constructed above.

dF

has rank n at each point of Mx 0. (u',...,um,y',...,yp,o,...,o)

g(x, v)

on

Use the

E which was

Then

Fg '(u, y) = fh-'(u) + (y'... yp]

A(h-'(u))

so that

and

A(h-'(u))tr

a(Fg ')/ay

a(Fg') /auk

=

,'

a(fh') /auk + ((y' ...yp] . a(Ah ' ) /auk) tr

The last term vanishes at points of

g(M x o),

since y - 0

at such points.

Hence

D(Fg') at points of g(M x 0).

_

[D(fh')(u)

A(h'(u))tr]

The columns of D(fh ') span the tangent plane t(x);

MANIFOLDS WITH BOUNDARY

$5.

the rows of A

span the plane

53

by choice of A these spaces are

A(x);

independent.

F : E - Rn is a homeomorphism when restricted to the sub-

Now,

furthermore, each point of M x o

space M x 0;

has a neighborhood which

F maps homeomorphically onto an open subset of Rn

(by the inverse func-

in E

It follows that there is a neighborhood of Mx o

tion theorem).

which F maps homeomorphically onto an open set in Rn

in E

There is also a neighborhood of Mx 0

(see Lemma 5.7).

on which dF

has rank n;

let V be the intersection of these neighborhoods and set W - F(V). There is a natural projection a

Cr diffeomorphism of V onto W,

tion of W onto

F it

F-1

- r

M.

Since

F

is

is the required retrac-

f(M).

Construct the fine

Exercise.

(a)

of M x Rn onto

it

Co neighborhood of the normal

map n required in the preceding proof. Use the fact that

Hint:

hoods of

n(xo)

5.6

and

is an open map to construct neighbor-

consisting of subspaces which are independent.

t(x0)

Let

Corollary.

x

There is a neighborhood W of

M non-bounded.

f : M -. N be a Cr imbedding; f(M)

in N and a Cr retraction r : W -

f(M) . Let

Proof.

hood U

of

(a)

is the required retraction,

Exercise.*

M and N non-bounded. simply when N ric.

Let

f : M

There is a neighbor-

of U onto

gf(M).

defined on W - g 1(U).

N be a Cr immersion, with r > 2;

The normal bundle E

of

f

may be described most

is a submanifold of RP having the derived Riemannian met-

In this case, let E be the subspace of M x RP consisting of pairs

(x, v)

such that v is a p-tuple representing a vector tangent to N and

normal to case,

N - Rq be a Cr imbedding.

in Rq and a Cr retraction ro

gf(M)

Then g Irog

g:

E

df(w)

for each w tangent to M at

is described as a subset of M x T(N).)

manifold of M x RP.

x.

(In the more genera).

Then

IR

is a

Prove the tubular neighborhood theorem:

Cr-f

010.

I. DIFFERENTIABLE MANIFOLDS

54

If f : M - N C RP is a Crimbedding, there is a neighborhood U

of M x 0

in the normal bundle E and a Cr-1 diffecenorphism h of U

with a neighborhood of

in N such that hIM x o

f(M)

Let F : E -y RP be the map

Hint:

space

X.

let

r

h . rF.

Let A be a closed subset of the locally compact

Lemma.

Let

let

F(x, V) - f(x) +-V*;

be a retraction of a neighborhood of N onto N;

5.7

f.

.

X -+ Y be a homeanorphism when restricted to A;

f:

of A has a neighborhood Ux which

pose each point x

phically onto an open subset of Y.

neighborhood of A

onto an open subset of

maps homeomor-

f

is a homeomorphism of some

f

Then

sup-

Y.

(X and Y are separable met-

ric, as always.)

Let U be the union of the neighborhoods Ux.

Proof.

locally 1-1 at each point of such that

fIB

is

1-1,

fIB

where xn and x are in

f(x),

(1)

If

prove there is a neighborhood of be the E/n-neighborhood of

and yn

from some

f(xn)

If

U.

where

C,

f is

of Un such that

on no set

1-1

f(xn) - f(yn).

Then

C.

locally

f(x)

f(y);

-

1-1

at

x,

(2)

If

C

since

is

f

so we could not have

Un be the a/n-neighborhood of

f

is

1-1

is

f

fin,

on no set

of A - Un such that

1-1.

Let Un

U1

is can-

there are points xn

x and y will be points on C,

1-1

c

and so that

f is

1-1

But

x - y.

for large

f(xn) - f(yn)

where

f

on

1-1

is

1-1

n. C U A,

on V U A.

is small enough that

on U1

is

f

U1

(using (1)).

there are points xn of Un - A and yn

f(xn) - f(yn).

numbering, we may assume xn -' x,

we

0,

By passing to subsequences and

C,

U1UA,

on

1-1

is a compact subset of U and f is

is compact and lies in U, If

is

f

we prove there is a neighborhood V of C such that Let

Since

onwards.

is small enough that

c

so it con-

x.

renumbering, we may assume xn -+ x and yn -. y; of

n

on whose closure

C

f(xn) -+

is open in Y,

f(Ux)

is a compact subset of U and

C

pact and lies in

Now

B.

xn must converge to

is a homeomorphism,

For let

is a homeomorphism.

tains all elements of the sequence fjUx

is

f

Furthermore, if B is any subset of U

U.

then

Then

By passing to subsequences and re-

where x

is a point of

C.

Then

f(yn) -

S 5. MANIFOIDS WITH BOUNDARY

Now

f(x). 1-1

at

fIC U A is a homeomorphism, so that yn -. x. so we could not have

x,

such that Vn is a compact subset of U and we may choose a neighborhood Vn.1

f

is a compact subset of U and

neighborhood V of

is

f

such that

Then

is

f

on the

1-1

There is a non-negative

let

Let

(U.).

be a partition of unity dominated by the

(mi)

dim M - m.

For each

x in Ui n Bd M.

is zero for

positive at

set in Hm about so that

k(x)

he (x)

at points of Let

mth coordinate function

Furthermore, if

onto an open set in

a(hY k-')/auJ - 0

D(hik ') is non-singular,

the

i,

k(x) - (u',...,um)

then

x,

k-')/sum

a(I

is

is a non-singular transformation of an open

For hi k-1

k(x):

g(x) - 0

be a locally-finite covering of M by

((Ui, hi))

Let

is another coordinate sustem about the point

Bd M.

By (2),

1.

coordinate systems;

since

on VnUA.

1-1

on Vn+tU A.

i-1

Let M be a Cr manifold.

has rank

Proof.

0,

is

An,

Cr function g on M such that for each x in Bd M,

dg(x)

hi(x)

Define VO

A.

