Differential Topology (North-Holland Mathematics Studies)

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DIFFERENTIAL TOPOLOGY

NORTH-HOLLAND MATHEMATICS STUDIES 173 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester N e w York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

DIFFERENTIAL TOPOLOGY

Juan MARGALEF-ROIG Consejo Superior de Investigaciones Cientificas Madrid, Spain

Enrique OUTERELO DOMINGUEZ Departamento de Geometria y Topologia Universidad Complutense Madrid, Spain

1992

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERSB.V. Sara Burgerhartstraat 25 P.O. Box211, IOOOAEAmsterdam,The Netherlands

ISBN:O444884343

0 1992 ELSEVIER SCIENCE PUBLISHERSB.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521,1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher, Elsevier Science Publishers B.V. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

To my parents and Maria d e l Mar and

To t h e memory of Natividad Gordo.

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vii CONTENTS

PREFACE .......................................................iX INTRODUCTION..................................................Xi CHAPTER 1: REAL DIFFERENTIABLE MANIFOLDS WITH CORNERS. 1.1. The Differential of Maps over Open Sets of Quadrants

of Banach Spaces..........................................l 1.2. Differentiable Manifolds with Corners....................18 1.3. Differentiable Maps ......................................34

1.4. Topological Properties of the Differentiable Manifolds...41 1.5. Differentiable Partitions of Unity 50 1.6. Tangent Space of a Manifold at a Point... 62

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

CHAPTER 2: THE WHITNEY EXTENSION THEOREM AND THE INVERSE MAPPING THEOREM FOR DIFFERENTIABLE MANIFOLDS WITH CORNERS. 2.1. The Whitney Extension Theorem............................73 2.2. The Inverse Mapping Theorem. Local Diffeomorphisms

......104 100 .................... ........................................ 110

2.3. Products of Differentiable Manifolds 2.4. Sum of Manifolds

CHAPTER 3: SUBMANIFOLDS AND IMMERSIONS

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

3.1. Submanifolds. 3.2. Immersions. 3.3. Embeddings

113

128 148

CHAPTER 4: SUBMERSIONS AND QUOTIENT MANIFOLDS

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

4.1. Submersions......... 159 4.2. Inverse Image of a Submanifold by Means of a Submersion.174 4.3. Regular Equivalence Relations. Quotient Manifolds.......l85

CHAPTER 5: SUBIMMERSIONS

........................................... 215 ................ 272

5.1. Subimmersions 5.2. Manifolds of Germs of Submanifolds......

CHAPTER 6: LIE GROUPS

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

6.1. Lie Groups 6.2. Homogeneous Spaces and Orbits...

283 293

CHAPTER 7: TRANSVERSALITY

..........309 ................................ 326 Manifolds............... Transversal Submanifolds. ............................... 337

7.1. Transversal Map to a Submanifold.............. 7.2. Transversal Family of Maps. Fibered Product of 7.3.

viii

Contents

CHAPTER 8: PARAMETRIZED THEOREMS OF THE DENSITY OF THE TRANSVERSALITY

...................345 Manifolds ............................................... 354 The Sard and Brown Theorems............................. 362 Smale’s and Quinn’s density Theorems.................... 380 Parametrized Theorem of the Density of the Transversality .......................................... 409

8.1. Lebesgue Measure Zero Sets in Rm. mrl 8.2. Subsets of Measure Zero in Finite Dimensional 8.3. 8.4. 8.5.

CHAPTER 9: SPACES OF DIFFERENTIABLE MAPS

......427 ............................... 435 . 468 Order Whitney Topology .................................. Infinite Order Jets. Whitney Topology of Infinite Order ................................................... 485 Special Open Sets in the spaces of Differentiable Maps .................................................... 496

9.1. Finite Order Jets between Differentiable Manifolds 9.2. Spaces of Continuous Maps 9.3. Topologies over the Spaces of Maps of Class p Finite 9.4. 9.5.

9.6. Continuity of the Composition of Differentiable Maps

....508

CHAPTER 10: APPROXIMATION OF DIFFERENTIABLE MAPS 10.1. Approximation of Differentiable Maps in the Case

Finite Dimensional

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

10.2. Elevation of the Class of a Differentiable Manifold

523

....538

CHAPTER 11: OPENNESS AND DENSITY OF THE TRANSVERSALITY

. .

.........543 . Theorems for Manifolds with Corners.................... 551 559 11.3. Whitney Immersion Theorems ............................. 11.4. Morse Functions ........................................ 565 BIBLIOGRAPHY ................................................. 575 INDEX........................................................ 589 11.1. Density of the Transversality R Thom Theorem 11.2. Multijets Density of the Transversality Mather

.

ix PREFACE Spaces of differentiable mappings between finite dimensional manifolds should be viewed as infinite dimensional manifolds. The traditional calculus of smooth mappings between finite dimensional spaces has a straightforward and powerful generalization to the realm of Banach spaces. Even the llhardll theorems like the inverse function theorem, the implicit function theorem (if one assumes the existence of complementary subspaces), and existence and uniqueness of solutions of ordinary differential equations continue to hold. But unfortunately spaces of differentiable mappings cannot be treated in a wholly satisfactory manner in the realm of Banach manifolds: a result of Omori and de la Harpe says that if a Banach Lie groups acts effectively on a finite dimensional manifold then it must itself be finite dimensional. Spaces of differentiable mappings can be viewed as manifolds modelled on more general locally convex spaces than Banach spaces and there is a satisfactory calculus for such spaces (see the book of Frolicher and Kriegl [F-K]), but all "hard" results fail to hold in general. Inverse function theorems are only possible if by a priori estimates one can generate enough rudiments of a Banach space situation such that some iteration process converges. Applications involving inverse function theorems thus are extremely technical and they boil down to Banach space techniques. So there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewehere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well defined subject; the possibility to fall back to it adds to the

X

Preface

feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry. Peter W. Michor. Wien, October 18, 1991.

xi INTRODUCTION Infinite dimensional manifolds appear naturally differential topology. One main example are manifolds differentiable maps.

in of

For some types of infinite dimensional manifolds it makes sense to introduce and study notions which are classical in the finite dimensional case. For instance, we can mention embeddings and immersions, transversality, Morse theory, topological degree theory, etc. Morse theory was generalized to Hilbert manifolds and later to Banach manifolds and this generalization provided a new method to study problems of variational calculus. Similarly, one can construct topological degree theories of Leray-Schauder type on infinite dimensional manifolds without needing finite dimensional approaches. It should be noted, finally, that infinite dimensional manifolds can be used to solve problems that arise in the finite dimensional setting. A nice example of this is S. K. Donaldson's result on the existence of 4-dimensional topological manifolds that do not admit differentiable structures. All this shows the interest of infinite dimensional manifolds, and the need of a complete study of their properties.

Historically, the concept of infinite dimensional manifolds can be traced back to the work of B. Riemann; ltber die Hypotheses, welche der Geometrie zu Grundeliegen, 1854. Differential Calculus on Euclidean Spaces is easily extended to real Banach spaces. Nevertheless the extension of this calculus to more general spaces, Frechet spaces for instance, involves additional difficulties. For this reason the manifolds considered in this book will be real Banach manifolds. The study of ordinary geometrical figures such as the closed

xii

Introduction

euclidean ball, the solid torus, etc., requires manifolds with smooth boundary. Thom cobordism theory and Smale handlebodies theory have been developed for this type of manifolds. Then the product of manifolds is often needed and of course it is not in this category of smooth bordered manifolds. Thus the manifolds with corners are the natural means to get rid of the technique of the so-called llarrondissementll of corners. Following this idea, the aim of the authors is to introduce infinite dimensional Banach manifolds with corners and prove, for these, some basic theorems of Differential Topology. The first part of chapter 1 is devoted to the introduction of infinite dimensional manifolds with corners and maps of class p. Differentiability of maps on open sets of quadrants of Banach spaces and the boundary invariance theorem for quadrants of Banach spaces are the most important tools. The second part of this chapter is concerned with the topological properties of the manifolds: dimension, differentiable partitions of unity, tangent vectors, inner tangent vectors and outer tangent vectors, etc. The Whitney Extension Theorem and the Inverse Mapping Theorem for manifolds with corners are the topics developed in chapter 2. The Whitney Extension Theorem has been generalized to the infinite dimensional case, when the closed set considered is a quadrant of a real Banach space. Of course, the closed set in the well known Well's counter example is not a quadrant of a real Banach space. The generalized Whitney Theorem makes the Inverse Mapping Theorem a corollary of the Implicit Function Theorem for manifolds with corners. Chapter

3

studies submanifolds, immersions and embeddings.

Introduction

xiii

Submanifolds are obtained as images of embeddings and the characterizations of immersions follow from the injective version of the implicit function theorem. Chapter

4

concerns submersions and quotient manifolds.

If we are given a manifold X of class p and an equivalence relation R on X I then the problem is to know under what conditions the quotient space X/R admits a differentiable structure. Submersions and the so-called regular equivalence relations give the solution to this question. Chapter 5 is devoted to the study of subimmersions and the construction of tubular neighbourhoods of Hilbert manifolds. The local constant rank theorem, the local representation theorem and the characterization by factorization of subimmersions are of course involved in the study of subimmersions. First, we construct the normal bundle associated with an immersion in a Hilbert space, a tubular neighbourhood associated with a closed embedding in a Hilbert space and a collar neighbourhood of the boundary of a Hilbert Hausdorff paracompact manifold. Then we prove the existence of a closed neat embedding of every Hausdorff paracompact Hilbert manifold with smooth boundary into Hx[O,m) , where H is a Hilbert space, such that the normal space at each point of a certain neighbourhood of the boundary is contained in Hx(0). This in turn gives a necessary and sufficient condition for a Hausdorff paracompact topological space to admit a differentiable structure of class m with smooth boundary. Chapter 6 is devoted to several basic results on Lie Groups including the construction of a differentiable structure on the space of orbits, in both the discontinuous and the free proper continuous cases. The topic of chapter 7 is transversality theory. Its topics are: maps transversal to a submanifold, transversal families of

x iv

Introduction

maps, fibered products of manifolds and transversal submanifolds. Since differentiable maps preserve zero Lebesgue measure sets, one can define zero measure sets in the finite dimensional case. Sardls density theorem says that the set of critical values of a differentiable map is a zero measure set and therefore, assuming some mild topological conditions, the set of regular values of a differentiable map is a dense set. The pattern used to define zero measure sets in Rn can not be followed in the case of infinite dimensional manifolds. Nevertheless, for Fredholm maps the set of regular values is dense. This generalization of Sard's theorem is due to S. Smale. Chapter 8 is devoted to the following topics and results: zero Lebesgue measure sets in Rm; subsets of measure zero in finite dimensional manifolds; Sard and Brown's theorems; Smale and Quinn's density theorems and the parametrized theorem of density of transversality. Finally, the set of differentiable maps between two manifolds is endowed with a differentiable structure. A first step in that direction is to endow it with topological structures. Jets and jet manifolds are powerful intrinsic techniques to do this. In chapter 9 we discuss this problem through the following topics: Finite order jets between differentiable manifolds; topological spaces of continuous maps; topologies on the spaces of maps of class p, finite order Whitney topology; infinite order jets; Whitney topology of infinite order; special open sets in the spaces of differentiable maps; continuity of the composition of differentiable maps. Chapter 9 contains the theorems of density of the maps of class p in the space of continuous maps. This study is completed in chapter 10 with the density of the maps of class s in the space of maps of class r (srrrl).

Introduction

xv

The finite case is treated using convolution techniques. The infinite dimensional case needs different techniques because convolution theory is not available. The main topics are: Approximation of differentiable maps in the finite dimensional case and elevation of the differentiability class of a manifold. Finally in chapter 11 we deal with openness and density of the transversality: Density of the transversality: Thom's Theorem: multijets; Mather's theorem: Whitney's Immersion theorem; Morse functions. The authors are grateful to Prof. Jesus M. Ruiz for his careful reading of the manuscript and accurate comments that have contributed to improve the book; to Prof. Peter W. Michor for kindly writing the preface; to Miss Patricia Aguirre for careful typing; and to DGICYT for its partial financial support through the grant PB89-0122. Madrid, September 1991

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1

Chapter 1 REAL DIFFERENTIABLE MANIFOLDS WITH CORNERS In this chapter we introduce real differentiable manifolds with corners, which include, in particular, manifolds with simple boundary and without boundary. Our manifolds will be modelled over real Banach spaces. Moreover we introduce the morphisms between manifolds, or in other words, the maps of class p. But since our manifolds may have corners, we need to establish before some results about Differential Calculus over open sets of quadrants of Banach spaces. In Global Analysis for instance in Morse Theory, we often have infinite dimensional manifolds with boundary or corners. For this reason we introduce the manifolds with such a generality. 1.1. The Differential of Maps over Open Sets of Quadrants of Banach Spaces.

In this paragraph we develop some results about differential Calculus over open sets of quadrants of Banach spaces, in order to introduce differentiable manifolds with corners. The method of exposition will be similar to the classical one in ordinary Differential Calculus on open sets of Banach spaces. But in our case there is an additional difficulty in the points of the boundary, whose resolution involves Whitney#s extension theorem. First we recall some definitions and elementary results about real Banach spaces and we introduce the notations that will be used in the book. Let E be a real linear space and 1 !:Ed a map. The map I is a norm if: llv11~0 for all VEE; llvll=O if and only if v=O; IIrvII=IrI llvl for all VEE, reR; IIv+uIIsIIvII+IIun, for all U,VEE. If E is a real linear space and 1 1 is a norm in El we say that (E,1 1 ) is a normed real linear space.

Chapter 1

2

Let ( E l 1 1) be a normed real linear space. Then it is easy to see that the map :ExE-+R, defined by d (VrU)=iV-Uur dll IIi is a metric in E. The topology of the metric d will be the Y U topology considered in E. Moreover we can see that the operations in E, addition and product by a real number, are continuous with the topology of E and hence E is a topological linear space.

H

is Let (El 1 1) be a normed real linear space. Then d Ill invariant by translations in E, that is, d III (u+z,v+z)=d, ,(u,v), and BE(v)=v+BE(0), Bi(v)=v+B;(O) for all veE and all &ER, &>O where BE (v) Be (0) are open balls and BZ(v) , Bi(0) are closed balls with the metric d I

n u-

If the metric space (E,d, I) is complete, we say that (E,II II), or E, is a real Banach space. Sometimes, if no confusion can occur, we shall use 1 1 to denote different norms. Let (E,! 1) be a real Banach space and F a closed linear subspace of E. Then (F,II is a real Banach space whose topology is the topology induced by E in F. Let (Eli I) be a real Banach space and F a closed linear subspace of E. Then it can be shown that ( E / F , [ 1,) , where 1 [v]Il=inf. ( UV-U///UEF) , is a real Banach space called quotient space of E by F. The topology of E/F is the quotient topology of E.

Let E be a real linear space and :EXEM a map. We say that is an inner product on E if the following conditions are satisfied: is bilinear, symmetric over ExE; >O for all veE , v+O

.

If E is a real linear space and an inner product on E, it can be easily shown that: a) For inequality)

all

U,VEE,

JI2s,

(Cauchy-Schwarz

b) The map 1 [ : E a r given by Ivl=+J, is a norm in E, called the norm associated to the inner product .

Real Differentiable M a n i f o l d s with Corners

3

From the condition a) , it follows that the map C,>:ExE+R is continuous with the natural topology induced by [v[=+J . Let E be a real linear space and c , > an inner product on E. We say that (El) is a real Hilbert space, if the norm 1 associated to the inner product satisfies that (El[ 1) is a real Banach space. Let (E,) be a real Hilbert space and F a closed linear subspace of E. Then (F,IFxF)is a real Hilbert space whose topology is the topology induced by E in F. For all ndN it is easy to see that Rn, with the usual structure of real linear space and with the product :R nxR n+R n is a real Hilbert space given by O such that (a(V)lSallvU for all vcE. Let

E,F be real Banach spaces. We consider the map given by IIaII=Sup( IIa(v) (/WE, Ilvllsl). Then it can be shown that (Y(E,F), 1 11) is a real Banach space.

11 11 :Y(E,F)+R

Another important result that we shall often use is the following:

Chapter 1

4

Let E,F be real Banach spaces and u r f ( E , F ) . Then we have: 1) ua(v) Isl[aulvi f o r all vcE. 2) lul=inf(kzO/la(v) Ilsk[vi for all VEE),

3)

/laU=Sup(/ a ( v )II/vrE, iv[=l)=Sup {J-!!f-/vsE-(O}

is another real Banach space and & ! t ( F , G ) ,

then ~ B o u l b p i . ~ a [ .

A l s o , it can be shown that an element u belonging to I ( E , F ) is an open map if a is onto. In particular a is a homeomorphism if and only if a is a continuous linear isomorphism.

Let I 1(11/1 112 be two norms in an real linear space E. We say that these norms are equivalent if the associated topologies coincide. are two n o m s in If E is a real linear space and 1 [ l r l l E, the following statements are equivalent: a) 1 I1and i2 are equivalent, b) There exist A>O and B>O such that for all y6E.

AUY~~S~~Y~~SBIYJ~

If E is a finite dimensional real linear space and 11 i l l 11 112 are any two norms in E, then it can be shown that these norms are equivalent. If (E,1 1 ) is a normed finite dimensional real linear space xn), and {el,...,en) is a basis of E l then the map vt+(xl, n where v= 1 x.e is a linear homeomorphism from E onto Rn. i=l 1 i'

...,

...

If Ep,...,E, are real Banach spaces, the map I (:E1x xEn+R ixnII) is a norm in the real defined by 1 (x,!. . ,xn)I/=Max{ixli, linear space E1x.. .xEn and (E1x.. .xEn,1 1) is a real Banach space, whose topology is the product topology of the spaces Ell... ,En'

.

. ..,

It can be shown that the maps $

I (XI,. - .'Xn) iil=uxlu+. . .+II~,II and

111[

n (Xll..

/2:~1~...~E n4 given by

I I ~ 2= ~..+II~,II I ~2, I I +.

are also norms in E1x.. .xEn and they are equivalent to the norm, defined above.

Real Differentiable Hanifolds with Corners

5

.

If Ell.. ,En,F are real Banach spaces, the set (f:Elx.. .xEn-+F/f is an n-linear map) will be denoted by n L (E1,...,En;F) and the set (f:Elx xEn+F/f is a continuous n n-linear map) will be denoted by I (Ell...lEn;F). If El= En=El

...

...=

the sets defined above will be also denoted by Ln(E;F) L"(E;F) respectively. We will often use the continuous multilinear maps.

following

and

characterizations of

.

Let u be an element of Ln ( E l l . . ,E,;F). Then the following statements are equivalent: a) a is continuous, b) a is continuous at 0, c) the set ( (la(xll... ,xn)p[xlll=l,. I ixn#sl) is bounded, d) There exists k>O such that ia(x ll...,xn)~~k~x,~...~xn~, for all (x,,.. .,xn )EE,x xEn.

..

...

If Ell...,En,F are real Banach spaces, then the map n 1 11:L (Ell... ,En:F)+R given by lall=Sup( Ila(xll.. .,xn)II/uxllldll.. . ,IIxnlllsl) is a norm in the real n n linear space L (E,,...,En;F) and (I (El,...,En;F),I 1) is a real Banach space. Moreover We have that

.

II~II=SUP( iia(xlI.. .,Xn)ii/iixln=l,.. I lxnn=l)= =Inf(k~O/~~a(xl,. ..,xn)IlskllxlII.. Hxnll for all x1"E1,.. ,xneEn). Hence if u is an element of Ln(E1,.. ,En;F) it can be shown that IIu(xl,.. .,xn)IIcllull IIxlll.. . IIxnII for all element (xll.. ,xn) of E1x. . .xEn.

.

.

.

.

If E1,...,En are finite dimensional real Banach spaces, then n L (E,,...,E n ;F)=2n (E,,...,En;F).

.

Let Ell.. ,En,F be real Banach spaces and ,En;F) the map defined by p:L(El,L(E2,.. ,L(EnlF). .))+Yn(E1,.. p ( u ) (x,,.. .,x,)=(. .. (a(xl)) (x2) .) (x,). Then cp is a linear homeomorphism and 1 p (a) 1 = 1 u 1 .

.

.

..

.

From the properties of the nonns of linear continuous maps that we have seen above it follows that:

Chapter 1

6

a) The map o : I ( E , F ) x k ? ( F , G ) + J e ( E , G ) given by o ( c X l 8 ) = 8 o 0 r iS a bilinear continuous map. b) The map e:Exl(E,F)+F given by e(v,a)=a(v) is a bilinear continuous map. ,En ;F)+F given by c) The map e:E 1x...xE nxi?n (El e(v ll...lvn' a)=a(v lI...,vn) is an (n+l)-linear continuous map.

,...

If E,F are real Banach spaces and n is an element of IN, then the set (f:En+F/f is an n-linear symstric map) will be denoted by L:(E;F) and the set (f:En+F/f is an n-linear continuous symmetric map) will be denoted by Y:(E;F). It can be shown that Y:(E;F) is a closed linear subspace of Lt(E;F) and hence (k?:(E;F) I 1 1) is a real Banach space. If E is a topological real linear space and V ( 0 ) is the system of neighbourhoods of 0 in El then U=(UVcExE/Vd(O)], where Uv={(x,y)~ExE/x-y~V), is a uniformity in E whose associated topology coincides with the topology of E. We shall say that a topological real linear space, El is complete if the associated uniform space (E,U) is complete. topological real linear space, E l is called Banachable if it is complete and there exists a norm, 1 11, in E whose associated topology coincides with the topology of E; of course (E,1 11) is a real Banach space. A

Definition 1.1.1 Let E be a real Banach space and A=(hll independent system of elements of I(E,R).

...,An)

a

linearly

..

Then the set (x~E/k~(x)tO,. ,hn(x)~O) is called A-quadrant of order n of E and is denoted by Ei or by Ei if A=(h). Obviously, ve have E

=

+ if [y Eii and EA=E

A=@.

1=1

If E=!R2 and A=pl where pl(xly)=x, then E~={(x,y)€R2/xrO). If E=R2

and

A= ( p1 ,p2 ) ,

where

p1 ( x , y ) =x

and

p2 (xI y)=y ,

then

Real Differentiable Manifolds vith Corners

7

+ 2 EA=((x,y)~R/ x ~ O , y ~ O ) . If E is a real Banach space and A=(Al,...,An) is a linearly independent system of elements of Z(E,R), then the closed linear n 0 0 or by EA if A=(A). subspace (1 ker(Ai) of E is denoted by EA 1=l

o n o

0

Obviously, EA=() Eai and EA=E if A=#. 1=1 Definition 1.1.2 Let E be a real Banach space and F a closed linear subspace of E. A linear subspace H of E is said to be a topological supplement of F if E=F+H, FnH=(O) and H is closed. In such case ve shall write E=FsTH and it can be shovn that the map 8:FxH+E given by B(x,y)=x+y is a linear homeomorphism. We shall often use the following facts: a) If two closed linear subspaces of a Banach space have the same topological supplement, then they are linearly isomorphic. If two closed linear subspaces of a Banach space have the same codimension and this is finite, then they are linearly isomorphic. b) Every finite dimensional linear subspace of a real Banach space, admits a topological supplement. c) Every closed linear subspace of a real Banach space whose codimension is finite, admits a topological supplement. d) Every closed linear subspace, F, of a real Hilbert space, I (HI), admits the subspace F =(uEH/=O for all veF) as a topological supplement. We shall also use the following result. e) Let E be a real Banach space and F a linear subspace of E. Then the following statements are equivalent. i) F admits a topological supplement in E. ii) There exists a linear continuous map, p:E+E, pop=p and Ker(p)=F.

such that

a

Chapter 1

iii) There exists a linear continuous map, q:E+E, such that q o q = q and irn(q)=F. L e m m a 1.1.3

Let E be a real linear space. Then ve have: a) Let (hl,...,X n ) be a set of linear maps from E to W. Then {Al,...lhn) is a linearly independent system if and only if there exist XI,. .. ,xnrE such that X . (x.)=aij, where B is the ij 1 7 Kronecker index, b) Let (yll...,yn) be a set of elements of E. Then {yl,...lyn) is a linearly independent system if and only if there exist linear maps ul,...,~nfrom E to R such that u.(y.)=8 1 J ij* using t h i s lemma it can be shown that: Proposition 1.1.4

a) Let E be a real Banach space and A=(hl,...,Xn) a linearly independent system of elements of 2(E,W). Then there exist linearly independent elements of E, xl,. . . ,xn, such that 0 Ai(x.)=aij and E=EA o L(xl ,xn)' vhere L ( x ~ ~ . . . , x ~is ) the 7 T linear subspace of E generated by xl,~..,xn.Moreover the map a:E AoxR n+E, defined by a(xo,rl,.. .,rn)=xO+rlxl+. +rnx n is a linear homeomorphism such that

,...

..

+

+

Finally, EA is homeomorphic to EA for all AEA. b)

Let

M=(qll...,qm)

be a real Banach space and h=(A l,... ,An), linearly independent systems of elements of Y ( E , R ) E

+ + Then ve have: 1) Eh=% 0 0 such that Eh=En. , 2 ) n-rn, 3 ) there exists a bijective mag, ~:(1,2,~..~n)+{1,2,...,n), such that for every + satisfying Ai=riqT(i). k { l , . . . , n) there exists r.ER 1 c)

Let

E

be

a

real

Banach

space

and

A=(Al,

... ,An),

M={TI~,...,~~ linearly ) independent systems of elements of Y ( E , R ) .

Real Differentiable Manifolds vith Corners Then there exists a linear homeomorphism, u:E+E, + 0 0 u(E + )=EM and u(E )=EM.n A A

9

such that

Now we introduce the differential of a map defined on an open set of a quadrant of a Banach space. Proposition 1.1.5 Let E,F be real Banach spaces, A=(Al,. . .,An} a linearly + independent system of elements of Z(E,IR), U an open set of EA, f:U+F a map and x a point of U. Then if u,v are elements of Z(E,F) such that

ve have that u=v. Since Proof. Let z be a point of Ei with z+O and & > O . + lim = O f there exists 6>0 such that B6(x)nEAcU and Y-X Y+X for all yeU satisfying O0 such that Df (x)(v)=lim f(x+tv)-f(x) t t+O x+tvcU and Df (x)(v)- f ( ~ + t ~ ) - ~ ( ~ & ) . Then for all yEg+B&(O) and f o r all aeA-F we have ly(a)1O for all A E A - A ~ . Therefore there w=(Og) A€&

.

I

(1

exists a>O such that p(x)+awE

Ei.

If we take a8O for all AEA1.

2)+1). Consider A1=(A~A/Ap(x)=O). If Al=#,

then v is a strictly

inner tangent vector at x. Suppose that A1+#. that w=(O:)

(v)e() AEhl

that p(x)+awe

(1

Then, from the hypothesis, it follows

+ intEA.

-1

On the other hand there is a>O such

+

intEA. If we take a'U-A;(()

75

(It is clear from

ncH KcXi the fact that if zcU-A, then d(z,A)>O).

*

c) For all new,

all KcX,

and all K'cXn

ve have that

Ih (U-A)*$, Ktn(U-A)+$, D (A,K)21/2" and D (A,K' ) O , there is 6>0 such that Bg(xO)cU and Hpk(xo,y) ~ ~ ~ ~ for ~ all y -yeB6 x (x,)~ ~ k Then f is a map of class r on U and D f=fk for all ke(0, r).

...,

(By induction, every applies) . o

.

fk is a continuous map and 2.1.7

Proposition 2.1.10 (The Whitney Extension Theorem)

Let A be a closed set of Rp vith AcU, vhere U is an open ball of diameter 1, let F be a real Banach space, f:A+F a map, r&b{O) and for all ke(O,...,r), fk:A+Pk(RPIF) a map with fo=f. S For every ke(O,l,...,r) Rk:AxA+Pi(RPIF) defined by Rk(xlY)'fk(Y)-

let

us

consider

r-k fk+i(x) ( (Y-x) i=o 1 i!

the

map

~

The Whitney Extension Theorem and the Inverse Happing Theorem

81

.

Suppose that for every ke(O,l,. . ,r), xOeA and c>O, there is 6>0 such that N R ~ ( X ~ , XI1~5s) cIIx1-x211 r-k for a l l x1,x2eA with IIx1-xo11nO

such

that

such

that

n0 mO 1 A B bp- C AkBkmObpIJO such that B, (xo)x[O,so)cV, where BE (x,) is an open ball of 0

0

Let 7 be the diffeomorphism of class

CO

from IR to

defined by y ( t ) = & o t / w . Then r ( [ O , + ) ) = [ O , e o ) h:B (xo)x[O,+)+F defined by h=g

(-&or&o)

and the map x7) is a

cO

map of class p. Since h is a map of class p, there are an open

Chapter 2

94

neighbourhood W xO of xo in B,

(x,),

6>0

with 6 < c o and K>O such

0

that : xO Ih(Y,t) IO 1 I Pn- 1 (PlI + 0 n-1 and an open neighbourhood Vxo of xo in EAx(R (Pll * IPn-1V

a ( x ) E [ Eix (R

...

--

X,

such that V ux[o,co)cV. Let r:R+(-cOlcO) be the diffeomorphism of class

OD

defined by

. we define r ( t ) = c o t / w . It is clear that 7 ( [ 0 1 + ) ) = [ 0 1 e o )Then the following map of class p: X

h=gJ

v

x7):V 'x[O,+)+F.

"(1 0

XIOdO)

v

0

Since h is a map of class p, there are an open neighbourhood Wxo

of xo in V x o l 6 > 0 with 6O such that h,DiDjh and Dkh Y t

The Whitney Extension Theorem and the Inverse Mapping Theorem 99 xO are bounded by K on W x[O,6), for i+jsp and ksp.

Consider the function # : W d

of the preceding lemma. Then

X

define a map 5:W

'XR+F

by (h(y,t), if Ost

where ( an n e ~ (uo and (bn)ncmu(0 ) are the sequences of Lemma 2.1.32. Now the proof follows like in 2.1.33.0 Proposition 2.1.35

...,

Let E be a real Banach space, h=(hl, An) a linearly independent system of elements of I(E,R), U and open set of Ei, F a real Banach space, f:U+F a map and pew. Then the folloving statement are equivalent: a) f is a map of class p, b) for every xeU, there are an open neighbourhood Vx of x in E and a map f:++F of class p (in the sense of Ordinary Differential Calculus) such that =f

I vxnu I vxnu

Proof. It follows from 2.1.33

and 2.1.34.0

Remark

If at every point there is an open neighbourhood on vhich all the derivatives of f are bounded by the same constant K, then the preceding proposition is also true for maps of class m. Theorem 2.1.36

...,

Let E be a real Banach space, h=(hl, An) a linearly independent system of elements of Y?(E,R), F a real Banach space, + + k f:EA+F a map, rePlv(0) and, for every ke(O,...,r), fk:EA+Is(E,F) a map, vhere f o = f .

Chapter 2

100

For

every

...,r)

consider the map i r-k fk+i(x) ( (Y-x) ) defined by Rk(x,y)=fk(y)-.C l! 1=0 kG(0,

+ + k Rk:EAxEA+Ys(E,F)

+

Suppose that for every k€{O,...#r), xO€EA and c>O, there is + 6 > 0 such that for x1,x2eE,, with I(x1-xo1/0 and l2 ' 1 2 A (x! +x! ) = O r t+j, )>0 and h i (xi +xi )>0 and this is l2 2 1 2

If A.(x!

a contradiction since

+ (z)=ind + ( z ) for all ZEEI;,. EA

Therefore

As0

and

a1=...=a k=O

and

y=O.

That

is

EA0nL{xi,...,xi)={O). Finally, from the fact that codim(EA)=k, 0 we 0 0 get EA+L(xi, x'k)=E, and EA+Er=E, which ends the proof.

...,

11)

*

I).

120

Chapter 3

Let x'EX'.

If x t ~ B O Xthen, , of course, x'eBOX'.

Suppose that x'€BkX with kzl. Let us consider a chart c=(U,p,(E,A)), A=(A1,...,Ak), of X adapted to Xr at x' through + (F,M), M=(pl,.. 'pn), that is x'EU, p(x')=O, p(UnX')=(p(U)nFM and

.

0 0 FLcEf;. Then we have that FMcEAI EA0=FM00TH and, from the hypothesis

0 E=F+EA. Moreover Then it holds

and 3.1.8.d), p . (v.)=aij. 1

3

0

0 F=FM 0 L(V~,...,V~)~ where T +cE+ that vl,.. ,vnEFM A'

.

0

E=F+H=FM+H+L(vl,...,v n )=EA+L(vl,...,v n ) and hence nrk. From

a (F;)

the

+ + = (FM)na (EA) *

hypothesis I

ax'= (ax)nX' ,

it

follows

that

0 Suppose that n>k. Then, of course, there is vi eEA. We take

1

0 If k=l then this point vim and construct the space EA+L(vi-). 0 E=EAaTL(vi

I

0 If k>l, then there is vi eEA+L(vi

I

) with il*i2. We 1 0 and construct the space EA+L(vi ,vi ) . If k=2 2 1 2 then E=E?~L(V~ ,vi 1. 1 2

).

1 take this point vi

2

rn

Since

k

is

finite,

there

are

such

that

0 vilf+ -l V*i k+ E=E @I L(vi , ,v ) Therefore vi +. .+v. Ea(FM)c8(EA) and there A T 1 ik 1 k' )=...=A. (v. ) = 0 is Ai€A such that Ai(vi +. .+vi ) = O . Then A .1 (v. 1 1 k 1 k' , which is a contradicti0n.o and L(vi '...,v ik )+E0cEo A hi 1

...

.

.

.

'

If X' is a submanifold of class p of X and c'=(UtIp',(ErIA')) is a chart of X' with (p'(x')=O, the natural question is: Is there a chart c=(U,(p,(E,A)) of X which is adapted to XI at x' through (E',A') and c' is the chart induced chart in X' by c? Proposition 3.1.11 Let X be a differentiable manifold of class p and X' a submanifold of class p of X vith ax'=#. Let us consider a chart c'=(U',p',E') of X' vith x'EU' and (p'(x')=O. Then there is a

Submanifolds and Immersions

121

c=(U,p,(E,A)) of X such that x'EU, UnX'cU', * where n=indX(xt)I p(x')=O, * is the third * n A=(pll.. ,pn)op3, p3 E=E'xGlxR , projection and p(y)=(p'(y) , O , O ) for ycUnX'. Moreover cp (UnX' ) =p (U)n (E'x ( 0) x ( 0) ) chart

.

.

Proof. Let us consider a chart cl=(Ul,pl, (EllAl)) of X which is adapted to XI at x' through F1. Then F1 is a closed linear subspace of El which admits topological supplement, GlI in Ell X'EU~, pl(x')=O, pl(UlnX~)=pl(Ul)nFl and F cE+ and therefore

F CEO

.

It is clear that

'Al

0

* G1=GlnEyA

is a topological supplement 1

.

of F1 in E

'Al

lA1

Suppose that Al=(A1l...l An) and let ( V ~ ~ . . . n ~ )V be a linearly independent system of elements of El such that 0 Then we have E =E A . (v.)=6 @TZ(vll...lv n ) and the map 1 J ij' 'Al

u :E~+F~XG:XR~ defined by v=vo+v'+r 1v 1+...+ rnvn, is a

* n+ ( F p p 1A,

off

=F xG*x (R") +

-1

1

1

u(v)=(voIv',rlI...Irn)l where linear homeomorphism. Therefore and upl(U1) is an open set

~P1'".~Pn~

A

of this quadrant such that (0,0,0)~ucp,(U,).

Let us consider two

* (Rn)+ VocFl and W ocGlx such that (P1'".'Pn) v0xw0cup1 (U1) and p1 (VoxWo)nX'cU'. Then we have the following chart of X: open

neighbourhoods

This chart is adapted to X' at x' through F1x(0)x(O)

* * * * cp (U nX')=pl (Ul)n(Flx(0)x(O))=V 1

1

since

0 x( O ) x ( 0).

Consider the diffeomorphism of class p I

*

p:v 0xw0+v

0x(o)x(o)xw

0

*

0

+p'

1 Urnxl ) -lxl w0 ,(u*,nx

* 0 (U1nX')xW ,

((Pi

1x1

W0

xwo-

whose image, p'(U1nX')xW , is an open set of the quadrant * n) + =(E'xG1xR * n) + * E'xGlx(R (P1l...'Pn) (Pll.. * rPn)"P3 Then we have the chart of XI * * * n c;*=(UllWlI (E'xGlxR I (Pll

* -. lPn)oP3) 1

I

122

Chapter 3

Thus @p;(y)=(p'(y)

*

,O,O) for y€U1nX'.o

Proposition 3.1.12 Let X be a differentiable manifold of class pI X' a submanifold of class p of X and x'~(intX)nX' (PEN if dimX,X'=+m and x'EaX').

c'=(U',p',(E',A')) be a chart of X' with x'EU' and Then there is a chart c=(U,p,E) of X such that x'EU, UnX'cU', E=E'xF where F is a real Banach space, ip(y)=(p' (y)'0) for yEUnX' and p(UnX')=p(U)n(Ei,x{O)).

Let p'(x')=O. p(x')=O,

Proof. Let us consider a chart c1=(U1,pl,E1) of X adapted to X' at x' through (FlIM). Then F1 is a closed linear subspace of El which admits topological supplement G1 in El, x'eU1, pl(x')=O, + Therefore E1=FlaTG1, a:E +F xG defined by pl(UlnX')=pl(Ul)nF 1 1 1'

.

a(v)=(vl,v2),

v1+v2=vI is a linear homeomorphism and apl(U1) is

an open set of (F1xG1) with (0,0)~ap,(U,).

0

0

Let A=V XW cFlxGl be

an open set of upl(U1) such that (0,O)eA and p1-'a-'(A)nX'cU' where V 0 , W0 are open neiqhbourhoods. Then we have the following

This chart is adapted to X' at x' through (F1xOI Mpl) and chart of X', where is

Consider

the

diffeomorphism

of

an

open

class

p,

set

of

6 : (VonF+ )xWo+(V onF+ )x(0)xWo (p; 1 u1nX' -'xlWo >(U1nX')xW * 0+ lM IM * 0 +cp ' (U,nX ' ) xW . - 1 * whose image p' (U,nX')xWo is an open set of the quadrant 1 + E',xG =(E'xG ) A 1 1.A'P1'

Submanifolds and Immersions

o + * Now we take h:V nF +pt(U1nX') that is @=hxl o I

123

, defined by h(y)=p'pl *-1 ( y , O )

-

lM

and a diffeomorphic extension of h, h:S1+S2

W

(see 2.1.35) , where S1,S2 are connected open sets of F1,E' 0 + * respectively such that OeSlcV , 0eS2, S2nEiIcp'(U1nX') and + h(y)=E(y) for all yes nF M'

.

Then we have the following chart of X: c**- -1 -1 0 1 OL ''1'( O)aplIH"P,

* HnX'cUlnXrcU',

-

and x'EH, p(x')=O, and (P(HnX')=F(H)n(Eifx(O)) .o If

ax+$,

-W

p(y)=(p'(y)

,O)

E'xG1) for all ycHnX'

aX'+qj, and x'nax+$, we have the following example:

Example 3.1.13 and X'=(R 2 ) + (htPl) (PllP,) X is a differentiable manifold of class of class rn of X. Set X= (

R ~

+

where h (x,y)=y-x. Then mI

and XI a submanifold

Consider the local diffeomorphism at ( 0 , O ) of class m, 3 p ~ : R 2 + l R 2 , defined by p' (x,y)=(x +x,y3+2y-x). Then p'(Xt)=Xl, p~(o,o)=(o,o) , pl(axl)=ax, and DpI(0,O) is a linear homeomorphism. Therefore, from 2.2.6, there are U',V' open sets of XI with (O,O)eUtnV' such that p'IuI:Uf+V' is a diffeomorphism 2 of class m. Then ct=(Ut,p'IUII(R , ( h , p 1 ) ) ) is a chart of XI. Suppose that c=(U,p,(R2 , ( h 1 , A 2 ) ) is a chart of class pr3 of UnX'cU' and pIunxI=p~IunxI. Then X with (O,O)eU, p ( O , O ) = ( O , O ) , 2 2 k k Dp(O,O)=Dpt(O,O), D p(O,O)=D p~(O,O),...,D p(O,O)=D p ~ ( O , O ) , . . . .