Lemma.

real-valued

covering

is empty.

of the compact set VnUAn+I

Let V be the union of the sets Vn.

and

n.

In general, suppose Vn is a neighborhood of

to be the empty set.

5.8

is locally

f

Let A be the union of the increas-

of compact sets, where A0

AOC A1C...

ing sequence

But

for large

f(xn) - f(yn)

We now prove the lemma.

(3)

Vn.1

55

Hm,

and

at points of FP-l,

h

k-'(u...... u1°-',0)

for

j < m.

Since

a(bY k_1)/21u m must be non-zero at points of

RD1-';

this derivative must be positive

is non-negative for all x, RTII-' .

g(x) - F.i Ti(x)

Furthermore, if

q(x).

Then

k(x) - (u1,...,um)

g(x) - 0

whenever x

is in

is a coordinate system on M,

then

(gk 1)/sum - Ei hi (,ik-.')/sum + L Whenever x

is in Bd M,

the first term vanishes, because

second term is strictly positive. Bd M.

a(hY k-')/sum

Hence

dg(x)

has rank

hT(x) - Oj I

the

if x is in

I. DIFFERENTIABLE MAIIFOIDS

56

5.9

phism p

There is a

Cr manifold.

of a neighborhood of Bd M with Bd Mx (0, is in Bd M.

x

whenever

(x, 0)

Let M be a

Theorem.

Cr diffeomor-

such that

1)

p(x) _

Such a neighborhood is called a product

neighborhood of the boundary. Proof.

Cr retraction

g on M

r : W

By 5.8, there is a non-negative

Bd M.

such that if

is in Bd M,

x

is in Bd M,

then

g(x) - 0

by the equation

f : W - Bd M x(0, e)

Define

f(x) _ (x, 0) and

and

has rank

dg(x)

is non-singular:

df(x)

Cr function

f(y) _ (r(y), g(y)).

be a coordinate system about

k(x) _ (u1,...,um)

Bd M in M and a

there is a neighborhood W of

By 5.6,

if

1.

x

For let

let h - (klBd M) x i.

x;

D(hfk 1)

one uses Theorem 1.11 to prove that each x in Bd M has a neighborhood

Ux which Since that

f

carries homeomorphically onto an open subset of Bd M x to, o).

is a homeomorphism when restricted to

f

is a homeomorphism of some neighborhood U of Bd M in W onto

f

a neighborhood V of Bd M x 0

in Bd M x to, oa).

2.6, we may construct a Cr function (x, t) of

a restriction of

5.10

t < 8(x).

Then p(U)

contains

(x, 0)

and

PO: U0 - Bd %x[o, 1)

of the boundary in MO

and

and let p1.

A

such that the point

Let

be the diffeo-

s

(x, t)

into

Bd M x to, 1),

and

Let M be a Cr manifold with non-empty boundary.

Definition.

M,

D(M),

(x, 1)

is the union of MO - M x0

identified whenever x

may impose a differentiable structure on D(M)

D(M),

on Bd M

p will satisfy the demands of the theorem.

Recall that the double of

with

Using Exercise (a) of

onto itself which carries

oo)

and let p = of.

(x, t/8(x)),

8(x) > 0

lies in V if

Bd M x to, a*)

morphism of Bd M x (0,

M x1,

Bd M, it follows from 5.7

and

and

M1.

M1 a

We

Let

be product neighborhoods

Let U be the union of

P : U -. Bd M x(-1, 1)

is in Bd M.

as follows:

Bd Mix(-1, o]

p1: U1

and

UO

and

U1

in

be the homeomorphism induced by po

Cr differentiable structure on D(M)

is well-defined if we

§5. MANIFOLDS WITH BOUNDARY require (1)

and

to be a

P

in D(M)

M1

57

Cr diffeomorphism, and (2) the inclusions of

MO

Cr imbeddings.

to be

Now the differentiable structure on the choice of the product neighborhoods

p0

D(M)

and

depends strongly on However, the differ-

p1.

entiable manifolds arising from two such choices are diffeamorphic, as we shall prove later (6.3).

(a)

D(H2)

Exercise*.

Impose two distinct differentiable structures on

by using different choices for the product neighborhood of

Bd H2.

Show that the resulting differentiable manifolds are diffeomorphic, but that

the diffeomorphism may not be chosen as an arbitrarily good tion to the identity on each copy of (b)

H2.

If M and N

Exercise*.

C1 approxima-

are

Cr manifolds, show that Mx N

has a Cr differentiable structure such that each inclusion M - M x y and N-, x x N is a Cr imbedding.

Theorem.

5.11

manifold

M,

then

Proof. empty.

If

9) is a

5) contains a

Cr differentiable structure on the

C0D structure.

We have proved the theorem in the case where Bd M

Otherwise, let us consider the non-bounded manifold D(M),

provide it with a

Cr differentiable structure, as in 5.10.

structure contained in this, and let D'

is

and

Choose a

Co

denote the corresponding differen-

tiable manifold.

Consider the map defined in 5.10.

The Cr structure on Bd M

note the resulting differentiable manifold by diffeomorphism of the

where

P-1: Bd M x (-1, 1) -D(M)

P

is the map

contains a C0D one; let us de(Bd M)'.

C00 manifold (Bd M)' x (-1, 1)

Then

P-1

is a Cr

with an open subset

U of the C0 manifold D' I. Now there is a Cr diffeomorphism h U, which is C0D in a neighborhood of (Bd M)' x (-1/2, 1/2). h(Bd M x 0)

of

(Bd M)' x 0,

(Bd M)' x (-1) 1)

and equals

(Here we use Exercise (b) of 4.2.)

is a C00 submanifold of

D'.

P-1

with outside

Then the set

58

I. DIFFRRENTIABIE MANIFOLDS

h induces a

Now merely define i

f(x) - hP(x)

f

if x is in U,

and the set

fi(M)

f(x) - x

C°° submanifold of

otherwise.

If

Cr imbedding of

is a

fi

we

onto itself;

D'

D'.

Let M and N be non-bounded

Lemma.

5.12

is a

of

and

is the inclusion mapping, then

: M - MOC D'

in D',

Cr diffecanorphism

M

Our result follows.