124

Chapter 3

Furthermore

and therefore hl=plI h2(x,y)=x+y and ~(x,O)=(a(x),-a(x)), where a is a function of class pz3 with a ( O ) = O and al(0)=l. From 2.1.29, we know that for every CU and that H=B6 (0)n ( R2 ) (PllP,)

E>O,

there is 6>0 such

+

D ~ P ( O , O( ) ( ~ ~ (2)) 0)

-~

~ ~ (( 0 ( ~~~ (3)) 0 0 )),=

~~P(xlo)-DP(olo)(xt0)2! 3! 3 =II (a(x)-x-x I-a(x)+x)IISEXP for all (X,O)EH. 3 p-3 or if we have taken E= 1 But Hence x 52cxP and -Bx 2E 5 ~ x ~for - ~ all (x,O)EH and pz3 is a contradiction. Hence the chart c cannot exist.

=.

Remark Let XI be a totally neat submanifold of XI x'EX'nBkX. Consider the following items: chart (U,p,(E,A)) of X adapted to X' at x' through (E',h') , E'=Ei!@TL(~;, ...,XI) k where (xi,. ,xi) is a linearly independent system of elements of El such that 0 h i (X! ) = 6 A'=(hi , . . . , h i ) and E=EA@TL(x l,...,~k) where J ij' (xl,...,xk) is a linearly independent system of elements of E A=(hl,...,hk). Then we have seen that such that hi (x. ) =6 0 I ij', ,x;C)=EI E',cEA and there exist a1 , 6 k ~ Rwith ai>0 EAoTL(xi A for all k(1, k) and a bijection ~ : ( l , k)+(l, k) such

..

,...

that for

(v

,...

...,

...,

...,

X'

=

x (k)} we have hi (w. t(l),.. w.=,'3 ) =6 ijl (we can

%(l). . 't(k) suppose xl=wl, ,xk=wk and then L (xi, ,xi)=L ( x1 , Therefore there exists a linear diffeomorphism L(xi '...,xi) onto L(xl, ,xk) defined by a(x;)=wl,

...

...

...

and

0

EA=E'i,@TF,

where

FnE'=(O).

...,xk)cE' ) . a from a(x')=w k k

...,

As

a

consequence,

the map

0 @ :E+EI A I xFxR

defined by @(v)=(v 1~v21r11***lrk)l where v +v +a(rlxi+...+r x')=v, is a 1inear VIEE'Al IVZEF, 1 2 kg k + and diffeomorphism. Moreover @(E;)=EtA,xFx(R ) (Pl,*.'tPk) 0

k ,( p l , . cl=(u,@~(p,(EtAIxFxR 0

..,pk)op3)),

where fj3:E'Al~F~R 0 k4k, is

Submanifolds and Immersions

a

chart k + (E'Atx(o)xR (p,,

of

0

X

with

BOP (x' 1=o,

X'EU,

- c(E'~,xFxR 0 k) +

...,pk)op3

125

(PI,

-

*

and

IPk)OP3

.

- In other words, c1 is (P11"*IPk)oP3 k adapted to X' at x' through, (E':,x(O)xR ,(pl, .,pk)op3).

B(P(UnX ) =B(P (U)n (EI : ,

x ( O)xRk)

+

..

We can summarize the preceding remark with the following proposition. Proposition 3.1.14 Let X'

be a totally neat submanifold of a manifold X and

x' EX' nBkX.

Then there (U,4,(GxFxRk ,(pl, ,pk)0p3)) k (Gx(o)xR J(P11"'lPk)"~3)'0

.. .

is adapted

a to

X'

chart at

of

x'

X, through

Proposition 3.1.15 Let X' be a totally neat submanifold of a manifold X, ct=(U',p',(E',At)) a chart of X' and x'EU' with cp'(x')=O. Then of X such that x'EU, there is a chart c=(U,p,(E'xF,h'opl)) UnX'cU', (p(x')=O,

1

(P (UnX' =(P

(U)n(E'x( 0 1 ) ;,pl=(P

+

(U)n(Ei,x( 0 1 )

and p ( y ) = ( p ' ( y ) , O ) for all ydJnX'.

..

k Proof. Let k=card(A') and cl=(U,4,(GxFxlR ,(pl,. ,pk)op3)) the chart of X constructed in 3.1.14, that is c1 is adapted to X' at x' through (Gx(0)xlR k ,(pl, ,p,)op,). Then x'EU,~(X')=O,

...

k + 4(UnX' ) =4(U)n [ Gx ( 0) x (IR ) ( P l t * * * I P k] ) and this set is an open set k + Moreover of Gx(o)x(R {pl,. ,pk) (UnXt,41unx,,(Gx(0)xR k ,(p,, .,pk)op3)) is a chart of X' with x'EUnX' and 4(x')=O. Then the set (~'(Unu') is an open set of Eif, k + *lunx,(unu') is an open set of Gx(O)x(R )(pl,,.,,pk), 41UnX,p' -1:(P'(U~U')+~~~~~,(U~U') is a diffeomorphism of class p

..

..

and (P

' (9I UnX

I

)-I.' 4IUnX t (UnU' 1+(P

' (UnU' 1

126

Chapter 3

is a diffeomorphism of class p. Then there is an open set U1 of U 0 0 0 k + such that x'eU1 and q (U,) =V1xV2xV3c9(U)cGxFx (IR ) (Plr***rPk) 0 0 Vlx(0)xV3cQ(UnU'). Therefore the map 6: U 1 "

q

0

0

0

xv xv 3+~'*

-1

0

0

0

(Vl~(0)~V3)~V2

is a diffeomorphism of class p and im(6) is an open set Then (U1,6,(E'xF,A'~pl)) is a chart of X such that X'EU~, + 6 (x' ) =O, U1nX'cU', 6 (UlnX') =6 (U,) n(Ei,x( 0)) and 6 (y)= (p' (y) 0) for all ycU1nXrcU'. o Propositions 3.1.11, 3.1.12, and 3.1.15 allow us to study the transitive property of the notion of a submanifold. Proposition 3.1.16 Let X'

be a submanifold of class p of X and let XI1 be a

submanifold of class p of X'. Then w e have: a ) If X' is a totally neat submanifold of X then XI1 is a submanifold of class p of X (see 3.1.15). b ) If XI1cintX, then XI1 is a submanifold of class p (PEP(

if

with dimXllXr=m and xI1EaX') of X (see 3.1.12). there is xw1~XI1 c) If ax'=$,

then XI1 is a submanifold of class p of X (see

3.1.11).

Proof. a) Consider XI~EX~~CX'CX and let c'=(U' r p ' (EtrA')) be a chart of X' adapted to XI1 at x1I through (F,M) Then x~~EU' r p r( X * ~ ) = O , (p' (U'nXIl) =p' (U') nFi and FACEAT. By 3.1.15, there is a chart c=(Urpr(ErxFl,A'pl)) of X such that xWJ, UnX'cU', p(xI1)=Or + ' x ( 0 1 1 + (E CP (UnX 1=cP (U) =(P (U)n (EI;~x ( 0 1 1 and CP (Y = (CP ' (Y1 r 0 ) for

.

A'P1

+

ycUnX'. Then it is clear that p(UnX1l)=p(U)n(F,x(0)) chart of X adapted to XI1 at x1I through (Fx(0),Mpl)

.

b) Consider x ~ ~ E and X ~let ~ c'=(U'~~',(E',A'))

and c is a

be a chart of X'

Submanifolds and Immersions adapted to XI1 at x1I through + cp' (U'nX1I)=cp' (U')nFM and FACEAT.

(F,M).

127

Then

x~~EU',~~(X~~)=O,

By 3.1.12, there is a chart of X, c=(U,cp,E), such that x"EU, (P(x~~)=O, UnX'cU', E=E'xF1, cp(y)=(cp'(y) ,0) for yeUnX' and

+

cp (UnX' 1=cp (U)n (EA,x ( 0

1*

Then cp (UnXll)=cp (U)n (FLx( 0)) and c is a chart of X adapted to XI1 at x1I through (Fx(0) ,Mpl).

c) Fix X ~ I E Xand ~ ~ let c'=(U',cp',E') to XI1 to xI1 through (F,M). + + cp' (U'nX1I)=cp' (U')nFM and FMcE'.

be a chart of X' adapted Then x~~EU', ( P ( X ~ ~ ) = O ,

From 3.1.11, we get a chart c=(U,cp,(E,h)) of X such that * n * where xIIEU, cp (XI@) =0, UnX'cU' , E=E'xGlxR , h = ( p l , . ,pn)op,, * n=indx (x") , p3 is the third projection and cp(y)=(cp'(y),O,O) for yEUnX'. Moreover cp(UnX')=cp(U)n(Etx(0)x(O)). Then cp (UnXll) =cp (U)n (FLx( 0)x ( 0) ) and c is a chart of X adapted to XI1 'at xn1 through

..

(Fx(0)x(O),MPl)

-0

Proposition 3.1.17

Let f:X+X' be a diffeomorphism of class p and XI1 a submanifold of c l a s s p of X. Then f(Xll) is a submanifold of class p of X'. Moreover, XI1 is a neat submanifold if and only if f(X") is a neat submanifold and XI1 is a totally neat submanifold if and only if f(Xll) is a totally neat sub manifold.^ Proposition 3.1.18

Let Y be a submanifold of class p of X, k d " ( 0 ) and yaBkY. Then there is an open set U of X such that YEU (BkY)nU=Bk(YnU)cBk,X, vhere k'=indX(y). In particular, if BY=#

Chapter 3

128

then there is an open set U of X such that yeU and YnUcBk,X (BOY=Y)

.

Proof. Since Y is a submanifold of X and yeY, there is a chart c=(U,p,(E,h)) of X with yeU,p(y)=O and there exists (F,M) such + + , p(UnY)=cp(U)nFM+ , card(h)=k' and card(M)=k. From that FMcEA FACE: we have that F ~ c and E ~ hence (BkY)nUcBk,X.o 3.2.

Immersions

There are several possible definitions of an immersion. We choose here one in which the function is, locally, an inclusion map. Definition 3.2.1 Let f:X+X' be a map of class p and xcX. We say that f is an immersion at x if there are a chart c=(U,cp,(E,h)) of X with cp(x)=O and a chart c'=(U',p',(E',h')) of X' vith cp'f(x)=O such that f(U)cU', E is a closed linear subspace of E' vhich admits a topological supplement in E', cp (U)CP' (U' 1 and -1 p'flUp :p(U)+p'(U') is the inclusion map (then, of course, E:cEl;f and EhcEA,). 0 0

If f is an immersion of class p at every point XCX, we say that f is an immersion of c l a s s p on X. ProDosition 3.2.2 Let f:X+X' be a map of c l a s s p. Then ve have that the set {xeX/f is an immersion at x) is an open set of X. Proof. Let xcx such that f is immersion at x. Then there is a chart c=(U,cp, (E,h)) of X with xeU and cp(x)=O and there is a chart c'=(U',cp', (E',A')) of X' with f(X)eU', cp'f(x)=O, such that f(U)cU', E is a closed linear subspace of E' which admits cp (U)CP' (U' 1 and topological supplement in E', p ' f ,ucp-l:cp(U)+cpt (U') is the inclusion map. Therefore Ef;cEAf. Let us consider yeU with x+y. Then cp(y)+O

and we take

Submanifolds and Immersions

Al=( &A/p(y)

129

.

On the other hand p'f (y)=p (y)+O and we take A;=(h*~A'/A'p(y)=o). It is clear that there is &>O such that for the open ball B&(p(y)) of E' it holds BE(p(y))nEi-=#, where E{-=(xEE'/~(x)sO), for all heA'-A; and B&(p(y))nE,=# for all heA-Al

=0)

and B&(p(y))nEA,cp'(U') +

and B&(p(y))nE;cp(U).

and

Let us consider &I O f where h={Al,.. ,Ak) 1 2

,

1

and

2

Ai u"(x +xi )>O and 1 il 2 which is a contradiction.

Submanifolds and Immersions

139 0

Then A#O, u 1= . . . = u k= O , y=O, !~!(x~,...,x~)~EI;,=(O) and E '=EI;Oe' T E ( x1 I ,xk) Moreover there are 6 1>0, ,ak>0 and there is a bijection T : (I,.. ,k)+(ll.. ,k) such that for

...

.

.

")

...

.

we find A! (w.)=Ziij. We can suppose that 1 3 yl=wll...lyk=wk. Then we have that E(y ll...lyk)=E(xl,...,xk)cFl n and E1=F1+G. If yEF1nG, then y=x;+ulxl+. .+ukxk, where x'EF" 1 lhu-l'

.

y-x;=u

x

+. ..+ukxk~EI;,n2(xl,...,xk) 0

1 1

and hence y-x;=O,

y=x;

and

y=O. Therefore E'=F 18 TG, GcEI;: and we take G=F2. Now we have that + + + =F xF since (F1XF2)h~e=Flh' 1xF2= (F1xF21hu -1p1 'hu -1 2 + where e:F1xF2+E' is the map defined by F+ - l=Ei,nF 1'

IF

hu

x )=x1+x2. Of course, the set ucp(U)xF2 is an open set of + and we can consider the map of class p xF =(F xF ) Flhu-l 2 hu-lp.

e(x

+

1' 2

defined by h ( y) =e-l(p f( p - l ( y)

.

ile

Since h (a(p (U)) ca ( FlxF2) , Dh(O)=B-lu and h(O)=(O,O) , from 3.2.10, we deduce that there are an open neighbourhood U1 of + (0,O) in (F1~F2)A,e I an open neighbourhood U; of (0,O) in + (F1XF2)h#e I a * diffeomorphism c(:U;+U1 of class p and an open * * neighbourhood U1 of 0 in p(U) I such that h(Ul)cU;, Dh(0) (U1)cU1 and the diagram

&

8-lu (U:) =Dh (0)(U:)

c&Ul

&

is commutative, which ends the pro0f.o Problem. Is the preceding theorem also true changing the hypothesis "ind(v)=ind( (Txf)(v))" by "2ind(v)=ind( (Txf)(v))"? Proposition 3.2.4 shows the connection between immersions and submanifolds. Next we study again that connection and obtain a further characterization of immersions.

140

Chapter 3

Proposition 3.2.13 Let f:X+X' be a map of class p and xcX. Then the folloving statements are equivalent: a) f is an immersion a t x. b) There is an open neighbourhood Vx of x in X such that f (Vx) is a submanifold of class p of X' and f:VX+f (Vx) is a diffeomorphism of class p.

.

Proof. a)+b) From the hypothesis, we get a chart c=(U,p, ( E , h ) ) of X, with XEU, p(x)=O and a chart c'=(U'~~',(E',A')) of X', with f(x)cU', p'f(x)=O such that E is a linear subspace of E' that admits topological supplement in E', f (U)cU', p(U)cp'(U') and p'fp-' is the inclusion map. Then we have the following commutative diagram f ,U' U

+ + Then p(U) is a submanifold of class p of Moreover EhcEI;,. EAT and therefore f(U) is a submanifold of class p of X' and f :U+f(U) is a diffeomorphism of class p. IU b)+a). is obvious.o We recall that in the first paragraph of this chapter we saw that under certain hypothesis any chart of a submanifold can be obtained as a restriction of a global chart. Next we shall study the same problem for immersions. Proposition 3.2.14 Let f:X+X' be a map of class p. Suppose that f is an immersion at xeX and f(x)Eint(x'). L e t c=(U,p,(E,A)) be a chart of X with XEU, p(x)=O. Then there are a chart c'=(U',p', (E',A')) * of X', vith f(x)EU', p'f(x)=O and an open set U of X, with * * xcu CU such that f(U )cU', E'=ExF where F is a real Banach space, p(U*)x(O)cp' (U') and the map u=p'fp -1 Iu*:p(U*)+p' (U') is

*

u(z)=(z,O) for a l l zep(U ) (PEN if dimxX=m and xcaX).

141

Submanifolds and Immersions

Proof. By 3.2.13, there is an open set Vx of X, with xcVXl VxcU, and f (Vx)cintX' , such that f (Vx) is a submanifold of class p of X' and f:Vx+f(Vx) is a diffeomorphism of class p. Then cl=(U1=f (VX ) , )-I, (E,h)) is a chart of the submanifold f(Vx) of X'. Then, by 3.1.12, there is a chart of int(X'), c~=(Ut,p8,E~) such that f(X)eU', p'f(x)=O, U'nf(Vx)cU1, E'=ExF where F is a ,O) for all ycU'nf (Vx) and real Banach space, p' (y)=(pf-'(y) -+ 9' (U'nf (Vx))=p' (U')n(Ehx( 0))

.

We end by taking the open set of U, U*=f-l(U'nf(Vx))

.o

Proposition 3.2.15 Let f:X+X' be a map of c l a s s p. Suppose that ax=# and that f is an immersion at xcX. Let c=(U,p,E) a chart of X vith xeU and p(x)=O. Then there are a chart ct=(Ut,prI(Et,h~)) of X', with * * and f(x)eU', cp'f(x)=O and an open set U of U, vith xeU * * * f(U )cU', such that E'=ExGl~Rn, vhere G1 is a real Banach space and

n=indx,(f(x))

u=ptf(p/u*)-l:p(u

,

cp(U*)x(O)x(O)cp~(U~)

* )-vpt(~t)is

and

u(z)=(z,~,~)for a l l

the

map

* zcp(u 1.

Proof. From 3.2.12, there is an open set Vx of X, with xcVXcU, such that f(Vx) is a submanifold of class p of X' and f:Vx+f(Vx) is a diffeomorphism of class p. Then cl=(U1=f(V X ) , )-',E) is a chart of the submanifold f(Vx) of X'. cp1='p

Ivx

Then, from 3.1.11, there is a chart c ~ = ( U ~ , p ~ , ( E ~ , h of~ )X' ), * such that f(x)eU', p'f(x)=O, Utnf(VX)cU1, E'=ExGlxR n , A'=(pl,...,pn)opjr where n=indX,(f(x)), pj is the third projection and pt(y)=(pl(y) ,O,O) for all yeU'nf (Vx). Moreover p t (U'nf (Vx)) = p t (U')n(Er:( O)x{ 0)). We end by taking the open set of u, U*=f-l(U'nf (VX)) .o Next we study the composition of immersions.

Chapter 3

142

Proposition 3.2.16 Let f:X+X', g:X'+X@* be maps of class p and XEX. Suppose that f is an immersion at x and that g is an immersion at f(x). Consider the following statements:

a)gf (x)Eint(Xt1). b) There is an open neighbourhood Vx of x in X, such that gf (VxnaX)caX1l and ind(v)=ind(Tx(gf) (v)) for all VE(T~X)~.

Then if some of these statements holds true, gof is an immersion at x.

Proof. We have that T,(gf)=T (g)oTxf and, by hypothesis, f (XI Tx(gf) is an injective map and im(Tx(gf)) admits topological supplement in T XV1,(see 3.2.9, a)). Then: gf (XI If b)

holds, the proposition follows from 3.2.12.