Cp manifolds;

let

f : M - N be a Cr diffe miorphism. Identify N with N xo in N xR for Given a positive function

convenience.

function

on M

s

mation to

Cp imbedding of M into N xR which is a e-approxi-

Then there is a

f.

there is a positive

on N x R,

such that the following holds:

g be a

Let

a

Cp diffeomorphism h

of N x R

onto itself,

such that (1)

h carries

(2)

h

g(M)

into

N.

is an e-approximation to the identity, and equals the identity

outside N x (-2/3, 2/3). Let

Proof.

and equals

t < 1/3

Let

k > 3.

enough that

P(t)

for

0

be a monotonic C0 function which equals

be the projection of

it

g(M)

lies in

N x

is a diffeamorphism of M onto

N x R

(-1 /3,

N x 0.

for all

k >I 0'(t)l

onto

Let

N x 0.

then

t;

be small

8

ng

1/3), and small enough that (By 3.7,

as closely as desired by taking

of - f

imate

Let

t > 2/3.

s

for

i

ng may be made to approxsmall enough.)

Then the general point of g(M)

is of the form

where

y

is in N and

function on
2/3, 9(IsO - 0 and h is the identity. 5, we may make

By proper choice of

For this, we note that

tion to the zero-function as desired.

using 3.7, we see that if g approximates

g(ag)-1(y, o);

of - f,

ng approximates

enough,

as close an approxima-

c(y)

approximates ff

1

approximates f 1,

(ag)-

Let

. identity.

g(ag)

is an

q)(y)

so(y) - min e(y, a)

N must be imbedded in some RP before this makes sense;

-1 < s < 1.

is imbedded in Rp+1.

then N x R

closely

and

be small enough that

8

e0(y)/2k approximation to the zero function, where for

f

(y, (p(y)) -

Now

I(p(y)I < eo(y),

so

II h(y, s) - (y, s) II - IY(y,, s) - sI < E0(y) Let v be a unit tangent to N xR at

as desired.

v2 be its components tangent to II

and y xR,

N x s

- v II < II dh(v1) - 'vi II

dh(

eo/2, since

The second term is less than

+

II

let v1

(y, s);

and

respectively.

dhCC2) - v2U

Ia!/as - 11

I1P(y)A'(1s1)I

The first term equals

eo/2.

HdY(v,) (I

-

IIo(IsOdrv(1) II < eo/2

as desired.

5.13

f : M - N

Theorem.

be a Cr diffeomorphism

is a Cp diffeomorphism Proof.

tion to

Let M and N be manifolds of class

f

and

Let

8

be small enough that if

then f

then

f1

is a

f1

is onto.

81

and

s2

to

as a sub-

By Prob-

f,

re-

and h : D(N) -

is a 82-approximation to the identity, then bg is a 8-approximation f.

Let g : M - D(N) to

Consider N

on M and D(N)

spectively, such that if g is a 51-approximation to D(N)

f.

Co 5-approxima-

is a Cr imbedding of M into D(N).

lem 3.7, there are positive functions

let

5(x) > 0, there

f1: M -+ N which is a 8-approximation to

f1(Bd M) C Bd N,

manifold of D(N);

Given

(1 < r < p < a).

Cp;

f.

If

81

be a Cp imbedding which is a

is small enough,

neighborhood Bd N x(-1, 1)

81-approximation

g will carry Bd M into the product

which is used to give

D(N)

its differentiable

I. DIFFERENTIABLE MANIFOLDS

60

Then we may apply the preceding lemma, to obtain a

structure.

phism h of D(N)

which carries

enough, we may make sure

Then

h

is a

maps M

hg - f1

D(N).

Since

into

5.14

there is a

Problem.`

N

onto

between

ft

for each

which is a 8-approximation to entiable isotopy t,

Bd M

into Bd N;

f

and

f1,

N

such that

t.

You will need to generalize Lemma 5.12 as follows:

Cr differentiable isotopy between two

each

and carries

our result follows.

Cp,

Cr differentiable isotopy

Hint:

be a

D(N)

Generalize the preceding theorem to prove that

is a diffeomorphism of M

ft

small

s1

M must be carried into the subset

is of class

fl

By making

Bd N.

s2-approximation to the identity.

a connectivity argument shows that of

onto

g(Bd M)

Cp diffeomor-

f

for each

gt

Let

Cr imbeddings of M into N x R t.

Then there is a

ht which is a diffeomorphism of N x R

which satisfies (1) and (2) for each

Cr differ-

onto itself for

t, and in addition satisfies

the conditions: for some

(3)

If

gt(M) C N

(4)

If

Fat is of class Cp

5.15

Problem*.

Let

t,

then

for some

is the identity map.

ht t,

f : M - N be a

so is

ht.

N

Cr map;

Prove that any sufficiently good strong approximation to ably homotopic to f. Specifically, prove that given such that for any Cr map

Hint:

a

Let

f,

for each

ft

between

f

and

f,

g which is an

t.

Cr retraction of a neighborhood of

N.

is differenti-

there is a s(x) > 0

N be a submanifold of'euclidean space

g along straight lines in RP, in

f

g : M - N which is a 8-approximation to

there is a Cr differentiable homotopy e-approximation to

e(x) > o,

non-bounded.

N

in RP,

and then apply

onto r

N.

RP;

r

let

Deform

f

be into

to make the image remain

MANIFOLDS WITH BOUNDARY

55.

Generalize Problem 5.15 to the case where

Problem*.

5.16

draw the additional conclusion that whenever both

a boundary;

g(x) lie in Bd N,

so does

ft(x)

for each

Imbed Bd N in Rp;

Hint:

it

of D(N)

is the projection of RP+1

f(x)

and

extend to an imbedding h : U - RP+1, so that

onto RP.

xh(U)

h(N),

Finally, choose an imbedding

in some euclidean space which agrees with h in a neighborhood using 2.11.

of Bd N,

traction r to D(N)

N has

t.

where U is a neighborhood of Bd N in D(N), where

61

at

Then apply the techniques of 5.15, choosing the re-

so that it carries a neighborhood of x in the plane normal

x

onto

This implies that r

x.

joining two nearby points of

h(Bd N)

carries the line segment

into h(Bd N).

Exercise*.

Provide an alternate proof for Problem 5.14.

(a)

*

5.17 Problem .

A space X is locally contractible if for each

point x of X and each neighborhood U of

x,

V of x

U

stant map

such that the inclusion map c

: V

is hcmotopic to the con-

: V - x.

Let M and N be Fr(M, N)

i

there is a neighborhood

Cr manifolds;

M

compact.

is locally contractible in the C1 topology.

Prove that Prove also that the

following spaces are locally contractible in the C1 topology: (1)

The space of all

Cr immersions of M into

N.

(2)

The space of all

Cr imbeddings of M into

N.

(3)

The space of all Cr diffeomorphisms of M

onto

M.

Uniqueness of the Double of a Manifold*.

§6.

Let M be a non-bounded Cr manifold, let W be a

Lemma*.