If a) holds, then 3.2.6 ends the proof. If c) holds, then 3.2.15

leads to the conclusion. Indeed, let c=(U,p,(E,A)) a chart of X with XEU, p(x)=O and ct=(U',p',Et) a chart of XI with f (x)EU' and ptf(x)=O, such that E is a linear subspace of El which admits topological supplement in El, f (U)cUl, p(U)cp# (Ul) and ptfp-':p(U)+pl (Ul) is the inclusion map. From 3.2.15 we deduce that there are a chart c ~ ~ = ( U (E1llA1l) ~ ~ , ~ )~ ~ , * of XI1 with gf (x)EU~', pV1gf ( x ) = O , and an open set U of U' , with * * n * f(x)EU* and g(U )cull such that E1l=E'xGlxR , where G1 is a real Banach space and n=indX,,(gf(x))

,

*

p' (U )x{ O)x( O ) C ~ ~ ~ ( Uand ~ ~ )the

is u(z)=(z,~,~) for all u=pl~g(p~ *)-':~~(u*)+~~I(u~I) * IU * zepl(U ) . Then (f'l(U nU),p~,(E,A)) and cl@verify the conditions for gof to be immersion at x.0

map

Submanifolds and Immersions

143

Corollary 3.2.17 Let X’ be a submanifold of class p of X and let XI1 be a submanifold of class p of X’ such that axltcaXand for all X ~ ~ E X ~ ~ ~ ~ ) ~ ,jll:X1l+Xis ind(vll)=ind(Txll(jll) (vll)) for all V ~ ~ E ( T ~ , , Xwhere the inclusion map. Then XI1 is a submanifold of class p of X. Proof. It follows from 3.2.4 and 3.2.16.0 Proposition 3.2.18 Let E,F,G be real Banach spaces, ucJe(E,F), and v&(F,G). Suppose that im(v0u) admits a topological supplement in G and that vou is a injective map. Then u is an injective map and im(u) admits a topological supplement in F. Proof. Of course u is an injective map. From 3.2.9, it follows that vou is an immersion of class m. Then by c) of 3.2.8 there are an open set U2 of E, an open set U;1 of G and a map q:U;1+E of class p such that 0eU2, vou(U2)cU;1 and qovou(y)=y for all yeU2. Therefore (TOq)(v0u)=lE, which implies that im(u) admits a topological supplement in F.o Proposition 3.2.19 Let f:X+X’, g:Xt+Xl1 be maps of class p and xeX. Suppose that gof is an immersion at x. Then f is an immersion at x, if some of the following statements holds: a)f (x)Eint (XI) b) There is an open neighbourhood Vx of x in X such that f(vxnax)caxt and for all ~ ( T ~ X ) ind(v)=ind(Txf) ~ , (v). Therefore, if f(X)cint(X’) p, then f is an immersion.

and gof is an immersion of class

Proof. It follows from 3.2.18, 3.2.6 and 3.2.12.0

Chapter 3

144 Corollary 3.2.20

Let X be a submanifold of class p of XI1 and differentiable manifold of class p such that XCX'CX~I.

X'

a

Suppose that the injection maps, j:X+X' and j':X'+Xll, are maps of class p and the topology of X' is the topology induced by the topology of XI1. Then X is a submanifold of class p of X', vhose induced structure is the structure of X as submanifold of XI1, if some of the following statements holds: a) Xcint(X') b)

axcaxl

and

for

all

XEX

and

all

VE(T~X)~,

ind(v)=ind(Txj) (v). Proof. Of course the topology of the manifold X is the topology induced in X by the topology of X'. Moreover, from 3.2.19, j:X+XI is an immersion of class p. Then, by 3.2.4.b), X is a submanifold of X' and the differentiable structure of X I induced by XI and XI1, coincide.0 Next we corollary.

see that

a),b)

are

essential

in

the preceding

Example 3.2.21 Let X1I=R2, X'=[O,+)xR

2 2 2 and X=( (x,y)~R/(x-l) +y =I).

In this case X is a submanifold of XI1, XI is a submanifold of XI1, j I : X t 4 X l 1 and j:X+X' are maps of class p I X is not j) (0)=1 and X is not a contained in int(X') I O=ind(O)+ind(T (OIO) submanifold of XI. Lemma 3.2.22

Let X be a differentiable manifold of class p. Then X is

Submanifolds and Immersions

145

connected if and only if int(X) is a connected set. Proof. If int(X) is connected, then int(X)=x is also connected. Suppose that X is connected and that int(x) is not connected. Then there exists a connected component, K, of intX with K+int(X). Let us consider C=int(X)-K. Since int(X) is locally connected, we have that K and C are open and closed sets of int (X) Moreover RuE=X, WE+# and bccaX. Let XE%E and let c=(U,p,(E,h)) be a chart of X with XEU, p(x)=O such that p ( U ) is + a convex set. Then p-'(p(U)nint(E,))=A is an open connected set of int (X) and A=Unint (X)

.

.

Therefore either AcK which is a contradicti0n.o

or AcC, but AnC=UnC+# and AnK=UnK+#

Lemma 3.2.23 Let K be a connected component of B X, where q e H u ( 0 ) . Then q X-Kcaq+l(X). Moreover, K is a submanifold of class p of X with aK=$. On the other hand need not be a submanifold of X. Proof. Since K is a connected component of B X I we have that q (x-K)nB X=$. On the other hand ik?caqX, which is a closed set. q Hence i?-Kcaq+lX. Let us consider xeK. Then xeB X-K since K is an q open set of B X. Let c=(U,p,(E,A)) be a chart of X with XEU, q 0 p(x)=O, such that UnB X-K=#; let E'=EA and A'=#, (3.1.4). It is g o + Therefore K is a clear that p(UnK)=p(U)nEh and EAf=E'cEA. submanifold of X and aK=#. Finally we quote an example due to Cerf, that justifies the last statement of the lemma: Let X be the manifold of class m defined by im(f) and its interior in the plan, where f:[0,+)+R2 is given by

-

f (t)=[+, q such that f(x)eK', where K' is a connected component of BqlX'. Then, following the proof of b), there is a neighbourhood Vx of x in intX, such that f(VX)cK' and hence VXcint(X)-G.

Submanifolds and Immersions

147

From a) ,b) and c) we conclude that G=int(X) .o Corollary 3.2.25 Let X' be a differentiable manifold of class p, let X be a connected submanifold of class p of X', qdNu(0) and K a connected component of B (XI). Suppose that int(X)nK+#. Then int(X)cK and therefore Xcx. Proposition 3.2.26 If f,g are immersions of class p at x,x' respectively, then fxg is an immersion of class p at (x,xI). Therefore if f and g are immersions of class p, then fxg is an immersion of class p.o Corollary 3.2.27 If A is a submanifold of X and A' is a submanifold of X', then AxA' is a submanifold of XxX'.o ProDosition 3.2.28 Let f:X+XI be an f(x)eint(XI) or that f an open neighbourhood ind(z)=ind(Txf) ( z ) for

immersion of class p at xeX. Suppose that verifies the following condition: there is Vx of x in X such that f(VxnaX)caXI and all ze(TxX) i

.

Then (fIg):X+XIxXIt is an immersion of class p at x for every map g:X+Xtt of class p, such that g(x)eint(Xn) (see 3.2.6 and 3.2.12) .o

Example 3.2.29 Let f:R+[O,+)cR

be the map of class

m

defined by f(x)=x 2

Then (lR,f) is not an immersion at 0 (see 3.2.13

.

or 3.2.24).

u a 3.3.

Chapter 3 Embeddings

The following example shows us that Proposition 3 . 2 . 1 3 is not true in a global sense, even if f is an injective immersion. Example Let f:R+R2 be the injective map of class

m

defined by

Then, for all tGR, Ttf is an injective map and therefore f is an injective immersion of class

W.

On the other hand im(f) . . , which is the lemniscate in R2 with axis y=x, is not a submanifold of class 1 of R 2 . Now we study the injective immersions whose images are submanifolds. Definition 3 . 3 . 1 Let f:X+X’ be a map of class p. We say that f is an embedding of class p if f is an immersion and f:X+f(X) is a homeomorphism. The next result is an important characterization of embeddings. Proposition 3 . 3 . 2 Let X,X’ be differentiable manifolds of class p and

f:X+X’

a map.

Then the following statements are equivalent: a) f is an embedding of class p. b) im(f) is a submanifold of class p of X’ and f:X+im(f) a diffeomorphism of class p.

is

Submanifolds and Immersions

149

Proof a)*b) Since f:X+im(f) is homeomorphism, it is clear that there is a unique differentiable structure [a] of class p in im(f) such that f:X+im(f) is a diffeomorphism of class p and the topology T is the induced topology in im(f) by the manifold [dl Xf. On the other hand the inclusion map j: (im(f) , [d])+Xf is an immersion, since f is an immersion. Then, by 3.2.4, (im(f),[a]) is a submanifold of class p of X‘. b)ea) It is clear that f:X+im(f) f is an immersi0n.o

is homeomorphism. Moreover

Corollary 3.3.3 Let X be a differentiable manifold of class p and X’ a subset of X. Then the following statements are equivalent: a) X’ is a submanifold of c l a s s p of X. b) X’ is the image of an embedding of class p.o

If an embedding f:X+Xf has closed image we shall say that f is a closed embedding. Of course in this case f:X+Xf is a closed map.

Next we see a characterization of the closed embeddings. Proposition 3.3.4 Let X,X’ be differentiable manifolds of class p and f:X+X’ a map. Then the following statements are equivalent:

Chapter 3

150

a) f is a closed embedding of class p. b) f is an injective immersion of class p and f is a proper map. This proposition is a direct consequence of the following result of general topology: "Let X,X' be topological spaces and let f:X+X' be an injective map. Then f:X+im(f) is homeomorphism and im(f) is a closed set if and only if f is a proper map, [M.O.P. 3 . 4 . 1 I g g . o Corollarv 3 . 3 . 5

Let X be a differentiable manifold of class p and X' subset of X. Then the following statements are equivalent:

a

a) X' is a closed submanifold of class p of X. b) X' is the image of a closed embedding of class p.0 Proposition 3 . 3 . 6

If f and g are embeddings of class p, then fxg is also an embedding of class p.0 We end this paragraph showing that every differentiable Hilbert manifold can be considered as a closed submanifold of a Hilbert space. Before we recall the construction of the Hilbert direct sum of Hilbert spaces and of Banach spaces. I) Finite case Let ( H l r ~ r ~ l ) r . . . r ( H n r ~be r ~ real n) Hilbert spaces. As we have seen in Chapter 1, there are several norms in H=H x.. .xHn 1 that make H a real Banach space whose topology is the product

151

Submanifolds and Immersions topology. Among these nonns we choose

I (X,l*..

2 'Xn) Il=(llx,II,+.

2 1/2 ..+IIxnIIn)

I

where iyji=+di.

product in H and

-

.

..

1 wl,. * 'Xn) Il=+dfor ~ all x,yeE. It is clear that is an inner icI product in E and IIxl12= for all xeE. Thus (El) is a real Hilbert space that will be called Hilbert direct sum of the family of real Hilbert spaces ((Ei,i))ieI. This Hilbert direct sum will be denoted by ( (B Ei,). ieI Theorem 3.3.7 Let X be a Hausdorff paracompact differentiable manifold of class p whose charts are modeled over real Banach spaces which satisfy the Urysohn condition of class p (1.5.4). Then there are a real Banach space, (E,II 11). and a closed embedding f:X+E of class p. Therefore the manifold X is diffeomorphic of class p to a closed submanifold of E, (3.3.2). Moreover it holds the following local property: For every xeX there are an open neighbourhood Wx of x in X, a closed linear subspace El of E and a quadrant (E ) + of El, such that dl :Wx+(E ) + is an embedding of class p and f(Wx) is a totally f IWX dl neat submanifold of (E ) + . dl

Submanifolds and Immersions

153

Proof. Since X is a Hausdorff paracompact space, X is a regular space and using 1.4.8 we get for every xeX a chart cx=(Ux,pxl ( E x l h x ) ) of X I with X E U ~ ,p,(x)=O, px(Ux) bounded in Ex a closed set of X if and only if p,(F) is a and such that closed set of

is an open covering of X I there is a Since U=(UJxeX) locally finite open refinement (Ui)ieI of 91. For all is1 we take u.cux Then X.EX such that

' i

1

,hi=h

J=(ci=(Uilpi=pxilUir (Ei=Ex

i

) ) )ieI

.

is an atlas of class p

xi

of X such that (Ui)ieI is a locally finite family and FcUi is a closed set of X if and only if pi(F) is a closed set of E i . Since X is a T4 space there is a contraction, V=(Vi)iEIl of (Ui)ieI. Vi=X. By the That is, VicUi for all ieI, Vi+# for all i e I and ieI same argument there is a contraction W=(Wi)ieI of V with Wi+# for all isI. Then for all ie1, pi(Fi) is a closed set of (Ei)i4cEi and p.(wi)cpi(Vi). 1 Having in mind 1.5.4, for all ieI L

is a function A i : E i + [ O l l ] of class p Ai( (Ei)Ai-pi(vi) + and b p i (wi)= t 1)

there

can

Then

) cpi (V,) cpi (ui) for all ie1.

A=Supp (A

It

such that

be

'ilx-p;l(suP??(AiI

easily

seen

+

( E i ) A,

))=O,

that

i' I U,I

is

of

-i

X=Uiu(X-pi (A)) and

1

class -1

pi (A)

is

ui

p, a

closed set of X. Therefore qi is a map of class p. Let 1 be the norm of EixR for ieI and let ( E l 1 11) be the Hilbert sum of the family of real Banach spaces ( (EixR, [ li) )ieI.

prove that

C l'i ieI

(x)igx

Then there is a unique differentiable structure of class p on U/Su such that i is a diffeomorphism of class p. Since i is a submersion of class p, qu is a submersion of class p and Su is regular in U.0 Lemma 4 Let X be a differentiable manifold of class p and S an :S+X is an open map. equivalence relation on X such that p 21S suppose that X= (JUil where Ui is an open set of X with ie I q-'q(Ui)=Ui for a l l i d and suppose that Su =(UixU.)nS is a 1 i regular equivalence relation on Ui for all ieI. Then S is a regular equivalence relation on X, (q:X+X/S is the projection

map).

From Lemma 2 it follows that q(Ui) is an open set of X/S with the quotient topology, Tc, and we consider the following commutative diagram:

Chapter 4

196

A

where aiqui (x)=q (x)

.

It is clear that the map ai:Ui/S

+q(U.) is surjective. This 1 ‘i way there is a unique differentiable structure [ai] of class p over q(Ui) such that ai is a diffeomorphism of class p. Then q:Ui+(q(Ui),[di]) is a submersion therefore an open continuous map for all iEI. By

the

definition

of

quotient

of

topology

class

we

see

p

and

that

On the other hand, q(UinU.)=q(Ui)nq(U.) is an open set of 3 3 both (q(Ui),[di]) and (q(Uj),[Bj]), for all i,jcI. Hence the maps [ a .3 ) and q:UinU.+(q(UinU.), 3 3 1 Iq(U.nU.) 1 3 q:UinIJ.+(q(UinU.), 3 3 [d.] 3 IS(UpJj)1 are surjective submersions of class p and therefore, from 4.1.6, we deduce [’i] 1 q(uinu. )=[’j J 1 q(uinuj) 3 Then, by 2.4.5, there is a unique differentiable structure [d] of class p on X/S such that for all i c I , q(Ui) is an open set of (X/S, [all and [dl Iq(ui)=[dil.

is a submersion of class p Moreover the map q:X+(X/S,[d]) and therefore S is a regular relation over X.0 Lemma 5

Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a submanifold of XxX and

Submersions and Quotient Manifolds the map p

197

:S+X is a submersion of class p. llS

Suppose that X= ()Ui, where Ui is an open set of X, and that ie I is a regular equivalence relation on Ui for all ieI. Then S SUi is a regular equivalence relation on X. Proof :S+X is open and by Lemma 2 the map q:X+X/S 21S is open and continuous, with respect to the quotient topology Tc on the set X/S. By Lemma 1, p

Thus, Vi=q-'q(Ui) is a saturated open set of X for all ieI and X= 0Vi. On the- other hand (VixVi)nS is a submanifold of ie1 Vixvi and the map pI1(vixvi)ns: (VixVi)nS+Vi is a submersion of class p. Then, from Lemma 3 applied to Vi, we see that Sv

is regular

i on Vi. Finally by Lemma 4, S is regular on X.0

Lemma 6 Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a submanifold of XxX. Let j:S+XxX be the inclusion map. Then for every xeX ve have Nx= ( veTxX/ (v,0)e (T is a linear subspace supplement Kx in TxX. Proof

Since AcS , s and T(xIx) (k)(T(x,x)A)cT (x,XI H=(T (XIx) Pl'T ! X I x)P2)T (x,x ) (j)(T(x,x)S ) cTx (X)xTx (X), where k: A+S is the inclusion map.

Chapter 4

198 Moreover

(T(x, x)PlJ( x,x)P2)T (x,x) ( j ) T (x,x) (k)(T(x,x)A)= - (T(x,x)p1 fT(x, x)P2)T( x,x! (i)(T(X? x)A) is the diagonal, D, of the space Tx(X)xTx(X),

where i:A+XxX is the inclusion map.

and Tx(X)x(0) are closed linear subspaces of Tx(X)xTx(X) such that Dn(Tx(X)x(0))=( ( 0 , O ) ) and D+(Tx(X)x(0))= BT ( Tx (X) x ( 0 ) ) =Tx(X)xTx (X) Let 0 be =Tx(X)xTx (X), we have that the linear homeomorphism 0:Dx (Tx(X)x( 0 ) ) +Tx (X)xTx(X) defined by 0( (u,u), (v,O))=(u+v,u). Since

D

.

On the other hand Nx is, also, a closed linear subspace of Tx(X) such that DnCNxx(0))=((O,O)) and D+(Nxx(0))=H. Hence D is a topological supplement of Nxx(0) and the map O1 :H-Dx (Nxx( 0)) defined by 0,(u,v)=((v,v),(u-v,0)) is a linear homeomorphism. admits a topological supplement in Tx(X)xTx(X), there is a linear continuous map, q: Tx(X)xTx (X)+Tx (X)xTx (X) such that q o q = q and H=im(q). Since H

Let us consider the surjective linear u=O1OqoO: Dx (Tx(X)x ( 0 ) ) +Dx (Nxx( 0 ) ) CDX(Tx(X)x ( 0 ) )

.

continuous

It is clear that a a = u and u((v,v),(o,o))=((v,v),(O,O)). us consider the commutative diagram:

where

Or

is the

u ( t+Dx ( ( 0 , O )

surjective

)=u(t)+Dx(

linear continuous map

( 0 , O ) ).

Finally consider the linear diffeomorphisms:

defined

map

Let

by

Submersions and Quotient Manifolds

199

Lemma 7 Let X be a differentiable manifold of class p, xeBn(X) and S an equivalence relation on X such that S[aX]=aX, S is a neat submanifold of XxX and p :S+X is a submersion of class p. llS Consider the space N ~ = ( v E T ~ ( x )(v, / 0 )E (T the space

P ~ , T ( ~P ,,) T(x,

(j1 (T(x,

S)

1,

H=(T (x,x) p1 (x,x)p2) (x,x) ( j (T(x,x)s 1 CTx (XIxTx (XI and a space Kx, which is a topological supplement of Nx in Tx(X), (see Lemma 6). Then dim(Xx)hn. Proof :S+X is a submersion of 21S class p. Moreover aS=Sn ( a (XxX)) =Sn [ (XxaX)u (aXxX)3 =Sn (aXxaX) and therefore p2(aS)caX. Then by 4.2.3 we have that )-l((x))=Yx(x) is a closed submanifold of S such that From Lemma 1 we deduce that p

(Yx(x))=dimxX, where j :Yx(x)+S and codim (x,XI map.

is the inclusion

:S+X is a submersion of ind (x,x)=n. Since p S llS class p and pl(aS)cCJX again by 4.1.14, pl(BnS)cBnX. Therefore YcBnX, Y is a submanifold of BnX and Y is a submanifold of X. By 4.1.14,

Chapter 4

200

Since p

11 (YX{X))

:Yx{x)+Y is a diffeomorphism of class p and

(x,x) (j)T(x ,x) (Yx(x) ) =kerT (x,x)(P2lS) we have (x,x) (Pl) (x,x) (k)(T(x, x) ( YX( X 1 1 1=TX( l )Tx (Y1=Nx where L:Y+X and k:Yx(x)+XxX

are the inclusion maps.

Finally, dim ( Kx)=codim (Nx)=codimx (Y)hcodimx (BnX)=n. Lemma 8 Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a neat submanifold of XxX, S[ax]=aX and p :S+X is a submersion of class p. Then, for llS every XCX there is a neat submanifold A of X such that XEA and Tx(j)Tx(A)8TNx=Tx(X), where j:A+X is the inclusion map and Nx is defined as in the preceding Lemma. Proof Let c=(U,p,(E,h)) be a chart of X centered at x, F the space

x -1

(Kx), where K, is a topological supplement of Nx, and G the x -1 If xeint(X) then space (Oc) (Nx) Then E = F Q ~ Gsince TxX=NXeTK,. -1 A=p (p(U)nF) is a submanifold of X with O(A)=# which fulfils the X requirements of the lemma: Oc(F)=Tx(j)(TxA)=Kx. (Oc)

.

Suppose that xsBnXcaX with nrl. Then, by the preceding lemma , dim (F)anal Therefore there are a closed linear subspace ,vn) of elements F1 of F and a linearly independent system {vl, of F, such that F=F 8 L(vl, ,vn). Thus E=(G+F ) 8 L(v l,...,~n) 1 T 1 T + and G+F1+E. Then int(EA) is not contained in G+F1 since otherwise

.

...

+

...

+

G+F 1-G+Fl>int(EA)=EA and G+F1=E which is a contradiction. Hence

+ (G+F1) and &>O such that BE (w,) cint (Eh) + and there are wlsint (Eh) B, (wl)n(F1+G)=#. Then the vector w1 fulfils the following conditions:

-

201

Submersions and Quotient Manifolds

c) If n>l, then Bc(wl)-[(G+Fl)+L(wl)] is not contained in 0

EA+L(wl) * Let w2 be an element of BE (w,) - [ (G+F1)+L(wl)] such that w dE0+L{wl). Then: 2

A

0

a) EAnL(W1 I w2 )=( 0

-s not contained in

c) If n>2, then Bc(wl)-[(G+F1 0

EA+L(wl ,W21 By

induction we obtain a linearly wn) of vectors of E such that:

independent

system

...,

(wl,

+

.

.

1) (wlf.. , W ~ ) C(wl)cint(EA) B~

2) E=(G+F1) @ TL(wl,...,wn), that is mT(F1+L(w lf...,~ n ))=E. 0 3) E=EAeTL(wl,

4)

ia(1,2,

0

...,wn ) .

...

...,

w ~ + ~ ~ [ E ~ + L (, w W~~,) ] W [ ( G + F ~ ) + L ( ~ ~ , wi)] 11-11.

...,

for

all

Let us consider the closed linear subspace of E, +L(wl,...,wn). Then AF*=(A */Ash) is a linearly independent 1 IF * system of elements of X(F , R ) .

* F =F

We know that 0 E=EA8TL(ul,...,un), Then we have:

where A.(u.)=6 1 3 ij' AicA.

Salnun I

0 0 where wi"EA.

+annun

Chapter 4

202

Since E 0nL(wl, A

...,wn)=(0),

la..I+O. Consider the matrix 13

-1

and the vectors w;=bllwl+

...+blnwn

w'=b n nlw 1+...+ bnnw n These vectors of are linearly independent. Then we have that Ai(w!)=b. A . (wl)+. .+b. +b nani=6 ij 3 11 1 inA .1 (wn )=bjla li+ and therefore the vectors A n l F* are linearly independent.

* F

.

...

[

Moreover (F*)f; cEf; and a (F*)+ F* ].F'

*

*+

Finally (p(U)nF =p(U)n(F ) A

W

*

=(F * ) A + naEA. + F*

=B is a neat submanifold of p(U)

and p-l(B) is a neat submanifoid of U, and therefore (p-l(B)=A is a neat submanifold of X.

.

x (F* ) =Tx( j ) ( TxA) and Ocx (F* ) eTNx=Tx (X) Moreover Oc Lemma 9

Let X be a differentiable manifold of class p and S an equivalence relation on X.

PIIS

Suppose that S[aX]=aX, S is a neat submanifold of XxX and :S+X is a submersion.

Then for every xeX, there are an open set U of X vith XEU, a neat submanifold W of U vith xeW and a map r:U+W of class p such that r(aU)caW and for every YEU, r(y) is the unique element of W such that (y,r (y)) ES

.

Proof. Let x be an element of X. From Lemma 8, we obtain a neat submanifold A of X such that XEA and Tx (j) Tx ( A ) eTNx=Tx (X), where j:A+X is the inclusion map and Nx is defined as in Lemma 7.

Submersions and Quotient Manifolds We

consider

submersion

of

the

class

set p

c=(AxX)nS.

with

Pl I :s+x

is

a

(P, I s) -l (A)=c

is

a

Since

p1 (as) caX,

203

that and is the inclusion map, (4.2.1). The map p

21c (x,x) . Indeed :

:c+X is a local diffeomorphism of class p at

a) Let vsTxX. Then, by Lemma 1, there is hcT (S) such (XrXI ) (h)=v. On the other hand (Xr X2I'( I s is the (T( x p x)P1tT (x,x)P2)T(x, x) (k)(h)= (s,t) , where k: S+XxX inclusion map and t=v. Moreover s=u 1+u 2' where u1eNX and Theref ore u2€Tx (j1Tx ( A ) that

.

(u1 0 ) (T(x, x)p1 IT( x,x)p2)T( x,x) (k)(T(x,x)S)=Hr (UZ~VIEH. Then ~ ~ ~ ~ ~ ~ ~ ~ LET

(s,v)EH and ~(k)(~t ), x where , x and Then tv)=u2

(xrx)s' -1 T (x,x) (PlI s ) (t)=T(x,x)p1 (T(x,x)p1 J( x, x p 2 ) 2'( and t=T LET (x,x) (.a)T(x, x) (.a) ( a ), where acT (Xr X) ) (a)=v and the map T Finally T ) (xrx) 2'( I (xrx) 2'( I s surject ive map.

(c)

is

a

~

204

Chapter 4 c) Of course (p

U1,U2 of X such that

) (a1)caX. By

2'c xeU1~U2and

2.2.6,

there are open sets

is a diffeomorphism of class p. Let f be the inverse diffeomorphism. It is clear that f(y)=(p(y),y) for all yeU2, where p=p;oiof, i:cA(U1xU1)+AxX is the inclusion map and p;:AxX+A is the Is'-projection. Then p:U2+A is a map of class p, U2cUl and p(y)=y for all ycU2nA. Let us consider the open set U = P - ~ ( A ~ U ~of ) X I the submanifold W=UAA of U, and the map of class p, r=plU:U+AnU2. Then xcW , r (U)cW , xcU , aW= (ax)AW= (aA) n U = h (ax)AU and r (aU)caw. If yau,

then r(y)cW

and

f (y)=(r(y) ,y)cCcS and

therefore

(Ylr(Y)) e S .

Lemma 10 With the same hypothesis and notations xcX, U, W, r of the preceding lemma, there is an open set U' of X, such that xewcu'cu and Su, is a regular equivalence relation on U' which verifies that Sw (aW)=aw. Proof If i:W+U is the inclusion map, then ri(y)=r(y)=y for all yew. Therefore r:U+W is a submersion of class p at every point of W. Then, by 4.1.15, the set UI=(yeU/r is a submersion at y) is an open set of U which contains the set W and r :U'+W is a I U' submersion of class p. Consider the commutative diagram:

Submersions and Quotient Manifolds r U'

I U'

205

+w

* d

U'/SU,

It is clear that qu, I w is a bijective map and therefore there is a unique differentiable structure of class p on U'/Su, such that q u , l w is a diffeomorphism of class p. Then q

I U'

is a submersion and Su, is regular in U ' . o

Now 4.3.14.b) and Lemmas 9, 10, 5 and 4.3.13 statement a) of 4.3.14.0 Corollary 4.3.15

prove the

(Main Theorem on Quotients and Regular Equivalence Relations for Manifolds whithout Boundary).

Let X be a differentiable manifold of class p vith ax=# and let S be an equivalence relation on X. Then the following statements are equivalent: a)

S

is regular. (Hence a(x/s)=@, 4.1.11.)

b) S is a submanifold of XXX with a(s)=#, and p lls:s-+x is a submersion of class p. c) S is a submanifold of XxX with a ( S ) = # and for every ( x , y ) ~ Sthere are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))~S for z~U.0 In the theorem 4.3.14, the condition S(aX)=aX is taken as an essential hypothesis. Next we shall investigate the condition S[intX]=X. Lemma 4.3.16

Let X be a differentiable manifold of class p and S a regular equivalence relation on X. Then S[intX]=X if and only if a(x/s)=#.

Chapter 4

206

Proof If S[intX]=X then, by 4.1.11, a(X/s)=#. Suppose, now, that a(X/S)=# and x&X. Then there is krl such that x€Bk(X). Since q:X+X/S is a submersion of class p on X and a(X/s)=#, by Lemma 4.1.17 there are a chart c=(U,p, (ExGxRk,Ap3)) of X centered at x, a chart c'=(U',p',E) of X/S centered at q(x), such that q(U)cU' and the map p,qp-':p(U)+E coincides with the map pl,p(u):p(U)+E, where pl:ExGxR k+E is the lSt-projection, p3:ExGxRk+Rk

...,pi)

is the 3d-projection, A=(p;,

the ith-projection (therefore, (ExGxRk)+

and pi:R k+R is

=ExGx (W k ) A +)

.

+ and there is Then p(U) is an open set of ExGx(R k ) A such that (O,O,a)€p(U). It is clear that aaint (Wk ) A+ y=p"( (O,O,a))aintX and p'q(y)=O=p'q(x). That is, q(x)=q(y) and consequently xSy o

.

Theorem 4.3.17

(Main Theorem on Quotients in Case a(x/s)=#).

Let X be a differentiable manifold of class p and S an equivalence relation on X. Then the folloving statements are equivalent: a) S is regular and a(X/S)=#. b) S is a neat submanifold of XxX, p

:S+X is a submersion lls

of class p and s[intx]=x. c) S is a neat submanifold, S[intX]=X and for every (x,y)sS there are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))~S for zaU. Proof In 4.3.14 b)=)

we actually proved b')ec')

and therefore we have

Submersions and Quotient Manifolds

207

a)+c). From the preceding lemma we have S[intX]=X. Since a(X/S)=#, the diagonal A of the manifold X/SxX/S is a submanifold without boundary. Then, by 4.2.11, we have (qxq)"(A)=S is a submanifold of class p of XxX such that a(s)=sna(xxx) and therefore S is neat submanifold of XxX. Let (x,y)eS. Since q is a submersion at y, there are an open neighbourhood V of q(y) in X/S and a map of class p, s:V+X, such that y=sq(y) and qs(z)=z for zaV. Then the set q-l(V)=U is an open neighbourhood of x in X and f=sq is a map of class p from IU U to X. Moreover f(x)=y and (t,f(t))aS for teU. b)+a). Since S[intX]=X, by Lemma 3 of 3.1.14, it suffices to prove that Sint is a regular relation on int(X). Since S is a neat submanifold of XxX, Sn(int(X)xint(X)) is a submanifold without boundary of int (X)xint (X) Moreover the map is a submersion of P1Isn(int(X)xint(X)) :Sn(int(X)xint(X))+intX class p. Then by Corollary 4.3.15, 'int (x) is a regular relation on int(X) .o

.

Examples 4.3.18 Let X=((x,y,z)eR 3/lsx2+y 2+z 244) and S the relation of equivalence on X defined by: (x,y,z)S(x',y~,z')ox 2+y 2+z 22 2 =x' +y' +zt2. Then we have that X is a submanifold of class m of R3 (4.2.1) and s[ax]=aX. On the other hand it is clear that S fulfils c) of 4.3.14 (4.2.1) and therefore S is regular. Moreover the manifolds X/S and [1,2] are diffeomorphic. A)

B) Let X be the differentiable manifold of class m, [-l,l]x[-l,l]xR, and S the equivalence relation on X defined by: (x,y ,2 ) s ( (x' ,y ' ,2 1 ) ) ox2+y2-2 z=z '2+y ' -2 z '

.

Then S is regular, a(X/S)=# and X/S,R

are diffeomorphic of

Chapter 4

208

class

m,

(4.3.17)

.

C) Let X be the submanifold of S 2 given by ((x,y,z)~S2 /zrO)

and R the relation of equivalence on X defined by: “zRyez=y or z=-yvl. Then X/R is homeomorphic to the projective plane P2 by the map a( [x],)=[x]

,

(recall that P2=S2/R1,

where R1 is defined by:

R1

XR 1yex=y or x=-y). In the example D) of 1.2, P2 was endowed of a natural differentiable structure [d] of class m such that a ( P 2 ) = # and is the quotient topology. Therefore there is a unique T [dl differentiable structure [dl] of class m in X/R such that u is a diffeomorphism of class m. Hence we have also that a(X/R)=# and is the quotient topology. Moreover R[intX]=intX+X. T [dl 1

2

Finally ( P 2 , [A]) is the quotient manifold of S by R1, (X/R,[dl]) is not the quotient manifold of X by R (4.3.17).

but

D) Groups of Transformations Let X be a differentiable manifold of class p and G a group. We say that G acts over X as a group of transformations if there is a map u:GxX+X such that: lo) For all geG, the map a *X+X defined by u (x)=u(g,x) is a 9 g‘ diffeomorphism of class p. 2’)

For all g,g‘eG and XEX, u(glu(g’rx))=u(gg’lx) (that is,

If G acts over X as a group of transformations through u and e is the unit of G, then u e=1XI (ueu e=u e ) and for all geG, -1‘ (a -l=Qe=lX and u -lug=ue=lX). g g g 9 If G acts over X as a group of transformation through u, then we say that:

Submersions and Quotient Manifolds

209

a) G acts effectively over X if: 'la (x)=x for all xGX+g=eI1 9 b) G acts freely over X if :"a (x)=x for some xcX*g=ell. 9 c) G acts discontinuously over X if for all XEX there is an open neighbourhood Vx of x in X such that VXngVX=# for all geG-(e}. Note that if G acts discontinuously over X I then G acts freely over X. Let G be a group that acts over X as a group of transformations through a. Then the relation SGcXxX defined by: "xS yothere is gcG with ci (x)=y" G 9 is an equivalence relation on x. Concerning groups of transformations we have the following results: Proposition 1 Let X be a differentiable manifold of class p and G a group that acts discontinuosly over X as a group of transformations through a. Then SG is a regular equivalence relation on X such that SG [ 8x1 =ax. Therefore X/SG has a differentiable structure of class p such that the natural projection q:X+X/SG is a submersion of class p and q (ax)=a (X/SG)

.

Proof Let (x,y) be an element of SG. Then there is gcG such that

The map ctg:X+X is (zIag(z))cSG for all Z G X . On the other hand

a

diffeomorphism

of

class

since A is

p

a

and

neat

210

Chapter

4

submanifold of XXX and 1 X U is a diffeomorphism of class p, we x g have that (1 X U ) (A) is a neat submanifold of XxX. x g Since G acts discontinuously over X, we deduce that for all geG there is an open set A of XxX such that (1 xu ) (A)cAg and g x g A n(l xu )(A)=# for all g'€G-(g). In fact we can take the open g x 4' set A of the form (I (VXxug(?)), where ? is an open g xex neighbourhood of x in X such that VxnglVx=$ for all g'eG-(e). Thus SG is a neat submanifold of class p of XxX. Finally, since u :X+X is a diffeomorphism of class p for all g gcG, S,[aX]=aX and Proposition 4.3.14 ends the pro0f.o The manifold constructed in the preceding proposition has a special property. In order to establish it we need a preliminary definition. Definition 2

Let X,X' be manifolds of class p. We say that a map q:X+X' is a covering projection of class p if it is surjective map and if every point x'EX' has an open neighbourhood Vx' such that -1 x' q (V ) = \ J Ui, where Ui is an open set in X for all k 1 , UinU.=# 3 1E I for all i,jcI with i*j and q :Ui+Vx' is a diffeomorphism of IUi

class p for all ieI. Obviously, a covering projection of class p is a local diffeomorphism of class p and for all x'EX', the subspace q-'(x') of X has the discrete topology. Proposition 3

Let X be a manifold of class p and let G be a group that acts freely over X as a group of transformations of X through u:GxX+X. Then SG is a regular equivalence relation on X and q:X+X/SG is a covering projection of class p if and only if G

Submersions and Quotient Manifolds

211

acts discontinuously over X. Proof The condition is sufficient. By Proposition 1, SG is a regular equivalence relation on X. For all [x]eX/SG there is an open neighbourhood Vx of x in X such that VXngVX=# for all gcG-(e). Then q -1 (q(Vx))=(J gVx, so that q(Vx)=V[xl is an open geG neighbourhood of [XI in X/SG. On the other hand q .gVx+V[xl is I gvx' a bijective submersion of class p, whence q -gVx+Vcxl is a I gvx' diffeomorphism of class p for all geG. Thus, q:X+X/SG is a covering projection of class p. The condition is necessary. For all xeX, there is an open neighbourhood VlX1 of [XI in X/SG such that q-l(v[xl)=(J Vi, _ic 1 _ where Vi is an open set in X for all isI, V.nV.=# for all i , j d 1 7 with i+j and q :Vi+V[x3 is a diffeomorphism of class p. Let ix IVi be the element of I such that xeVi It follows that V. ngVi =#

.

x

=x

X

for all gcG-(e).o Pronosition

4

Let X be a differentiable manifold of class p and G a group that acts discontinuously over X as a group of transformations through a. Suppose that for all (x,y)eXxX-SG there are open neighbourhoods Vx and Vy of x and y respectively such that VXngVY=# for all g e ~ . Then the quotient manifold X/SG is a Hausdorff manifold.

Proof From the hypothesis it follows that X is a Hausdorff space and therefore A is a closed set of XxX. Thus SG=(J ( (1 u ) (A) ) X' g geG is a union of closed subsets of XxX. The

hypothesis

implies,

also,

that

the

family

Chapter 4

2 12

(lXxag)(A) )gEG is a discrete family of closed sets of XxX and therefore SG is a closed set of XxX. (

Thus, by 4.3.11, space.

the quotient manifold X/SG is a Hausdorff

Particular cases

I) The n-dimensional Torus Let

a ( ( z1

us

consider

X=Rn,

G=Zn

and

cf:GxRn+Rn

...,zn) , ( x1 ,...,xn) ) = ( xl+zl ,...,xn+zn) .

given

by

Then the group Zn

acts discontinuously over R" as a group of transformations through a and for every (xIy)~XxX-SG there are open neighbourhoods Vx and Vy of x and y respectively such that VXngVY=# for all gEG. The quotient manifold, Rn/S n r will be called n-dimensional Z torus and denoted by Tn. n n Since a(R )=#, a(T )=#,

(see Proposition 1).

Moreover Tn is a Hausdorff compact manifold (see Proposition 4) *

11) The Moebius Strip

Consider X=R2, G=Z and a:GxR 242 given by (x+z,-y) if z is odd

[

a ( z r (XrY)) =

(x+z,y) if z is even Then the group Z acts discontinuously over R2 as a group of transformations through a and for every (xIy)~XxX-SGthere are open neighbourhoods Vx and Vy of x and y respectively such that VxngVy=# for all geG. The quotient manifold, Since a(R 2 )=#,

2

R /Sz,

will be called Moebius Strip.

a(R 2/Sz)=#. Moreover the manifold

R2/Sz

is

Submersions and Quotient Manifolds

213

Hausdorff and non-compact. 111) The Klein Bottle

Let x=R2, G=Z 2 and

a:GxR

242 be given by (x+zl,-y+z2) if z1 is odd

a((z1,z2) I (xlY))= (x+zl,y+z2) if z1 is even Then the group Z2 acts discontinuously over R2 as a group of transformations through a and for every (xIy)~XxX-SGthere are open neighbourhoods Vx and Vy of x and y respectively such that VxngVy=# for all gcG. The quotient manifold, R2/S 2, will be

z

called Klein bottle. Since a ( R 2 )=#,

2

a ( R / S ,)=#.