6.1

neighborhood of M x 0

in M x R+, where R+ - [0, a').

f be a Cr

Let

imbedding of W into M x R+ which equals the identity on M x 0. is a Cr diffeomorphism

f

of W onto

f(W)

which equals

f

There

in a neighbor-

hood of the complement of W, and equals the identity in a neighborhood of

M x 0. Let

Proof.

denote the general point of M x R+. We may as-

(x,t)

sume that W is the set of points is a positive

0(x) < 1

Step 1.

Since

and

Cr function on M. f

Let

is non-singular,

Cr function

Choose a positive

for which

(x,t)

is positive.

such that

aT(x,t)/at > 0

for

o < t < E(x)

1 > E(x)IaT(x,t)/atI

for

o < t < 0(x)

We define a diffecmorphism g of W with itself by the equation

Y(x,t) - (1 - a(t/0(x))) E(x) - t + a(t/0(x)) a(t)

t < 1/3 cause

g(x,t) -

where

(x, Y(x,t)),

Here

where

f(x,t) - (X(x,t), T(x,t)).

aT(x,0)/at

on M

c(x) < 1

0 < t < 0(x),

is, as usual, a monotonic

and equals

C00 function which equals

for

2/3 < t.

g(x,o) - (x,o),

and

g(x,t) - (x,t)

Let f1 which equals

f

1

for

t > 20(x)/3,

- (Y)E(X) + a + (1 - E(X))(1/0(X))

fg. Then near W -W.

f1

0

for

The map g is a diffecmorphism, be-

1

aT(X,t)At =

t

and

CO > E(X) > 0

.

is a Cr diffeamorphism of W with

Furthermore, if we set

f(W),

f1(x,t) . (X1,T1),

then 0 < aT1(x,t)/at < 1

for

t < 0(x)/3

This follows from the fact that T1(x,t) - T(x,E(x)t) that

aT1(x,t)/at - E(x) aT(x,E(x)t)/at,

by choice of

for

t < 0(x)/3,

so

which is positive and less than

E.

Step 2.

Now we define a diffeomorphism h of W with itself by

the equation h(x,t) - (x, cp(x,t)), V(x,t) - a(2t/0(X)) Again,

.

where t + [1 - a(2t/0(X))) T1(x,t)

h is a diffecmorphism because 63

c(x,0) - 0,

and

q,(x,t) - t

for

1,

64

DIFFERENTIABLE MANIFOLDS

I.

and

t > 9(x)/3,

This last inequality

t < 9(x)/3.

for

acp (x,t)/at > 0

follows from the following computation:

tt

acV(x,t)/at - a + (1-a) aT1(x,t)/at + (1 - T1(x,t)/t) aT1(x,t)/at > 0

which is positive because

,

aT1(x,t*)/at < 1,

T1(x,t)/t

0 < t* < t.

where

Let

f2 = f1 h-1.

which equals

f(W)

and

a'/13

near W - W.

f

is a

f2

Furthermore, if we set

f2(x,t) _ (X2,T2),

in M x R+.

If M is compact, the completion of the proof is easy.

Step 3.

small enough that Mx [0,s]

8

Cr diffeomorphism of W with

in a neighborhood Y of M x o

then T2(x,t) a t

Choose

Then

where

f3(x,t) _ (X3,T3),

T3 - T2

is contained in Y,

and define

and

X3(x,t) = X2(x, a(t/b).t) f3

M

is a diffeomorphism of

is a diffeomorphism because the map x -. X(x,t0) for

and

to < s,

T3(x,t) - t

for

t < 8.

If M is not compact, more work is involved.

We need the follow-

ing lemma:

Lemma*.

Let U be an open set in Rm whose closure is a

6.2

ball.

Let

be a compact subset of U; let V be a neighborhood of

C

whose closure is contained in U.

Let

f be a Crimbedding of U x [o,a] into

Rmx [o,a) which equals the identity on U x 0, such that f(x,t)

There is a

Cr diffeomorphism

f1

_ (X(x,t),t).

with

of U x[o,a)

- (X1,t)

such that

f(Ux (O,a))

is the identity on U x o

(1)

f1

(2)

X1(x,t) = X(x,t) If

(3)

C

f

and on

for some

C x[o,s],

8 > 0.

outside V x(0,a/2).

is the identity on some set x x (o,b),

then

is the

f1

identity on this set also. Proof.

and equals

outside

0

which equals Let

Let

1

for

(p(x) be a C°° function on Rm which equals V.

Let

t < 1/3

r(t)

Define

0

for

E < a/2

C

2/3 < t.

f1(x,t) = (X1(x,t),t),

X1(x,t) = X(x, t(1 - .(t/E) q)(x)]) Here

on

be a monotonic C00 function on R

and equals

P(x,t) _ (X(x,t),t).

1

is a positive number yet to be chosen.

where

Now for x

V,

outside

in

and

and

C

X1(x,t) - X(x,0) - x,

t < E/3,

Condition (2) is clear, since

is satisfied.

tion (1)

for

y(t/E) - 0

identity on x x[o,b).

If

[o,b],

then ht that

is the

so that

be an imbedding of U

By Exercise (b) of 3.10, there is a function 8(x) > 0

t.

such that if

for x

is a diffeomorphism, and for

f

ht(x) - X1(x,t)

this it will suffice that the map

on U

f

t(1 - y(p),

so is

so condi-

Thus condition (3) holds.

All we need now to do is to show

for each

m(x) - 0

Finally, suppose

t > a/2.

is in

t

X1(x,t) - X(x,t(1 - y@)) - x.

in Rm,

65

UNIQJENESS OF THE DOUBLE OF A MANIFOLD

§6.

gt(x) - X(x,t),

is a 8-approximation to the map

ht

Then there is a positive constant

is an imbedding.

e0

such

ht will be an imbedding if 1X1(x,t) - X(x,t)f < 80

and i

X1 (x,t) /ax - ax(x,t) /axl < 80

for all x in U. By uniform continuity of X and such that

if x is in j and

faX(x,t)/at ap(x)/axi

minimum of

co

and

IaX(x,to)/ax - aX(x,t)/axl < 80/2

for

x

in

it, y(t/E). W(x)l < E,

where

so that

The first term is within

so/2

to - t(1 - yT).

of

of Ui with a ball in Rm.

that (Ci) covers

aX(x,t)/ax,

as desired.

Also, .

and the second is within

which equals

f

Let Ci be a compact subset of Ui such

fi - (Xi,Ti)

near Y - Y,

(1)

T1(x,t) - t

on Y,

(2)

Xi(x,t) - x

for x in

c

Cover M by a locally-

for which there is a diffeomorphism

Ui,

For convenience, assume

M.