Moreover the manifold R 2/ S

z

is

z

Hausdorff and compact. IV) Lens Manifolds Let

m,L1,.

..,Lnc(N

be

such

that

el,.

..,t n

are

coprime

2n-1 and (p: (H/mZ)xS2n-1+S given by 2ni [ k]! 2ni [ k]! l/m , . . . , z ne n'm). Then the p([kl,(Z1,...tZ n ) ) = ( z 1e group Z/mZ acts discontinuously and effectively over S2n-1 as a group of transformations through p, (note that H/mZ is a finite numbers,

group)

X=S2n-1

.

The

quotient

manifold

S2n-1/S

manifold and denoted by L(m,ll,.. Since a ( S

2n-1

)=#,

a(S

will

be

called

lens

"m;e ,en).

.

/SZ,mz)=#.

211-1

E) Stiefel F-manifolds

-

The equivalence relation, on S(F,E) , given in the example H of 1.2 is regular and the map a:S(F,E)/ +(FIE) is a diffeomorphism of class m.

This Page Intentionally Left Blank

Chapter 5

2 15

SUBIMMERSIONS 5.1. Subimmersions. Tubular and Collar Neighbourhoods. Definition 5.1.1 Let f:X+X' be a map of class p and xcX. We say that f is a subimmersion at x, if there are a chart c=(U,p,(E,A)) of X centered at x, a chart c'=(U',p',(E',A')) of X' centered at f(x) with f(U)cU' and a continuous linear map u:E+E' such that ker(u) admits a topological supplement in E, im(u) admits a topological and p'ofluop -1=uI supplement in E', u(p(U))cp'(U') P(U) * We say that f is a subimmersion, if f is a subimmersion at every point XEX. ProDosition 5.1.2 If f:X+X' is a map of class p, then the set G=(xaX/f is a subimmersion at x) is an open set in X.

Proof Let xcG. Then there are c,c' and u as in the definition of subimmersion. Let ycU with y+x. Then p(y)+O and we consider the set Al=(A~A/Ap(y)=O). Of course Al can be empty. On the other hand we consider the set A;=(A'EA'/A'p'f(y)=O). It is clear that there exists E > O such that BE(p'f (y))nEi,=# for all A'EA'-A;, where + E~,=(v'EE'/A' (v')sO), and BE (p'f (y))nE;,cp' (U') and there exists, 6>0 such that B,(p(y))nEh=# for all AEA-A~, also, Bg (9(Y)1 nE;ccp (U) and u(B,(p(y)))cB,(p'f(y)) I since U(Cp(Y) )=p'f(y) In particular, we have B~(p'f(Y))nE~'f=B~((P'f(Y))nE;,+ and B,(p(y))nEdB,((p(y))nEA1. +

-

-

1

Chapter 5

2 16

(u

p;f Iulp;l=uI(P

.

1

1

)

.

Therefore

f

is

a

subimmersion

at

y

and

XEUCG o ExamDles 5.1.3 a) If f:X+X' is an immersion at XEX, then f is subimmersion at x. Hence every immersion is a subimmersion.

a

If ax'=# and f:X+X' is an injective subimmersion of class p, then f is an immersion of class p , since in this case the linear map u of Definition 5.1.1 is an injective map, (3.2.6). b) If f:X+X' is a submersion at x and ind(f(x))=O, then f is a subimmersion at x, (4.1.17). c) If f , g are subimmersions at x1,x2 respectively, then fxg is a subimmersion at (x,,~,). Hence if f,g are subimmersions, then fxg is a subimmersion. d) Let E,F be real Banach spaces. Then u&(E,F) is a subimmersion of class m if and only if ker(u) admits a topological supplement in E and im(u) admits a topological supplement in F. Consider the set SB (E,F)=( uc2 E,F)/u class m).

is a subimmersion of

By 3.2.9,b) the set I(E,F) is an open submanifold of class 03 of Y(E,F) and by 4.1.16,b) the set Y(E,F) is an open submanifold of class m of Y(E,F).

Subimmersio m

217

Next we shall see that the set SB(E,F) admits a differentiable structure of class m without boundary such that the inclusion map j:SB(E,F)+I(E,F) is an immersion of class m. Let uOcSB(E,F), Eo a topological supplement of Go=ker(uo) in E and Ho a topological supplement of Fo=im(uo) in F. Let us consider the linear homeomorphisms 8:EOxGO+E, given by e(x,y)=x+y, and s:FOxHO+F given by T(x',Y')=x'+Y'. Thus for every ueI(E,F) I consider as a matrix

the map r-lu~:EOxGo+FOxHO can be

[; where cf:EO+FO is given by 7 (x)=pZT-1u€3(XI0) I 6 :GO+HO is

cf

#3 :Go+Fo given

3

(x)=plT-lue(xlO)I r:EO+HO is given by is given by

i.e.

-1

= (uef( E I F)/a :E ~ + Fis ~ a 1 inear

We define the set V

EolHo) the

I,@

homeomorphism) = ( U€V

and

U

set

(u,, EolHo)=

/u(E)eTHO=F). Then we have that: (uoI EolHo)

.

st

is an open neighbourhood of uo in Y(E,F) V(uo,EoIHo) :EO+FO is a linear homeomorphism and Indeed, since u 01% ' U plt-luO~(xlO)=uO(x)for all xcEoI it follows that u ~ E V ( ~ 0' 0' 0) .

On the other hand the map A: I( E l F)+2(E0, Fo) defined by A(U)=plt-lUejo, where jO:EO+EOxGO is jo(x)=(x,O)I is a continuous V

(uoI Eo IHO)

zd)

V l H )=A-1 (Iso(Eo,Fo)). Io'( 0 0 is an open set of P(E,F). map

Let ucv

and

(uoI Eo I Ho)

-

Then UEU

Therefore

if and only if (uoI Eo I Ho) is clear that

Indeed I it 6=7u-16 . u (Eo)=T (t'lue) e-' ( Eo)=t( ( c f ( X) I 7 (x)) /xeEo)= =t((x',~oL -1 (x'))/x'~F~)=(x'+7cf-~ (X')/X'EFO)

and

therefore

Chapter 5

218

u(EO)+HO=F. Moreover F=u(EO)aTHO since u(EO)nHO=( 0) and u(Eo) ,Ho are closed sets in F. Thus ucU

if and only if u(Go)cu(Eo). That is EolHo) if and only if for every ycGOl there is x1€F0 such

(Uo1

U€U

(uoI Eo I Ho) that u(y)=x'+ra-l(x1)=B(y)+6 (y) Therefore ucu if and only if g = ~ a - ~ S . I,@ EolHo)

.

cSB(E,F).

Indeed,

if

it

is

clear

that

UCU

then EolHo) ' u(E)=u(Eo) 1 F=U(E)@~H~=U(E~)@~H~ and ker(u)=(xl+x2~E/xlcEolx2~Gola(xl)=-~(x2) and r(xl)=-6(x2) ) = =(-a-1 ,9(x2)+x /x EG ) . Therefore it follows that E=ker(u)oTE0 and 2 2 0 W 0 I

ucSB(E,F). 4th)

If ucv

No Eo,Ho)

then F = U ( E ~ ) @ ~ H and ~ , ueU

I

(uoI Eo I Ho)

if

and only if u(Go)cu(Eo). 5th) If UEU

(uoI Eo I Ho)

then F=u(E)oTHo and E=Ker(u)eTE0.

Let us consider the maDs:

defined

and

r

[; i].

e

where r-1ue(xllx2)=(xllx2)

Then #1 and #2 are diffeomorphisms of class p inverse each other I

since

Analogously ##,, Thus

c=[V

u 1=# 1 (u)=t

if

(u)=u. EolHo) and #,(u,)=u

0

(UOIEOIHO)

1 .

,#llf(EIF)is

(U0l

u EV

then

p

e

0 1

-1

a

chart

of

I(E,F)

with

Subimmersions Let

us

consider

-

the

set

219 K=(ual(E,F)/6=0),

where

:).

Then we have that K is a closed linear which admits topological supplement, )nK. UO€KnU( u ~ , E ~ , Hand ~ ) 41(u(uo,Eo,Ho) )=V (u,, o, 'ue(x,y)=(x,y)[; subspace of l(E, F)

T

set SB(E,F). We shall see that the collection B cm-atlas over SB (E,F)

.

*

of these charts is a

Let us consider the Grassmann manifolds G(E),G(F) of E and F respectively and the set z (E,F)=( (G1,ul, F1)EG (E)xl(E, F)xG (F)/ uld(E,F), G =ker(ul) and F1=im(u ) ) . Then r(E,F) is a 1 1 submanifold without boundary of class m of G (E)x l (E,F)xG (F) Indeed :

.

Let (GO,uO,FO)~t(E,F).Then we take the index (GO,EO)eI(E), the

chart

c=(V

c =(U

Eof%o,Eo) ,@l,l(E,F)) of 1

(u,, Eo,Ho) constructed and

,Z(GO,EO)) l(E,F),

G(E) ,

that

we

the have

chart before

(FoI Ho1 , W O , H O ) ) of G(F) determined by the index (FO,HO)€I(F) Now, let c1xcxc2 be the product chart of G(E)xZ(E,F)xG(F) and let us consider the maps A:l(GO,EO)xV'O'(

the chart c2=(UH0

of

.

E 0,,H0 )xL(FO,HO)~Y(GO,EO)xV('of E 0 ,H0)xL(FO'HO)

Then the maps A and A l of class m are inverse each other and therefore A is a diffeomorphism of class m. Let us consider the chart of G (E)x l (E,F)xG (F),

220

Chapter 5

and

closed

the

subspace

I.

:)8-',v)/p=6=v=O

E (Go,Eo)XE(E,F)xE ( Fo,Ho) ,

of It

is

clear

that

topological supplement in Z (Go,Eo)XE(E,F)XE( Fo,Ho) Then

'9(UE

we

have

that

[

9 (UEoxV

,o'(

KO

admits

a

.

E ,H )xUHo)nr(E,F))= 0

0

0xv (uo,E0,H0)XUHO)nKO'

Thus t(E,F) is a submanifold without boundary of class G(E)xE(E,F)xG(F), (3.1.4). Moreover p of class

m

x]

[; 10)

Indeed, the diagram

e-l,0)=t e-', is commutative. Hence, is an immersion at (Go,uo,Fo) (3.2.61, Pal t(E,F)

where

i 0,t

of

:t (E,F)+E (E,F) is an injective immersion

I~(E,F)

such that p2(t(E,F))=SB(E,F). P, I

[ [

m

.

by

Then there is a unique differentiable structure [ a ' ] of m without boundary over SB(E,F) such that class P, I :t(E,F)+SB(E,F) is a diffeomorphism of class m. I~~(E,F) Therefore the map j:SB(E,F)+E(E,F) is an immersion of class m . ~

We note that the preceding diagram shows that 8* * cm-atlas over SB(E,F) and [ d l ] = [ 8 ' ] . On the other submersion of class

hand the map * since c3 =

m,

p13:t(E,F)4(E)xG(F)

is a

is

a

Subimmersions

221

and

Finally, from the equality #,(U deduce

that

the

set

SB(E,F)

(UO,EO'HO) (uo,E 0 ,H0)nK, we =(uESB(E,F)/U(E)@,H,=F) is a

HO submanifold without boundary of class is an open set of SB(E,F). SB(E,F) HO

m

of I(E,F).

Moreover,

Lemma 5 . 1 . 4

+

Let EA be a quadrant of a real Banach space E vith card(A)s3 and let F be a closed linear subspace of E that admits topological supplement in E. Then the set FnEi is a submanifold + of class OD of EA.

Proof lst)If A=#,

the result is obvious.

+

zd)

If A=(A) and h l F = O , then F c E ~ c Eand ~ therefore EAnF=F and the result is also obvious. 3d)

result

A;=(A

I

If

A=(h)

follows F).

4th)

and

from

If A=(Al,A2)

+

for

3.1.4.c)

+

c1=(EA,i,(E,A)),

and the system

+

+

independent, then EAnF=FA

+

nFA 'IF

=F

+

+ +

EAnF=EAnFAIF=FAIFand

AIF+O, then

+

(A

'IF

,A

21F

)

E;=F

th

is a quadrant of

'IF'

*O

21F

and A

and

is linearly

F. 5 ) If A = ( h l , h 2 ) , A

the

+

+

=0, then EAnF=FA

llF

21F

.

Chapter 5

222 th

6 ) If A = ( A l I A 2 ) , A

llF 7

8

+

th

th

)

+

+O and A

)

21F

llF

+

If A=(A1,A2),

+

+

gth)

+

A

.

If A=(Al,A2),

A

+O

and

A

+O 21F =FA 0

and

A

21F

+nF=FA + nFA + =F0 "FA+ EA A 'IF 21F 'IF 21F loth) If A=(h1,A2,A3)

'IF

21F

llth)If A=(A1,A2,A3),

IF

+bA

(A

+

IF' "I F r A 3IF

)

is

+

+

+

and arO,brO, then EAnF=FA nFA =F 'IF 21F

th

(AllF,A21F)is linearly independent,

=aA +bA and a < O , b>O, then ( A ,A 3iF 3tF + + independent system and EAnF=F ,A )* ('31~

IF

aO, then

21F

and the system

+

with

.

nFA 0

'IF

=aA

linearly independent, then EAnF=F

=aA 3iF

.

If I I = ( A ~ , A ~ A) ~ =O and A =0, then EAnF=F llF 21F

EAnF=FA nFA =FA 'IF 21F 21F

A

+

=0, then E nF=FA A

IF

IF

)

is a linearly

IF

is linearly independent, A A

=aA 3iF

IF

and

aO and a map u:U+ElxE2 such that: Ist) u ( O ) = ( O , O )

E.

and u is a diffeomorphism of class p from U

E .

Zd) If O:FlxF2+F 0 (x,Y 1=x+y ,

then

is the linear homeomorphism defined by the

maP

of

class

P

Chapter

228

5

verifies O ' l f IUa-l(x,,x,)= E. E, (xl),n(xl,x2)), where n:BEl(0)xBE L (0)-+F2 is a map of (0)xBE E2 (0)+F1xF2

=(Df(O)

I

class p such that n ( O , O ) = O and D(n) ( O , O ) = O " . Theorem 5.1.8 was announced with the most general hypothesis and the preceding corollary only deals with the case in which the spaces have not boundaries. Next we shall see an intermediate situation that will be used in the characterization of subimmersions. Corollary

+ FA, + be quadrants of E and F respectively, V an open Let EA, + + set of EA with OEV and f:V+FAJ a map of class p with f(O)=O and ('('1

+

(FA*

*

Suppose that ker(Df (0))=E2 admits a topological supplement El in E and im(Df(0))=F1 admits a topological supplement F2 in F such that (EIXEl)AOl=E + + XE28 (F1XF2)A#O=F1AixF2 + + and Al=A;o(Df(0)

I

),

lA1 where 01:E1xE2+E is the linear homeomorphism

given by 0 ( x ,x )=x1+x2 and O:F1xF2+F 1 1 2 homeomorphism given by O(y1,y2)=y1+y2.

is

the

Then there are an open neighbourhood U of 0 in V, E2

bijective map a:U+(BE l e t )a ( O ) = ( O , O )

+

(0)xBE (0))n(ElxE2)A01

linear

E>O

and a

such that:

and a is a diffeomorphism of class p.

P, verif ies and n:(BE (0)xBEE2 (0))n(E1xE2)AOl+F2 + n ( O , O ) = O and D r r ( O , O ) = O . o

is a map of class p such that

Subimmersions Theorem 5.1.9

Let f:X+X'

229

(Characterization of Subimmersions)

be a map of class p and asX such that ra(f)=nExG

is clear that @ is a bijective map of class p, D@(z,v) (w,u)=(w,D1(P,@) (z,v)(w)+D2(P2@) (z,v)(u)1 , 2 D (p2@) (z,v) (U)=@~(U), D@(z,v) is a linear homeomorphism for all (z,v)~cp(U)xH and @(a(cp(U)xH))=a(cp(U)xH). Hence @ is a diffeomorphism of class p (2.2.9) and @-l(z,u)= It

Subimmersions

Then

we

can

take

* *, c =(U~H,@-~o(cpxl~)=cp *

*

241

the chart (EXH,APl)1 1

of

class and

and

YxH, have f

cp ((UxH)nT f (Y))=cp (UxH)n(EAxE)=cp(U)xE, +

* + =cp (UxH)n(EAxG)=cp(U)xG

p of we

* cp (UxH)=cp(U)xH.

Thus using 3.1.4 we see that Tf(Y) of YxH, submanifolds of class p * * (ExE,hpl)) is a chart p c ' l I(UxH)nT f (Y) t

cp* ( (UxH)f-d (Y)1=

and Nf(Y) are * f cl=((UxH)nT (Y), of Tf(Y) and

*- * ,(ExG,hp;)) is a chart of N f (Y). "-'I these (UxH)nNf (Y) that Tf (Y) and Nf (Y) are totally clear, using charts,

c;=( (UxH)nN f(Y),

It is neat submanifolds of YxH and in a(T f (Y))=(T f (Y))n(ZJYxH) and a(N f (Y))=(N f (Y))n(ZJYxH).

particular

The last statement follows from the fact that locally t is the identity. b)

We

have

tl(a(Tf(Y)))=a(Y),

diagrams (UxH)nTf ( Y ) U

"11

(UxH)nNf ( Y ) U

=21

are commutative. Then b) follows from 4.1.13.

t2(a(Nf(Y)))=aY

and

the

Chapter 5

242

>(UxH)nTf (Y)

PI

UxH I

I

I

cp* and ,(UxH)nN f (Y)

Ql

UxH

are commutative and P,Q are maps of class p.

*

a)

*

d) We take the charts c and c2 constructed in the statement * *1 f f * * Then C ~ X C ~ = ( S (UxHInT =( (Y))x( (UxH)nN (Y)1 r(P1~P2i

((ExE)x(ExG).AP*,uAp;) 1

is

a

chart

of

T~(Y)~N~(Y),

H'=( ((u,v) ,(u,w))/u~E,vsE,w~G)is a closed linear subspace of (ExE)x(ExG) that admits a topological supplement in (ExE)x(ExG) * and A(plIH,) is a finite linearly independent system of elements of le(H',R).

H'+* c[(ExE)x(ExG)]+ * *, it follows that T f (Y)xyNf (Y) is a Apl I H' AP luAP3 f f submanifold of class p of T (Y)xN (Y). Finally, from a) , 3.2.27 and 3.1.16, we deduce f (YxH)x(YxH). T (Y)xyN (Y) is a submanifold of class p of

that

f

e) It is clear that u is a bijective map of class p and that -1 is a map of class p, hence Moreover, by c) and d) , u u is a diffeomorphism of class p.