Induction hypothesis:

Choose

It - tot -

Hence our desired result follows.

finite collection of open sets

f(Y)

Hence

t y(t/E)[aX(x,t0)/at a(p (x)/ax]

-

Completion of the proof of the theorem.

with

to be the

e

IX1(x,t) - X(x,t)I < 80,

6X1(x,t)/6x = aX(x,t0)/ax

hi

Choose

Ti.

so /2M.

Now X1(x,t) - X(x,to),

of zero.

co

Let M be the maximum value of the

It - tol < co.

entries of

80 /2

and

IX(x,t0) - X(x,t)l < 80

there is a constant

aX/ax,

U1

and U2

is a

are empty.

Cr diffeanorphism of Y

such that

and Ci

and

so that Uii x(o,c)

o < t < 8j,

for

is contained in Y.

J < i.

By &"I"

DIFFERENTIABLE MANIFOLDS

I.

66

the lemma, we may obtain a diffeamorphism outside

for some choice of

ei+1.

we will have Xi+1(x,t) = x

lemma,

on

extend it to U by letting it equal that

?

on Ui+1x o

and on

for all

Ci x[0,81)

f2

fi

Because of condition (3) of the

?(x,t) - limy _. . fi(x,t)

We then define

which equals

° (Xi+,,t)

such that Xi+1(x,t) - x

Ui+1 x(0,c),

Ci+1x[0,81+1)+

fi+1

outside

for

j < I.

in Y;

(x,t)

we

It is easily seen

Y.

satisfies the requirements of the theorem.

6.3

The double of a Cr manifold M is uniquely deter-

Theorem*.

mined, up to diffeomorphism. Proof.

and D'(M)

Let D(M)

be two differentiable manifolds ob-

tained by using different product neighborhoods of Bd M to define the differentiable structure.

Than P : V -. Bd M x(-1,1)

are diffecmorphisms of the open subsets V and V' respectively, with Bd M x(-1,1). Bd M xo

Now

PIP-'

into Bd M x (-1,1);

in Bd M x ( - 1 , 1 )

P': V'- Bd M x(-1,1)

and of

and

D(M)

maps a neighborhood W of it equals the identity on

and is a diffeomorphism when restricted to the subsets

Bd M x o,

(Bd M x [ o,1)) n W = W.I.

and

(Bd m x (-1 , o)) n w = W_

of

By the preceding

W.

which equals

theorem, there is a homeomorphism g of W with P'P 1(W) PIP-1

near Bd W,

and W_,

is a diffeamorphism when restricted to the subsets W.

and equals the identity in a neighborhood of Bd M xo. f - (P')-1g P is defined on the neighborhood

Then

Bd M in D(M),

of D(M)

with D'(M)

(a)

Hence

f may be extended to a homeomorphism

is a diffeomorphism of D(M)

f

Exercise*.

onto D'(M).

If

f

and

g are in Diff M,

f

weakly diffeotopic to g if there is a Cr diffeomorphism F

relation.

F(x,t) - (f(x),t)

in a neighborhood of M x(-o,o)

in a neighborhood of M x Let

i(M)

P-1(W).

Let Diff M denote the space of all Cr diffeomor-

phisms of M onto itself.

such that

of

near the

by letting it equal the identity outside

One checks readily that

I

P-1(W)

and equals the identity map of D(M) - D'(M)

boundary of this neighborhood.

(g(x) ,t)

D'(M),

is said to be

of M x R and F(x,t) -

Show that this is an equivalence

denote the equivalence classes.

The composition of two

UNIQUENESS OF TEE DOUBLE OF A MANIFOLD

§6.

diffeomorphisms is a di£feomorphism, so that Diff M this group operation makes

between

ft

g

are weakly diffeotopic. Exercise*.

(c)

Prove that

(g(x),1).

Exercise*.

(d)

group of

f

or

and

f

let G be a

Let M be non-bounded;

G(x,o) a (f(x),0)

and g

Cr diffeo-

and

G(x,1)

are weakly diffeotopic.

Let M be orientable;

let

r(M)

denote the sub-

generated by orientation-preserving diffeomorphisms.

r(M)

r(M)/r(M) 2gZ,

that

then

t,

(The converse is an unsolved problem.)

with itself such that

morphism of M x I

show that

into a group.

T(M)

g which is a diffeomorphism for each

and

f

is a group;

Show that if there exists a differentiable isotopy

Exercise*.

(b)

67

0

Show

according as M possesses an orientation-

reversing diffeomorphism or not. Exercise*.

Show that

(e)

Hint:

is a diffeomorphism of

f

If

r(Rm)

the linear map which carries

into Df(o)

x

Rm,

first deform

into

f

then deform this into the

x;

identity map. *

Exercise .

(f)

of the set of all

x

If

the support of

f : M -+ M,

for which

Let

f(x) / x.

ments of Diff M having compact support; let

f

is the closure

Diffc M denote those ele-

rc(M)

denote the weak diffeo-

topy classes of DiffcM, where the support of the diffeotopy is required

to have compact intersection with Mx I.

If M

is orientable, let

rc(M)

denote the subgroup generated by orientation-preserving diffeomorphisms. Milnor has proved that that in general

is non-trivial (see [5) and [10)).

Prove

is abelian.

rc(Rm)

Given

Hint:

rc(R6)

f

and

g,

Let

f

deform them so their supports are dis-

joint. Exercise*.

be a diffeomorphism of

Bd M

onto

M U£N denote the non-bounded manifold obtained from

M U N

by identify-

(g)

Let

ing each x in

Bd M with f(x).

Bd N.

Put a differentiable structure on M U1N

such that the inclusions of M and

N

are imbeddings.

Show that the re-

sulting differentiable manifold is unique up to diffeomorphism. Exercise*.

(h)

Bd N.

Let

For any diffeomorphism

element

f

1

0

f

of Diff(Bd M).

f0 f

be a fixed diffeomorphism of Bd M

of Bd M

onto

Bd N,

let

onto

f denote the

Show that up to diffeomorphism, the

68

I.

DIFFERENTIABLE MANIFOLDS

differentiable manifold M Uf.N

of f.

depends only on the weak diffeotopy class

CHAPTER II.

TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS

Cell Complexes and Combinatorial Equivalence.

§7.

In this section we prove the theorem that two finite polyhedra in euclidean space may be subdivided into simplicial complexes in such a way that their intersection is a subcomplex of each of them (7.10).

For this

purpose, it is necessary first to define what is meant by a rectilinear cell complex, and to study some properties of cell complexes.