-

u '=(P,Q).

f) We have (yo,O)s(UxH)nNr(Y) and the diagram (UxH)nNf(Y) el +H

243

Subimmersions

Since

DT ( 0 0 ) (u1r~2 )=D(*-') ( 0 )( ~ 1+)(B* * (Yo))-l(u2) Dy(0,O) :ExG+H is r

(YO) 1

( ~ 2=)

=@(Yo) (U,)+(B I a linear homeomorphism. Therefore e is a local diffeomorphism of class p f f at (yOrO)eN (Y), (2.2.6), as a(N (Y))=#.o In fact we have the more general situation: "Let (H,) be a real Hilbert space, AH a finite linearly independent system of elements of I(H,R), Y a differentiable manifold of class p+l (pzl), f:Y+H+ an immersion of class p+l, AH + + ,1 ,(H,AH)) the natural chart of HA and c'=(H,lH,H) the c=(HA H ' A H natural chart of H (note that jf:Y+H is also an immersion of + class p+l, where j:HA 4 H is the inclusion map, and H +

For

every

yeY

let

-1

Tf(Y)=(0cf(Y,)

Y

Of(y):H+T

T (f)T YcH Y Y

and

us consider f f J cH Ny(Y)=[Ty(Y)]

the

sets

(note

that

(H+ ) is the natural isomorphism, T f f(y) AH Y injective map, Nf (Y)e Tf (Y)=H, [O,(y)] -1 Ty(f)[TyY] Y T Y C

is

an

Y H+ ) i=HA + is a quadrant of T Y (Tf(y) AH H ow we if f(y)eHA0 and Tf (Y)~H; if ye(int(Y)) and f(y)EHo 1 ) . Y H AH H and take the sets Tf (Y)={ (y,v)~YxH/veTf (Y))cYxH f f course, Y f T (Y)=TY jf (Y)r N (Y)=( (y,v)~YxH/veN (Y) )cYxH, (of Y Y , Tf(Y)=Tjf(Y) and N f (Y)=Njf(Y)). Ny(Y)=Ntf(Y) f Then : a) Tf (Y) and N f (Y) are totally neat submanifolds of class p

244 of

Chapter 5

YxH.

Moreover

the

[ [ocf(y))-lTy(f)

e(Y/v)= YI

map

!:T(Y)+Tf(Y)

defined

by

(v)) is a diffeomorphism of class p from

T(Y) onto Tf (Y). b) The maps tl:T f (Y)+Y, t2:N f (Y)+Y defined by tl(y,u)=y, t (y,u)=y are submersions of class p. 2 c)

The

maps

P:YxH+T f (Y)

J

and

Q:YxH+N f (Y)

J

defined

such that PoP=P, QoQ=Q and p2Q=p2-p2P, where p2:YxH+H 2-projection.

by

is the

f

f (YlV))/(y,uWf (Y) I (Y/V)EN (Y)1 is a submanifold of class p of Tf(Y)xNf(Y) and it is also a submanifold of (YxH)x(YxH). d) Tf W x y N (Y)=(

((YlU) I

e) The map a:T f (Y)xyNf (Y)+YxH defined by a( (y,u), (y,v))=(y,u+v) is a diffeomorphism of class p whose inverse is a-1 (y,v)=(P(y,v),Q(y,v)). Therefore, Tf (Y) and Nf (Y) are closed submanifolds of YxH. ) ) and there exist an open AH neighbourhood G of aY in Y and an open neighbourhood Vo of 0 in H such that [VOnN f (Y)]+f(y)cHi for yeG and [VOnN f (Y)]+f(y)caHA+ Y H Y H for y&Y. Then there is an open neighbourhood A of ( (y,o)/y~Y) in + Nf(Y) such that the map of class p, e:A+HA , defined by H e(y,v)=f(y)+v is a local diffeomorphism of class p at every (~,o)~A~N~(Y).

f) Suppose that a(Y)=f”(a(H+

If

then Tyf (a(TyY)ilea( [Tf(y) H+ AH Ii)

and

(T f) (int(T Y)i)cint([T H+ 3 i) for yeY. Y Y f(y) AH

Lemma 5.1.17 (R. Godement) Let g:U+M be a local homeomorphism, X a closed set of M and

Subimmersions

245

a continuous section of g, that is ges=lx. Suppose that g ( U ) is a Hausdorff paracompact space. Then there is an open neighbourhood W of X in M and there is a prolongation of s to a continuous sections, s:W+U, of g such that s ( W ) = U , is an open set of u. s:X+U

Proof The map g is open, g(U) is an open set of M and X c g ( U ) . Let X-3(U)

*

a) If x e X , then there are a point ydJ such that g(y)=x, an open neighbourhood Vy of y in U and an open neighbourhood Vx :Vy+Vx is a homeomorphism. of x in g ( U ) such that VxnX=$ and g I VY 'l:Vx+VycU is a local section of g. Then sx= g IVY)

[

b) If XEX, then S ( X ) E U and there are open neighbourhoods Vs(x)cU and V x c g ( U ) of s ( x ) and x respectively such that * V s ( x ) - W x is a homeomorphism. Since s is a continuous map, g l V w*

x x there is an open neighbourhood V1X of x in g ( U ) such that VlcV and s (V:nX) cVs

.

Consider Vs

=g"

(V:)

the nVs

open

. Then

neighbourhood nvs (X)

of

s(x)

in

U,

x=vlx

=vXng 1 ( V S (X) =vXnv 1

and is a local section of g. From the statements a) and b) we obtain a family of such that continuous local sections s A : V A r g ( U ) + U of g : U + g ( U ) S =S and ( V A ) A E A is an open covering of g ( U ) . A I vAnx I vAnx Since g ( U ) is a paracompact space, there is a locally finite open refinement ( U . / j e J ) of ( V h / A s A ) . For every jeJ we choose 3 A . e h such that U . c V A and then we put s .=sA : U . + U . Since g ( U ) 3 l j jluj 3 is a Hausdorff paracompact space, g(U) is a normal space and there is a contraction ( W . / j E J ) of ( U . / j e J ) . 3 3

Chapter 5

246

.

Let us consider the set W=(yeg(U)/yEw.np +.s. (y)=s (y)) Of 3 k 3 k course XcWcg(U) and we take the map s:W+U defined by s(y)=s. (y) 3 where yew Then it is clear that gOZ=lw and Z j* I X=s' Now we shall see that W is an open set of g(U). Indeed, let xcW. Then S(X)EU and therefore there is an open neighbourhood V of s(x) in U and there is an open neighbourhood Vx of x in g(U) such that g ,v:V+Vx is a homeomorphism. We can choose an open set A of g(U) such that:

.

1) xeAcVXcg (U)

2) H=(j1,.

The

set

..,jk)cJ.

3

3 ) xcsj

.

4) AcU

.

n...nR jl jk

n...nU jl jk

is

H=(jcJ/Ani+i.+t$)

s (A)cV for ji s (x)=. .=s (x)=S(x)N. j,I jk

all

5)

a

finite

iE( 1,

set;

say

...,k] ,

since

s(x)

in V such

.

Then there is an open neighbourhood Vs(x) of

-

that

glvS(x)

-.Vs(x)+A

is a homeomorphism.

Finally we note that S

-.

..=s

Jl{AjklA and xcAcW. Hence W is an open set of g(U).

-

~ is ~ a continuous ~ = s 3 map, goE=lw and EIx=s. Therefore is a continuous map since (w.nW/jcJ) is a locally finite family of closed sets of W. 3 For the map s:W+U

we have:

~

,

~

s

Let zcW. Then there are an open neighbourhood VS(') in U and there is an open neighbourhood Vz of z in

of s ( z ) W such

~

Subimmersions

that g

-

247

*Vs(z)+Vz is a homeomorphism. Since

-

Ivs(z)

s

I VZ

:Vz4J is a

continuous map, there is WzcVz such that s(Wz)cVs(z), where Wz is an open neighbourhood of z in Vz. Let

-

WS(Z)=

[. @(

us

consider

-

-' ( Wz)cVs(z)

.))

the

open

-

of s ( z ) in VS(').

neighbourhood

-

If tsWS('),

then

g(t)eWZ and sg(t)eVS(Z). Thus gsg(t)=g(t) and Eg(t)=tes(W)=U,.

-

We

have proved that Ws(z)cUo and therefore U, is an open set in U.o Corollary 5.1.18 Let f:Y+Y' be a differentiable map of class p and X a closed subset of Y'. Suppose that: l S t )Y'

is a Hausdorff paracompact space.

d

2 ) There is a continuous map, s:X+Y such that fos=lX.

s(x)

-

3*) For every xeX, f is a local diffeomorphism of class p at

Then there are an open set , U of Y and an open set W of Y' with XcW such that f :Uo-IW is a diffeomorphism of class p and I uO

Proof By 2.2.9, a), the set U=(yeY/f is a local diffeomorphism at y) is an open set of Y and s(X)cU. Thus s:X+UcY is a continuous section of f From the preceding lemma we get an open U' neighbourhood W of X in Y' and a prolongation of s to a continuous section s:W+U of f such that s(W)=U, is an open set IU of U. Then f :Uo+W is a bijective local diffeomorphism of class IuO p and therefore f :Uo+W is a diffeomorphism of class p,

I

I uO

Chapter

248

(2.2.9.b))

5

.o

Proposition 5.1.19

+

Let (H,) be a real Hilbert space, a finite linearly independent system of elements of Y(H,R), Y a differentiable manifold of class p+l (prl) and f:Y+H+ a closed embedding of AH class p+l. Suppose that f-'[aH&]=aY and there exist an open neighbourhood G of aY in Y and an open neighbourhood V 0 of 0 in H such that [VOnNy(Y) f ]+f (y)cHAH + for a l l ycG and

[VOmY(y) f ]+f (y)ca[HiJ

for a l l yeay.

Consider the totally neat submanifolds of class p of YxH, f f f + T (Y) and N (Y), and the map e:AcN (Y)+HA of class p defined by H e(y,v)=f(y)+v, (by the generalization after 5.1.16, e is a local diffeomorphism of class p at (yl0)~AcNf(Y) for a l l yeY and for a l l ycY vith f(y)cH 0 ), [Ty(Y)]L f =Ty(Y)nH+ f Y AH

+

Then: a) There are an open set RA of AcN f (Y) vith Yx(0)cRA and an + vith f(Y)cW such that e .R +W is a open set W of HA H In*' A f diffeomorphism of class p and eo[=f, vhere [:Y+N (Y) is defined by E(Y)=(Y,o). b) f(Y) is a neat submanifold of H+

and the map n:W+W

%I

defined by

is a map of class p such that n(W)cf(Y) and nf(y)=f(y) for a l l yeY. Hence n:W+f(Y) is a submersion of class p a t every f(y)Ef(Y). Finally n(aW)ca(f(Y)) and there is an open set W1 of W such that f(Y)cW1 and n :Wl+f(Y) is a submersion of class p, Iwl (4.1.15).

Subimmersions c)

Suppose

that

for

every

249

yeB(Y)

there

is

an

open

+ and +f(y)cHA H Then for every yeY, there is an open

neighbourhood Wo of 0 in H such that

Y

+

+f(y)cBHA , H neighbourhood Vo of 0 in

(Ty(Y))i and

Y

there is

neighbourhood Vy of y in Y such that f ( y ) + for

all

-

ey(u)=f n' p, vhere

urV

Y

and

the

map

[f(y)+ [oCf(Y))-lTy(f)(u)],

C= [HiH,i, (H,AH))

an

open

-lTY(f) (u)eW1

e *Vo+Vy

Y' Y

defined

by

is a diffeomorphism of class

is the natural chart.

d) Suppose that there is an open neighbourhood Wo of 0 in H such

that

for

all

and H Then there exist an open set Ak of

yeB(Y)

.

+

Won

*

*

(TyY)i and an open set Ak of Bk(Y)xY such that A Bk (Y)cAk'

C YeBk (

( ( y , o)/YEBk(Y)IcAk

and

the

map

-1

TY (f)(v)) is a

Ek(y,v)=(y,f-ln(f(y)+(OE(y)]

diffeomorphism of

*

class p from Ak over Ak (krO). e) If H=Rq, then there is an open set W2 of W1 such that H+ >W>W1>W2>WZ>f(Y)and T[ %I I W, :W2+f(Y) is a proper map. Proof

(y,O)

a) For every yeY, there is an open neighbourhood V ( y ' o ) of in AcN f (Y) and there is an open neighbourhood V f(Y) of

e(y,O)=f (y) in H+

%

such that e:V(Y'o)+Vf(y)

is a diffeomorphism

of class p. Let U=(J V f ( ' )

us +

CH%I'

M=() V(YFo)cA and YCY Then the map elM:M+U is a local diffeomorphism of

consider

the

open

sets

YEY class p and therefore a local homeomorphism. On

the

other

hand

Chapter 5

250

f(Y) is a closed set in U, the map E:Y+AcN f (Y) defined by c(y)=(y,O) is a map of class p and the map s:f(Y)+M defined by s(z)=EOf-l(z) is a section of class p of e Then using I M' Godement's Lemma there are an open neighbourhood W of f(Y) in U and a prolongation of s to a continuous section s:W+M of e such IM that s(W)=RA is an open set of McA. Thus e :RA+W is a bijective I RA :R +W is a local diffeomorphism of class p and, by 2.2.9.b) , e IRA A diffeomorphism of class p, such that eOE=f and ), a closed embedding g:X+Hx[O,+) of class p with g-l(Hx(0))=a(X) (that is g(a(X))=g(X)n(Hx(O))) a collar neighbourhood (f,A) of such that class p of a ( X ) in X and an open set G in a(X)x[O,+) a(X)x(O)cG, the diagram A

AI'

>Hx[O,+)

is commutative, f(G)=G1 is an open set in X with a(X)cG1 and for all xeG1, N~(X)CHX(O)=(HXR)~ (see the remark after 5 . 2 . 1 6 ) . p2 Proof

By 5.1.28 there exists a collar neighbourhood (3,U) of a(X) in X of class p. Then ?(a(X)x[O,+))=U is an open set in X with a(X)cU and ?:a(X)x[O,+)+U is a diffeomorphism of class p such that ?(x,O)=x for all xd(X). Since X is a normal space and a(X) is a closed set in X, there is an open set V in X such that U>%V>a(X). Therefore

(?)-'(V) is an open set in a(X)x[O,+) and from 5.1.20 we get a map r:a(X)+lR+ of class p+l such that (x)~[O,r(x))c(?)-~(V) for all xca (X)

.

We consider the maps of class p

a:

u d a(X)x [ 0,+) -8P1 r a :U-m+

(X),

0

and

p:

u e a (X)x [ 0,+) -[p2

0 ,+)

.

3 Then (?)-'(x)=(a(x) ,p(x)), U1=(xcU//3(x)O and Abl-Aal-l>O, bbm-Aam-l>O. Then applying the

.

+.

...,

Parametrized Theorems of the Density of the Transversality preceding argument to these m m 1 1 IT (hb -hai-l) B s IT ( hbi-hai+l) +. 1 i-1 i=1

cubes we conclude that m + IT (hby-hay+l) Then dividing i=l

.

..

.

m 1 1 1 1 IT (bi-ai- x ) s IT (bi-ai+ x)+...+ IT (by-a?

m

i=l i=1 and therefore, since we V(C)SV(Cn )+. .+V(Cn 1 * 1 P

.

i-1 can

351

choose

A

i)

arbitrarily

large,

m

Whence

C V(Cn)zV(C) and C cannot have measure zero in Rm.o n=l

Corollary 8.1.8

If M is a measure zero set in Rm, then fi=@ and Rm-M is dense in Rm. In particular if C is a closed measure zero set in Rm,

0

then C is novhere dense in Rm i.e. [email protected] Proposition 8.1.9

If p(bl-al)+.

..+(bn-a,).

...

Chapter 8

352

Proof Suppose that alsa2s.. .sari. From the hypothesis we deduce sbn. Moreover [a ,b ] r ~ [ a ~ + ~ , b ~ + ~for ] = # lsksn-2. that blsb2s k k Hence (b1-a 1)+(b3-a3)+(b5-a5)+. .!(T nI)+. .+l(T nI). al ar

.

a

Hence

the

family

(

- i /i=l,.. (Ta n1)xC.

i 3 closed m-rectangles covers the set M and

Whence M has measure zero in Rm.o

.,r,j=1, - ‘”ai) * *

of

Chapter

354

8

Note that after a linear change of coordinates the hypothesis "for every a&, Mn( ( a)xRm-') has measure zero in Rm'llg can be changed into "there is a straight line of Rm such that for every hyperplan H of W m orthogonal to r, HnM has measure zero in

#-L. 8.2. Subsets of Measure Zero in Finite Dimensional Manifolds

Definition 8.2.1

Let X be a differentiable manifold of class p (prl) and McX. Suppose that dimx(X) O b-a and EC-. 2 Hence im(z)=im(ci)=f(J)q(K) is a compact set in X, ci being a continuous map. Since X is connected and Hausdorff, f(J)ug(K) is an open set and ';ii is a bijective map. Then f(J)ug(K)=X and iji:H/S+X is a homeomorphism. Then by 4.3.14 S is a regular equivalence relation and therefore there is a unique differentiable structure [a] of class p on H/S such that the map p:H+H/S is a submersion of class p. Moreover the manifold (H/S,[d]) has not boundary and by 4.3.6 -ci:(H/S,[d])+X is a map of class p. From the definition of ci diffeomorphism of class p. Hence

we see that ci is a local a is a diffeomorphism of class

P. Finally it is clear that (H/S,[A]) is CP-diffeomorphic to S 1 with the natural differentiable structure, which ends the proof of the lemma (the case of negative slope has a similar treatment) .n Now we shall finish the proof of the theorem.

.

suppose that X is not diffeomorphic to S1 Then we must prove that X is CP-diffeomorphic to an interval of LR which is not a single point. First we note that for every XEX, the map 1 (x,.) 1 :Tx(X)+R is a norm in Tx(X) Let x be a point of X and

.

Parametrized Theorems of the Density of the Transversality

377

c=(U,(p,( R , h ) ) a chart of X such that xeU and U is a connected set (A=# or A=(A)). Then p(U) is an interval I of W. We denote by a the map of class p, p-':I+U. For every t d let us consider the map L

where co is the natural chart of I. It is clear that norm in R . Thus for every tsI there is g(t)>O such that Then

g:IM

is

a

map

of

class

p.

1

is a

it-g(t).l

Consider

t 7(t)=/ g(s)ds for every te1. Then r'(t)=g(t)>O

1 it

tOEI

I. and

and therefore

7:1+7(1)=J is a diffeomorphism of class p and J is an interval of R.

Let B:J+X be the map defined by @=a7-'. Now we claim that the pair (JIB) is a parametrization of X by the arc-lenght. We know that J is an interval of W, B is a map of class p, B(J)=U is an open set of X, B:J+B(J) is a diffeomorphism of class

= g ( r - ' ( s ) ) . Ill=g(v-'(s)). Hence the pair (J,B) parametrization of X by the arc-lenght, as claimed.

is

a

Let P be the set ( (J,f)/ (J,f) is a parametrization of X by the arc-lenght). Now one defines the following relation over P "(J,f)s(J',f') o JcJ' and f' IJ=f" It is clear that M#, s is a partial order and inductive set.

(P's)

is an

Chapter 8

378

Let (L,h) be a maximal element of (P,s). Then h(L) is an open set of X. Moreover h(L) is a closed set of X. Indeed, if xeh(L) and x@h(L), then we construct (J,S) as before. Thus xeS(J) and B(J)nh(L)*@. Since X is not diffeomorphic to Sl, by the preceding lemma, there is a parametrization of X by the * * * * * arc-lenght (J ,f ) such that LcJ , f*IL=h and f (J )=h(L)uS(J)

*

*

and therefore there is teJ such that f (t)=x. Then t@L, M J * and (L,h) is not maximal, which is a contradiction. Hence xsh(L), h(L) is a closed set and h(L)=X, since X is a connected space.0 The following lemma allows to approximate differentiable maps by continuous maps. Using this lemma, the preceding classification theorem and Brown theorem we shall prove Brouwer's fixed point theorem. Lemma 8.3.13 Let X be a manifold of class p that admits partitions of unity of class p, E a real Banach space, f:X-+E a continuous map, A=(Mi/ieI) a locally finite family of subsets of X and (ci)iEI a family of real numbers with ci>O for every icI. Then there is a for map g:X+E of class p such that for every i d , ~~f(y)-g(y)~~0 if ( iEI/WxnMif#)f# and 6 X=1 if (ieI/WXnMi*#)=$. Since f is continuous, there is a neighbourhood Ux of x in Wx such that IIf (y)-f (x) for every yeUx.