7.1

If v0,...,vm are independent points of

Definition.

the simplex

a - v0...vm they span is the set of points

x = E bivi,

where bi > 0

and

barycentric coordinates of center of

a

E bi a 1.

The point

x.

and is denoted by

v.

the interior of

Bm;

If

A

a

a.

such that

The numbers bi E vi/(m+1)

are called the

is called the bar -

A face of a simplex

spanned by a subset of the vertices of to

x

The simplex

is the simplex

a

a

Is homeomorphic

is called an open simplex.

and B are two subsets of

Rn,

the loin A * B

B is the union of all closed line segments joining a point of point of

Rn,

B,providing no two of these line segments intersect

sibly at their end points. Then

of A and A and a

except pos-

a - v0 * (V1 * (v2 * ... )).

A (simplicial) complex K is a collection of simplices in Rn, such that (1)

Every face of a simplex of K is in

(2)

The intersection of two simplices of K is a face of each of them.

(3)

Each point of

many simplices of

K.

JKJ

K.

has a neighborhood intersecting only finitely 69

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

II.

70

Here

denotes the union of the simpliees of K and is called the

IKI

polytope of

K;

sometimes it is called a polyhedron.

(It would be more

general to let the simplicea of K lie in R°° = Un Rn, will suffice for our purposes.

but this definition

We are in fact restricting ourselves to

finite-dimensional complexes.)

A subdivision each simplex of

of K is a complex such that

K'

is contained in a simplex of

K'

K.

and

IK'I = IKI

A subcomplex of K

is a subset of K which is itself a complex.

If x is a point of the interiors of all simplices subset of fine J

denoted by

IKI,

St(S,K)

the star of x

IKI,

such that

at

If

St(x,K).

S

to be the union of the sets

is a subcomplex of

x

in K is the union of

lies in

Is any subset of IKI, we de-

for all x in

St(x,K)

it is convenient to write

K,

It is an open

s.

If

S.

for

St(J,K)

St(IJI,K) Let

a

be the simplex v0...vm.

f(x) = f(E bivi)

plexes, a map f each simplex of

for all x

E bi f(vi) e

IKI

A map in

a.

K linearly into a simplex of L.

the phrase - f : X - L is linear. K'

is linear if

e - RP

If K and L are com-

is linear relative to K and L if it carries

ILI

if for some subdivision

f

of

The map

K,

f

:

We often shorten this to

f : K - L is pi8oewise-l132ear

K' - L is linear.

a linear isomorphism if it is a homeomorphism of

IKI

onto

: K - L is

f

ILI

and carries

each simplex of K linearly onto one of L. It was long a famous unsolved problem (often called the Hauptvermutung)

whether the existence of a homeomorphism between

IKI

plied the existence of a piecewise-linear homeomorphism of Yes, if dimension K = 2 [11]; yee, if

Partial answers were: manifold

T1, 71;

the answer is no if dimension K > 6

(a)

St(x,K),

IKI

im-

onto

ILI.

IKI is a 3-

[6].

Recently, Milnor has shown that It is still unknown if the ans-

is a manifold.

Exercise.

Let K be a complex.

denoted by 5t(x,K),

We sometimes use

IKI

ILI

yes, if K and L are smooth triangulations of diffeo-

morphic manifolds (this we shall prove).

wer is yes when

and

Show that the closure of

is the polytope of a finite subcompleX of

St(x,K) to denote the complex as well as the polytope,

where no confusion will arise.

K.

S7. CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE

Show that if

Exercise.

(b)

homeomorphism of

f : K- L is a piecewise-linear

ELI, then they have subdivisions

onto

IKI

71

K' and L'

f : K' -- L' is a linear isomorphism.

such that

Let

7.2 Definition.

of all points x

c

consisting

be a bounded subset of Rn,

satisfying a system of linear equations and linear in-

equalities Li(x) - Ej such a set set of

c

Rn.

xJ > bi

aij

1 - i,...,p ;

;

is called a (rectilinear) cell. (Convexity of a set

It is a compact convex sub-

means that

c

contains each line seg-

c

ment joining two of its points.)

The dimension m of al plane

d'

containing

points, but not m + 2.

c;

is the dimension of the smallest dimension-

c

this means that

Since

contains m +

c

homeomorphism of

c

independent

must contain the simplex spanned by these

c

m + 1 points, it must have interior points as a subset of (P.

denote these points; let Bd c

1

be the remainder of

Int c

We show there is a

c.

with the m-ball BF carrying Bd c

Let us adjoin to the system defining

Let

Sm-1:

onto

a set of equations for I?

c

Some of the inequalities of the system may now be redundant; let L1(x) > bi,

i - 1,

..., p,

be a minimal system of inequalities which, along with

the equations for 6), serve to determine

p, the hyperplane

Li(x) - bi

c.

intersects 61

for otherwise, this hyperplane would contain

We note that for

in a plane of dimension m-1; (P and the corresponding in-

equality could be discarded without changing the set

Furthermore,

c.

equals the set A of those points for which each of these inequali-

Int c

ties is strict:

if xo bj,

i - 1, ...,

Clearly,

A is contained in Int a.

is a point of the intersection of

than arbitrarily near x0

so that

x0

Z.

d' with the hyperplane

are points of

La(x) -

(P for which Li(x) < bj,

does not lie in Int c.

Let c be a point of ning at

On the other hand,

Int a

and let

The intersection of r with

and convex; i.e., a closed interval.

necessarily a point y

of Bd c.

c

r be a ray in P beginis non-degenerate, compact,

One end point is

Each point x

c;

the other is

of the open line segment

72

II. TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

cy necessarily lies in so that

1 - 1, ..., p,

for

Int c,

Li(c) > bi Hence

Li(x) > bi.

c

and

L1(y) > bi

for

is equal to the join of

c

with Bd c. To complete the proof we need only show Without loss of generality we may assume

Sm-1.

The map

origin.

x

carries Rm- 0

x/IIxlI

homeomorphic with

Bd c

is Rm and c is the

(P

continuously onto the unit

sphere; it is necessarily a homeomorphism when restricted to

Let

Lemma.

7.3

finite number of

be an m-cell.

c

m-1 cells, each the intersection of an m-1 plane with

Let (P

Proof.

is the union of a

Then Bd c

These cells are uniquely determined by

Bd c.

Bd c.

by equations for 6,

c.

be the m-plane containing

let

c;

c be given

along with a minimal set of inequalities

L1(x) > bi,

i . 1, ..., p.

di

Let

then di

denote the set of points

is a cell, and Bd c

of dimension m - 1. the subset of for all

S

with

Furthermore,

(P is an m - 1 plane

is precisely di.

denote

L,(x) > b

is convex

S

for which L1(x) >

It also contains a point y Li(y) < bi,

for which Li(x) = b1;

in particular, it

Int c;

contains a point x

that

Let

Then

c

is a cell

di

(P for which i.

j

and contains

bi.