Parametrized Theorems of the Density of the Transversality

379

By the hypothesis there is a partition of unity of class p I (hx/xcX) subordinated to the open covering U=(UX/xeX) of X.

c hX(y) .f(x) for every XCX y ~ x .It is clear that g is a map of class p. Moreover if im(f)cC, then g(y)€C for every yeX, because of that C hx(y)=l. Let g:X+E be the map defined by g(y)=

xax

(since IIf (y)-f (x)U < G X ~ e i for ycUx and hx(y)=O for yeUx.o Theorem 8.3.14

(Brouwer Fixed Point Theorem)

S l ) f:Dn+Dn a continuous map. Let Dn be the set ( x e W n / ~ ~ x ~ ~and Then there is xeDn such that f(x)=x. Proof Suppose that the theorem is not true. Then, since Dn is a compact set, there is 6 > 0 such that IIx-f (x)IIrS>O for every xeDn. If we

put

X=D,,

E=R

n

,

C=Dn, #=(Dn)

preceding lemma, then we find a map of class

and m,

{&=

i} in

the

g:Dn+Dn such that

6

llf (x)-g(x) for every x‘Dn. It is clear that the map g has not 6 fixed points, since if g(x)=x, then llf(x)-xl[=llf(x)-g(x)II 0 , (p is a map of class O D . Then, by x-9 (XI Brown theorem 8.3.11, there is a regular value of (p, yesn-'. Let M=(p-l(y)cDn. For every XEM, (p is a submersion of class m at x. Thus by 4.2.11, M is a closed submanifold of class 00 of Dn, M i n t (D,) *#, a (M)=MSn-l and dimz (M)=1 for every zeM. Since (p(y)=y, yeM and therefore yEa(M). Let K be the Y connected component of y in M. Then K is a compact connected and Y Hausdorff 1-dimensional manifold. Then, by the classification theorem 8.3.12, Ky is Cm-diffeomorphic to a closed bounded interval of R which is not a singleton, since yEB(K ) Hence there is zca(K ) with z+y, which Y Y is a contradiction since (p (z)=y and (p ( 2 ) =z , 2 d - l .

.

Notice that the preceding reasoning with the function proves the following result:

(p,

"Let X be an n-dimensional compact Hausdorff manifold of class pr2 with smooth boundary. Then there is not a map (p:X+a(X) of class p such that (p I a (X)=la (X)* In the preceding proof, we have used a common technique of Differential Topology. Namely, a continuous function f is approximated by a differentiable function g and after picking a regular value x of g, it is the submanifold g-'(x) which gives some relevant information. The generalization of this will consist of substituting the regular value x of g by a submanifold transversal to g at x.

8.4. Smale's

and Quinn's

Density Theorems

In this paragraph we generalize Brown's theorem to infinite differentiable manifolds and obtain Smale's and Quinn's density theorems. In order to do it, we introduce the so-called

Parametrized Theorems of the Density of the Transversality 381 Fredholm differentiable functions. Definition 8.4.1 Let E,F be real Banach spaces and u:E+F a continuous linear map. We say that u is a left Fredholm operator if ker(u) is a finite dimensional linear space and im(u) is a closed set of F. If u is a left Fredholm operator, the number dim(ker(u))-codim(im(u))eZu(-a) will be called index of u and will be denoted by ind(u). If u is a left Freholm operator and ind(u)eZ, then u will be called Fredholm operator. The set (u:E+F/u is a Fredholm operator} will be denoted by @(E,F) and the set (u:E+F/u is a Fredholm operator with ind(u)=keZ) will be denoted by Ok(E,F). It is clear that @(E,F)=(j Bk(E,F). The set (u:E+F/u is a left Fredholm operator) keZ will be denoted by d(E,F). It is clear that e @ (E,F)=@(E,F)UO-~(E,F), where @-=(E,F) is the set (u:E+F/u is a left Fredholm operator with ind (u)=-a). Proposition 8.4.2 Let u:E+F be a linear continuous map. Then ud(E,F) if and only if ker(u) is a finite dimensional linear subspace of E and im(u) has finite codimension in F. Therefore if ue@(E,F), then ker(u) admits a topological supplement in E and im(u) admits topological supplement in F, (we shall shortly say that u is double splitting).

Suppose that dim (ker(u)) . Suppose now that Y is a T1 space. Then f€Vf c. ( (Y- ( y)) xY ' ) u( f-l (G' ) xG ' ) The following proposition gives another description of the basis as. Proposition 9.2.2 Let

Y,Y'

be

topological

spaces

and

consider

the

set

Spaces of Differentiable Haps *

437

f

/faC(Y, Y') ,E=(ci)iaI is a locally finite family of closed sets of Y, U=(Ui)ieI is a family of open sets of YxY' and for every i d , r cUi)' where Vr =(gac(Y,Y')/r (EIU)

Bs= { V

(C,U)

-

every kI).

*

Then BS=Bs. Proof

*

f f f Hence fsScfss. If VW€Bs, it is clear that Vw=V ((Y) I (W)) *

*

asS. Then for every i€II the set Let us consider V f (GIW wi=uiv[(Y-Ci)xY~] is an open set in YxY' and rfcWi. For every

. .

ieI, YxY' -Wi=C. XYI-U1CC1xY, and ( CixY } ieI is a locally finite 1 ~~~ a locally finite family family in YxY', so that { Y X Y ' - W ~ ) is of closed sets of YxY'.

(YxY'-Wi) is a closed set in

Therefore iaI

YxY' and W=() Wi is an open set in YxY'. 1€I f =vf .o VWaBs, and we conclude that V (EIU) w

Finally rfcW, hence

Corollary 9 . 2 . 3

Let Y,Y' be topological spaces, E=(Ci)ieI a locally finite family of closed sets of Y and 8 ' = { G ! 1 } i d a family of open sets for every k I ) is of Y'. Then the set .

Then for every yeY, there 9(Yi) h(Y)EV and exists id such that yhCicVYi , 9(Yi) f f (Y,h(Y))€VYiXV cW. Hence rhcW, hd7, and g e S I O r > c V W , which ends the proof. Remark 9.2.5

In the preceding proof the open sets Vg(y) can be chosen in a fixed basis of Y'. Then the following result is also true: "Let Y,Y' be topological spaces and B' a basis of the topology of Y'. Suppose that Y is a paracompact regular space. Then ~~*=(=() ]=l j j by 9.2.4, TscTc. b) If the maps of C(Y,R) distinguish points, then Y is Hausdorff. Hence if, moreover, Y is compact, then by a) Tc=Ts. Conversely, suppose that the maps of C(Y,R) distinguish points, Y * is a T1 space that contains an arc, T c=T s and Y is not compact.

Let y;=u(O), and the rfcW and

u: [ O,l]+Y* be a homeomorphism from [0,1] onto im(u) and y;=u(l). Consider the constant map f :Y+Y' of value y; open set of YxY', W=Yx(Y'-(y;)). Then it is clear that f VWeTs=Tc.

Let K1,. of Y'

..,Kn be

such that

compact sets of Y and G ; ,

fe be an element of ZccTC, where K is a compact set in Y and GI is an open set in Y', such that fr. Thus few' (f)ccK,G'> and TccT ,.o C

.

Proposition 9.2.16 Y be a TI s p a c e w h i c h i s n o t c o u n t a b l y c o m p a c t , Y' a d i v i s i b l e T1 s p a c e and y'cY'. Suppose t h a t y' i s not i s o l a t e d and h a s a c o u n t a b l e o p e n b a s i s of n e i g h b o u r h o o d s (V~'/ndti). Let

Then Tco +Ts'

Pro0f Since Y is a T1-space which is not countably compact, there exists a infinite countable set C=(yn/nd4)cY which is closed discrete. Let f :Y+Y' be the constant map defined by f (z)=y' for every and let U be the open set of YxY' defined by [ (J ((y,)xV:')]u[ (x-C)xYt]cYxYt. Then rfcU. Moreover we can new suppose vY' SVY' for every new. n+l n zcY

Let W f be an open neighbourhood of the diagonal A, of Y'xY'.

Spaces of Differentiable Maps

451 I

Then

(y',y')cA'

and

there exists

nOcH

t

such

that Vy xVy CW'. "0 "0 Consider neH with nznO, y>V:'-V:l, and g=C ,:Y+Y'. Of course Yn f gcw' (f) but g N u since (yn+l,g(yn+l))=(yn+lly~)eU and therefore f f and VUeT o.o r is not contained in U. Thus VUcTs 9 C Corollary 9.2.17 Let Y be a paracompact Hausdorff space and Y' a divisible T1 space that contains an arc. Then T O=Ts if and only if Y is a C compact. Moreover if Y is compact, then TCO=Ts=Tc.

Proof If T co =Ts' by 9.2.16,

we have that Y is countably compact

and therefore compact (see [M.O.P. X.4.211). If Y is compact, T =T O = T c . ~

s

then Ts=Tc

,

(9.2.6)

,

and by

9.2.15,

c

Next we shall study the Baire properties of the spaces of continuous maps. Definition 9.2.18

Let Y be a topological space, (Y',d') and H a subset of C(Y,Y').

a pseudometric space

We say that H is uniformly closed if for each sequence (fn)ncN of elements of H which converges uniformly to fcC(Y,Y') it follows fcH. If X is a set, (X',U') a uniform space and M is a non-empty subset of X, then the set BM=(W(M,U')/U'sU') is a basis of a uniformity on the set A(X,X') of all maps f:X+X', where W(M,U')=( (f,g)EA(X,X')xA(X,X')/(f

(x),g(X))EU' for every xcM).

Chapter 9

452

Moreover if fcA(X,X') and S=(SdaA(X,X'),daD,s) is a net in A(X,X'), then S converges to f in (A(X,X'),T ) if and only if

54

for every U'EU' there exists dOED such that Sd(x)aU'[f(x)] every xaM and every dbdol(U'[f(x)]=(x'aX'/(f(x) ,x')EU')).

for

If M=X, then the uniformity Ux is called uniform convergence uniformity and the topology T on A(X,X') uniform convergence uX topology. The space (A(X,X'),T ) will be denoted by A (X,X'). uX tU If X is a topological space and (X',U') then the set C(X,X')

a uniform space,

is a closed set of (A(X,X'),T

)=Atu(X,X'). uX a pseudometric space, (X',Ud,) the Let X be a set, (X',d') associated uniformity and (Ud,) the associated uniform convergence uniformity on A(X,X'). Then (A(XIX'),(Ud,)x) is a pseudometrizable space and the uniformly closed sets are closed sets in the uniform convergence topology. The notion of uniformly closed set is attached to the equivalent metric considered in the space (Y',d') as shows the following example: Let us consider Y=R,(Y',dl),(Y',d2) usual metric and d2(xly)= 1+

- &I,

x

if

where Y'=R,

dl is the

f=lR and for each naN, x K ,for every nem.

f

Then for every ndN there exist xneKn and pneN such that (xn)+f(xn)I (PnInep( is strictly increasing and ( x ~ is) a~

Pn sequence of pairwise different points. Then it is clear that ( x ~ has ) no ~ cluster ~ ~ point and therefore ( x ~ is) closed ~ ~ and ~ discrete. Thus by the Tietze extension theorem, there exists a continuous map c :Y4R' such that c ( xn)=d' ( f ( xn) ,f ( xn) ) >O Hence Pn f ':Be (f) for every ndN, which contradicts a) .o Pn

.

Corollary 9.2.22 Let X be a metrizable connected compact space, Y a locally compact Lindel6f Hausdorff space, ( Y r l d l )a pseudometric space and f:X+(C(YrY'),Ts) a continuous map. Then there exists a

~

~

Chapter 9

456

compact

subset

K of

Y such that the map

f :X+Y'

defined by

Y

f (x)=f(x) (y) is a constant map for every yeY-K.

Y

Proof Since Y is a locally compact Lindelof Hausdorff space, Y=(J fin where Kn is a compact set and K n + l ~ K nfor every new. new For every XCX there exist cx>O and a compact subset Kx of Y such that for every yeY-Kx, the map Z H ~(z)(y) is constant over d (x), where d is a metric in X that describes its topology.

Bc,

Indeed, if it is not true, then there exist xOeX, a sequence (xn)new in X that converges to x, and a sequence (yn ) new in Y without cluster points such that f (x,) (yn)+f(x,) (y,) for every new. By the preceding proposition, the sequence of continuous maps f (xn) does not converge to f(x,) in the topology Tsl which contradicts the continuity of f. Since

x

is a compact space, X = B ~ (x,)w.

d ..uBcx

x1 end the proof by taking K=K

,

(x,)

and we

P

u...uKx .n x1 P

Next we shall see with an example that Proposition 9.2.21 is not true if we weaken the hypothesis IIY is a locally compact Lindelof Hausdorf f space" to read I*Y is a Hausdorf f paracompact space". Example 9.2.23 Take Y=Q, Y'=R, fn 0 if Ixlrl/n

defined

by

for every nCN and f=O. Then (fn)new converges topology T, but 1) of b) in 9.2.21 fails

to

f

in the

Spaces of Differentiable Maps Let Y be a topological space and H(Y,Y)={fcC(Y,Y)/f homeomorphism). Our aim is to study the space (H(Y,Y),Ts).

Proposition

457

is a

9.2.24

Let Y,Y' be two homeomorphic spaces. Then the map a: (H(YIY') ,TS)--(H(Y',Y) ,Ts) f -f-l is a homeomorphism.

Proof Let fcH(Y,Y')

and W an open set of YxY' such that rfcW. If

we denote the set {(y',y)cY'xY/(y,y')eW) by W", then W-l is an open set of Y'xY. Moreover r cW-l. Indeed, for every f-l Y'EY', (Y',f-l(Y'))=(f(Y) rf-lf(Y) )=(f(y) ,Y) and (Y',f-l(Y') Finally a(WnH(Y,Y'))=W-lnH(Y',Y) .o Proposition

9.2.25

Let Y be a regular paracompact space. Then (H(Y,Y),Ts) is a topological group for the composition of homeomorphisms. Proof

It remains to prove that :H (Y,Y)xH (Y,Y)+(H(Y,Y) 0

(fl9)

-of

is a continuous map with respect to the topology Ts.

Chapter 9

458

Let fO,gO%H(Y,Y) and W an open set of YxY such that

r

cW. gof0 90f0(Y)

For every y%Y there exist open neighbourhoods Vy and V g0fo (Y) g0fo (Y) such that VyxV CW and gOfO(Vy)CV Since Y is paracompact, there exists a locally finite open refinement U=(Ui/i%I) of (VY/yeY). Moreover since Y is normal, there exists a contraction V=(V.1 ) i d of 21 and another W=(Wi)iEI of Y.

For

every

icI,

consider

y.eY 1

W . C ~ ~ C V ~ C ~ ~for C U every ~ C V ~k I~ . 1

VLO=nH(Y,Y) is an open neighbourhood of fo

Let us consider feVfo and gcVgo. Then for every y%Y there exists iOEI such that yewi Consequently f(y)Efo(Vi )cfo(vio) 0

.

0

and gf (y)sgOfo(Ui,). Hence

Thus

0

f (V OxV

)cVw 'fO'

and

0

is continuous at (fo,go). a

Ouestion Is H(Y,Y) an open set of Ts? Proposition 9.2.26 Let Y be a regular paracompact space and (Y',d') a locally compact metric space. Then the set Prop(Y,Y')=(f:Y+Y'/f is a proper map) is open and closed in Ts.

Spaces of Differentiable Maps

459

Proof Suppose that feC(Y,Y') is not a proper map. Then there exists a compact set K' of Y' such that f-l(K') is not a compact set.

...,

UA Since Y' is a locally compact space, there exist Ui, open sets of Y' such that K'cUiu.. .uUA and are compact. n and &=d ' (K' ,Y ' (Uiu. uUA)) >O Then L'=(x'eK'/d'(x',K')& c-'u uti; is a compact set of Y' that 2 ) u1 contains K'.

...

",.. .,u'

..

-

is an open neighbourhood of

It is easily checked that B:'(f)

2

f

in

Ts.

Now

d'(f(y),g(y)) of f in Ts gewlDWl>h(Y ) and rlvl:wl+h(Y') is a proper map. & f: Y4R

the continuous map defined by 2 c f (y)=min(l,d(hf(y) ,R (m+l) -Wl)) for every yeY and let d' be the metric in Y' (which describes its topology) defined by d ' ( y ' , z ' ) = ~ ~ h ( y ' ) - h ( z ' ) S . By 9.2.9.a), f€Bd'(f)eTs. Let g be an E,& d' element of BE (f). For te[0,1] and yeY we have f Let

+

be

such

462

Chapter

9

2

Consider the continuous map F1: Yx [ 0,l]+IR(m+l) defined by Now we shall see that F1 is proper. F1(y,t)=thf (y)+(l-t)hg(y)

.

Let K' be a compact set of W (m+1)2 and S=( (Y~,~~)EF;~(KI) ,dcD,a) -1 a universal net in F1 (K'). Then the universal net converges to zOcK'. Since [0,1] is F1 (Sd)=tdhf(yd)+(l-td)hg(yd) compact we can suppose, without loss of generality, that the universal net (tdE[ 0,1] ,daD,a) converges to to. On the other hand

1 hf (Yd

-' 0 1 ' hf (Yd - ( tdhf ( Yd +

-'O 1 "+ hg (Yd)1' 1 tdhf (Yd)+ ( l'td) hg (Yd)-'O 1 "f (Yd)+ 1 F1 +iF1(sd)-zO~. Thus there exists do such that for every dcD with dosd, hf(yd)cBi(zo). Again, since hf is proper, we can suppose that the net (ydldcD,a)converges to yo in Y. Thus S converges to (yO,tO)cF;l(Kt) in Yx[O,l] and Fil(K') is a compact set. +(l 'td)

Finally, the map F=h-lr

F :Yx[O,l]+Y'

lrj 1 1

is a proper map such

that F(y,O)=g(y) and F(y,l)=f(y) for every ycY. In particular g is proper and':B ( f)cProp (Y,Y' ) o f

.

Definition 9.2.29 Let Y,Y' be manifolds of class p, F a closed set of Y and f:F+Y' a map. We say that f is a map of class p, if there exist an open set U of Y with FcU and a map T:U+Y' of class p such that 3, F=f. Proposition 9.2.30 Let Y be a manifold of class l ' p that admits partitions of unity of class p, let Y' be a Hausdorff paracompact Hilbert manifold of class p+l with a(Y')=# and let F be a closed set of Y. Then CP(F,Y')=(g:F+Y'/g is a map of class p ) is a dense set in (C(FtY') ,Ts)

Spaces of Differentiable Haps

463

Proof Let f be an element of C(F,Y') in (C(F,Y') ,Ts).

and Vf a neighbourhood of f

By 3.3.7, there exist a real Hilbert space (H,) and a closed embedding of class p+l, h:Y'+H. On the other hand by 5.1.19 there exist an open set W of H such that H+W>h(Y') and a retraction r:W+h(Y') of class p. Let d f be the metric in Y', (which describes its topology), defined by d'(y',z')=llh(y')-h(z') 11. By 9.2.9 there exists a + d ' continuous map n : F 4 such that Bn(f)cV f and B;'(f)eTs. Since Y is a Hausdorff paracompact space, Y is normal and by the Tietze theorem there exists a continuous map nl:Y4+ such that

I F=='

On the other hand hof:F+H is continuous and by the Dugundji extension theorem there exists a continuous map F:Y+H such that

Since F(F)=hf(F)ch(Y'), set of Y and U1>F. For every XEF,

TI

1

we have that U1=F-'(W)

is an open

(x)=n(x) and by the continuity of n1 there

exists a neighbourhood Vx of x such that VxcUl and n l ( y ) > w for every y€VX

.

Moreover since hf(x)EW,

rhf(x)=hf(x) and r is continuous H =I (XI = and there exists €ix>O such that B8 (hf(x))cW, 2

1 r (y)-hf (x)1