Li(x) = bi

whose intersection with Bd c We prove that

of

is the union of these cells.

the intersection of the hyperplane T1,

x

such

since otherwise

would lie entirely in the region Li(x) > bi,

inequality the set

Then 611.

so that discarding the L1(x) > bi

c.

S n 61 Since

from the set of inequalities for

By convexity,

S

is not empty; s n Mi C di,

contains a point since

di

S

is open in

z

c

would not change

such that (P,

S n 6) 1

L1(z) = bi.

is open in

must be a cell of dimension m - 1.

V.

CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE

The uniqueness of the cells

is easy:

di

union of p planes of dimension m-1.

Bd c

73

is contained in the

Hence the intersection of any other

m-1 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

Definition.

into which Bd c di

If

decomposes is called a face of

is called a m - 2 face of

is also a face of

7.5

is an m-cell, each of the m -

c

as is

c,

Lemma.

Let

c

And so on.

c.

cells

1

di

each m - 2 face of a

c;

By convention, the empty set

itself.

c

be a cell given by a system of linear equations

and inequalities. Replacing some inequalities by equalities determines a c, and conversely.

face of

We proceed by induction on

Proof.

m - 0, the lemma is trivial.

m > 0, let

If

c; adjoin to the system determining

c

m, the dimension of

equalities for minimal set. If

a set of equations for

face let

e

Lj(x) = b,

if it does not intersect be the intersection of

the intersection of

e

let

q

a

e

of

y

(P

Otherwise,

it is a cell. 1

Now

plane, so that ar-

for which

L,(y) < b,.

c.

is convex, it must lie in some m-1 face

be the smallest integer such that

be a point of dq.

e

lies on the boundary of Since

it gives the empty face.

are points

Lj(x) _

this gives the trivial

(P with this hyperplane is an m -

bitrarily near each point of Hence

c,

d',

satisfying

c

with this hyperplane;

c

form a

< i < p

as we have just proved.

dj of c,

contains

be the in-

La(x) > bi by an equality.

consider the subset of

j > p,

If the hyperplane

1

This

(P.

Li(x) > bi

Then let us replace some inequality

In the case

c;

Let

c, so numbered that those for which

j < p, this determines an m-1 face

bj.

be the m-plane containing

60

we clearly may do without loss of generality.

If

c.

not in d1U...U dq_1;

e

let

lies in y

d

i

otherwise,

:

d1U...U dq.

be a point of

Then LI(x) > bi

for

i < q

and

Lq(x) - bq

Li(y) > bi

for

i < q

and

Lq(y) > bq

a

Let x not in

II.

74

lies in

z e (x + y)/2

The point that

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS but

e,

does not lie in d1U...U dq,

z

Li(z) > bi

adjoined, determine the

m - I

Furthermore,

e

an equality.

By the induction hypothesis,

cell

di

c,

with the equation

which contains

e.

is obtained from this system by replacing La(x) > bi e

is a face of

by

and of

di,

c.

A (rectilinear) cell complex K is a collection

Definition.

7.6

so

i < q,

contrary to hypothesis.

The system of equations and inequalities for Li(x) e bi

for all

of cells in Rn such that (1)

Each face of a cell in K is in K.

(2)

The intersection of two cells of K is a face of each of them.

(3)

Each point of

many cells of K.

JKJ

(Here

IKI

has a neighborhood intersecting only finitely denotes the union of the cells of

K.)

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

along with its faces, constitutes a cell complex.

c,

This follows

from the preceding lemma (see Exercise (a) below).

A subdivision of a cell complex K is a cell complex and each cell of

(K'l - IKI

K'

is contained in one of

K.

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

K,

denoted by Kp,

dimension at most

(a)

such that

The dimension The p-skele-

is the collection of all cells of K having

One checks that Kp

p.

K'

is a subcomplex of

K.

If K consists of a single cell, along with its

Exercise.

faces, show that K is a cell complex. (b)

Show that the simplex

Exercise.

a - vo ... 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 7.4.

Shaw that a simplicial complex is a special kind of

cell complex.

7.7 I.

Then

Lemma. K1

and

Let K2

K1 and

K2

be cell complexes such that

have a common subdivision.

JK11I

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

Proof.

cl

L. be the collection of all cells of the form

Let

cell.

is a cell of

and

Ki

and

is a cell of

c2

where

c1n c2,

Then L is a cell complex,

K2.

1K1! _ IK21 - ILI.

Exercise.

(a)

where

eln e2,

(b)

Exercise.

7.8

Lemma.

is of the form

and conversely.

ci;

Show that L is a cell complex.

Any cell complex K has a simplicial subdivision.

K is already a simplicial complex.

L is a simplicial subdivision of is an m-cell of

K,

the m -

Km-1,

I

a1 * c,...,ap * c.

of

skeleton of

ai * a

(Recall that

denotes the join of

K'

of

K,

which is a subdivision of

K.

It is often convenient to choose be canonically defined.

K'

the barycentric subdivision of

Exercise.

division of

c

as the centroid of If

K

c

c,

ai

the

in

is already simplicial, the

centroid is the same as the barycenter, and this subdivision

(a)

If

K.

adjoin to the collection L the aim-

c;

result will be a simplicial complex

order that

If

K.

In general, suppose

If we carry out this construction for each m-cell

c.)

of

al,...,ap be the simplices of L lying in Bd c.

let

Choose an interior point c plices

c,n c2

We proceed by induction on the dimension m

or m - 1,

m = 0

Show that any face of

is a face of

ei

Proof.

and

75

CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE

§7.

K'

is called

K.

Check that

K'

is a complex, and that it is a sub-

K.

7.9

1K1! _ JK21,

7.10

Corollary. K,

and K2

Theorem.

If

K,

and K.

have a common simplicial subdivision.

Let

K,

and

K.

in Rn. There are simplicial subdivisions respectively, such that

are simplicial complexes such that

K,1 U K2

be two finite simpliciat amgzlexes

K,

and

K2

is a simplicial complex.

of K 04.

TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS

II.

76

L such that plex of

We prove that some subdivision of

ILI . Rn.

a rectilinear triangulation Let

angulation

of

Rn:

a1,...,ap be the simplices of

K1.

L1

of Rn which contains

J1

affine transformation the collection

is a subcom-

K1

Choose a rectilinear tri-

One way to construct such a

a1.

of Rn which carries one of its simplices onto

h1

will be the required complex.

h1(J)

(An affine trans-

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

Ji be a rectilinear triangulation of Rn containing ai;

a common subdivision of J1,...,Jp tope of a subcomplex of

J1

of Rn and find a non-singular

is to take any rectilinear triangulation J

a1;

is a simplicial complex

A ,rectilinear triangulation of Rn

Proof.

Then

(using 7.9).

1K11

Similarly,

let

is the poly-

and this subcomplex is a subdivision of

L1,

be

L1

K1.

Similarly, let L2 be a rectilinear triangulation of Rn containing some subdivision of

vision of

and

L1

having polytopes

and

K2

and

IK1I

of the sets

IK11

IK2I,

is the set

Let

St(K1,K).

in R2

[0,1] x o

and

IK2I

is the union

.

be a subcomplex of the simplicial complex

K1

K1

The proof of Lemma 7.8 generalizes to the follow-

be a simplicial subdivision of

tending K

be the subcomplexes of L

respectively.

for n . 1,2,...

Definition.

ing situation:

K2

L be a common subdi-

You will need some further hypothesis to avoid

[0,1] xl/n

7.12

and

Kj'

Let

Generalize this theorem to the case in which

are not finite.

the case where

K;

and let

L2,

Problem .

7.11

as a subcomplex.

K.

to a subdivision of

let

We define a canonical way of ex-

K1. K,

K;

without subdividing any simplex outside

We will call it the standard extension of

K,'

to a subdivision of

K:

Every simplex of

K;

plex of K which is outside in A . St(K1,K) - IK1I

ces of K lie in A.

belongs to

K',

of course, as does every sim-

The simplices whose interiors lie

St(K1,K).

are subdivided step-by-step as follows: Each 1-simplex

a

No verti-

of K whose interior lies in A

§7.

CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE

is subdivided into two 1-simplices, the barycenter of

Let

For any m-simplex

its boundary has already been subdivided.

a,

a be its barycenter, and for each simplex

s

a itself.)

join of the empty set with a is

Exercise.

(a)

Problem.

K.

IKI.

Let K be a simplicial complex;

locally-finite collection of subsets of

IKI.

Let

quence of simplicial subdivisions of K such that This means that any simplex of

simplex of simplices integer

Ki+1 a

Na.

(a)

as well.

such that

a

K1

belongs to

equals

K1+1

Ki

positive continuous function on such that for any simplex

a

minimum of

in

for

x

SKI.

of a.

K',

be a

be a se-

K1, K2, ...

Ki

Ai

outside is a

denote the collection of those for all

i

greater than acme

Let K be a simplicial complex;

Exercise.

(Ai)

which does not intersect

limi,y m Ki

Let

let

Show that this collection is a subdivision of

8(x),

the

Check that the collection of simplices obtained in

this way is a complex whose polytope is

7.13

(Convention:

a.

After a finite number of

steps, we will have the required subdivision of

Ail

of the subdivision of

s * a be a simplex of the subdivision of

let

Bd a,

being the extra

a

In general, suppose the m - 1 simplices have already been subdi-

vertex. vided.

77

K.

let

There is a subdivision the diameter of

a

8(x) K'

be a

of K

is less than the

Immersions and Imbeddings of Complexes

S8.

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

f : K- M,

where K is a simplicial complex and M is a differentiable manifold, and we develop a theory of such maps analogous to that for maps of one manifold In particular, we define (8.2) the differential of such a map

into another.

and use this to define (8.3) the concepts of immersion, imbedding, and triAs before we define

angulation (which is the analogue of diffeomorphism). (8.5) 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 JK)

into Bd M.)

From now on, we restrict ourselves to simplicial complexes and subThe integer

divisions, unless otherwise specified.

r

(t < r < co)

will

remain fixed, for the remainder of this chapter.

8.1

Let K be a complex.

Definition.

relative to K if

The map

f

:

differentiable of class

Cr

each simplex

a

We usually shorten this to the phrase,

is of class

Cr.

of

K.

The map

equal to the dimension of

f a,

flu

is of class

is said to be non-degenerate if for each

a

in

IKl - M Cr,

is

for

f : K

flu

M

has rank

K.

We wish to generalize the notions of immersion and imbedding to this situation.

take a

As an analogue to a

Cr imbedding of a manifold, one might

Cr non-degenerate homeomorphism of a complex.

79

That is, until one

80

II.

TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS

looks at the homeomorphism

The crucial

of the accompanying illustration.

f

property of imbeddings - that any sufficiently good strong C1 approximation to an imbedding is also an imbedding - fails here, since arbitrarily close to

f

are maps like

This example shows us that we must seek further to

g.

find the proper generalization.

Definition.

8.2

b

of

Let

define the map

a,

f

a

x

and

Given the point

Cr map.

dfb: a -. Rn by the equation Df(b)

dfb(x)

Here

Rn be a

(x - b)

.

are written as column matrices, as usual; and we choose

b

some orthonormal coordinate system in the plane in which to compute

The map

Df.

in the plane of Let

dfb(x)

is independent of this choice of coordinates

dfb

a:

be some ray in

R

curve in Rn,

a

beginning at

Then

b.

fIR

is a

which we suppose parametrized by are length along

is merely the tangent vector of this curve at

11 x - b U.

in order

lies,

a

Hence

f(b),

Now

multiplied by

does not depend on the choice of coordinates,

dfb(x)

since it involves only the distance function in the plane of more, it follows that

R.

Cr

depends only on

dfbjR

fIR,

Further-

a.

not on any other values

of f. Now if ed for each

a

two simplices of

f : K

Rn

is a

Cr map, we have maps

dfb: a -+Rn defin-

in 3f(b,K).

These maps agree on the intersection of any

a''E(b,K),

since either (1) one is a face of the other, or

(2) their intersection lies in the union of rays emanating from

b.

Hence

the map

dfb: 'ST(b,K) -y Rn is well-defined and continuous.

By analogy with the situation for differen-

tiable manifolds, we call it the differential of

8.3

Definition.

submanifold of

Rn.

Let

The map

f f

: K- M be a

f.

Cr map, where M

is a

Cr

is said to be an immersion if

dfb: 5(b,K) - Rn is one-to-one for eaqh b.

An immersion which is a homeomorphism is called

§$.

m4msioris AND D Ef3DlivGS OF CGMPL cES

81

an imbedding; if it is also a homer fiorphism Onto, it is called a

Cr triangulat' on of

M.

If x and

hslorig to the simplex

b

of 1,

a

then the require-

ment that dbja be one-to-one implies that the matrix D(fla) b

equal to the diuensio