Beginner's Course in Topology: Geometric Chapters [1 ed.] 3540909702, 3540909931, 3540135774, 0387135774

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Fuks Rokhlin Beginner’s Course in Topology

Springer-Verlag Berlin Heidelberg New York Tokyo

NUNC C O G NOSCO EX PARTE

THOMAS J. BATA LIBRARY TRENT UNIVERSITY

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/beginnerscourseiOOOOrokh

Universitext

D.B. Fuks

V.A. Rokhlin

Beginner’s Course in Topology Geometric Chapters

Translated from the Russian by A. lacob With 17 Figures

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Dmitrij Borisovich Fuks Laboratory of Bioorganic Chemistry Moscow State University, Moscow, USSR Vladimir Abramovich Rokhlin Department of Mathematics and Mechanics Leningrad State University, Leningrad, USSR Andrei lacob Department of Theoretical Mathematics. The Weizmann Institute of Science. Rehovot 76100, Israel

Title of the Russian original edition: Nachal’nyj Kurs Topologii: Geome tricheskie Glavy. Publisher Nauka, Moscow 1977 This volume is part of the Springer Series in Soviet Mathematics Advisers: L.D. Faddeev (Leningrad), R.V. Gamkrelidze (Moscow)

AMS Subject Classification (1980): 54-01, 54A 05, 54Bxx, 54Cxx, 54Dxx, 54Exx, 55Pxx, 5 5 Qxx, 55Rxx, 57-01, 57M xx, 57Nxx, 57Rxx, 57Sxx, 58Axx, 58C 25, 58D10, 58D15, 58E 05

ISBN 3-540-13577-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13577-4 Springer-Verlag New York Heidelberg Berlin Tokyo

Library of Congress Cataloging in Publication Data. Rokhlin, V. A. Beginner’s course in topology. (Springer series in Soviet mathematics) (Universitext) Translation of: Nachal’nyi kurs topologii / V. A. Rokhlin, D. B. Fuks. Bibliography: p. Includes index. 1.Topology. I. Fuks, D. B. II.Title. III. Series. QA611.R6513 1984 514 84-10657 ISBN 0-387-13577-4 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustra­ tions, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort” , Munich. © Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and bookbinding: Beltz Offsetdruck, Hemsbach 2141/3140-543210

Preface

This book is the result of reworking part of a rather lengthy course of lectures of which we delivered several versions at the Leningrad and Moscow Universities.

In these lectures we presented an

introduction to the fundamental topics of topology:

homology theory,

homotopy theory, theory of bundles, and topology of manifolds.

The

structure of the course was well determined by the guiding term elementary topology, whose main significance resides in the fact that it made us use a rather simple apparatus.

^n this book we have retained

ihose sections of the course where algebra plays a subordinate role. We plan to publish the more algebraic part of the lectures as a separate book. Reprocessing the lectures to produce the book resulted in the profits and losses inherent in such a situation:,,

the rigour has

increased to the detriment of the intuitiveness, the geometric descriptions have been replaced by formulas needing interpretations, etc.

Nevertheless, it seems to us that the book retains the main

qualities of our lectures: pedagogical features.

their elementary, systematic, and

The preparation of the reader is assumed to be

limited to the usual knowledge of set theory, algebra, and calculus which mathematics students should master after the first year and a half of studies. exercises.

The exposition is accompanied by examples and

We hope that the book can be used as a topology textbook. The most essential difference between the book and the

corresponding part of our lectures is the arrangement of the material: here we have followed a much more orderly succesion of topics.

However,

from our experience, a lecture course in elementary topology which exaggerates in the last respect is rather tedious and less efficient than one which mixes geometry with algebra and applications.

This

remark may serve as a warning to the teacher who would like to use our book as a guide. in its order;

In fact, it is by no means necessary to read the book

a reader who is interested in getting to the homotopy

groups or to any other topic sooner, can easily do so.

VI

Concerning the terminology and notation, we have tried to stick to standard usage, and have permitted ourselves only a few reforms.

For example, we do not use the terms "simplicial complexes"

or "CW-complexes", but simplicial spaces and cellular spaces?

not

"cofibrations", but Borsuk pairs;

not "fiber bundles"

products"), but Steenrod bundles.

There is even one term which we do

not use in the generally

accepted way:

(or "fibered

for us, a connected space

refers to what usually is called a linearly connected (or path-connected) space (we do not have a special name for the spaces which are usually called connected).

Furthermore, we have avoided using non­

standardized notations for standard objects.

In fact, in the majority

of cases, our notation is just an abbreviation of the corresponding term and can be understood by itself: projection, skeleton,

in - for inclusion,

for example,

pr

dim - for dimension,

stands for ske - for

bs - for base, etc. Topology requires a very precise set-theoretic language, and

this compelled us to devote a special attention to this language; is illustrated by pp. 1 “ 4.

this

we emphasize that on these pages we only

list the terms and notations, assuming that the objects themselves are known. In this book we rarely refer to the history of topology.

We

have even departed from the tradition that some theorems bear the names of their real or imaginary authors.

In return, we willingly have used

names of topologists in the terminology and notations. The organization of the text and the system of references may be briefly described as follows.

Each Chapter is divided into Sections,

each Section - into Subsections, each Subsection - into Numbers.

The

chapters, sections and subsections have numbers and titles, while the numbers are denoted only by their numbers.

Each fact announced without

proof is called Information, and is distinguished from the rest of the text by this title.

To refer to a section, subsection, or number

within the same chapter, we do not indicate the number of the chapter, and references within a section or subsection are similarly abbreviated. Examples:

the entries

§1.2

(Section 2 of Chapter 1), Subsection 1.2.3

(Subsection 3 of Section 2 of Chapter 1), and 1.2.3.4

(No. 4 of Sub­

section 3 of Section 2 of Chapter 1) are abbreviated, within Chapter 1, as § 2 , Subsection 2.3, and 2.3.4, respectively; entries is abbreviated within §1.2 is abbreviated within §1.2 respectively.

the second of these

as Subsection 3;

the third entry

and Subsection 1.2.3 as 3.4 and 4,

The Authors

Contents

Set-Theoretical Terms and Notations Used in This Book, but not Generally Adopted.................................

Chapter

1

TOPOLOGICAL

§ 1.

§3.

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

5

5 8 10

Topologies............................................. M e t r i c s ............................................... S u b s p a c e s ............................................. Continuous M a p s ....................................... Separation Axioms ..................................... Countability Axioms ................................ Compactness...........................................

21

CONSTRUCTIONS.............................................

26

1. 2. 3. 4. 5. 6. 7. 8. 9.

26 27

Sums................................................... Products............................................... Q u o t i e n t s ............................................. G l u e i n g ............................................... Projective Spaces .................................... More Special Constructions............................ Spaces of Continuous M a p s ............................ The Case of Pointed Spaces............................. E x e r c i s e s .............................................

11 15 19

31 34

38 42 46 49 55

HOMOTOPIES.................................................

56

1. 2. 3. 4. 5. 6. 7. 8.

General Definitions ................................ P a t h s ................................................. Connectedness and k-Connectedness .................... Local Properties...................................... Borsuk Pairs........................................... CN RS-Spaces........................................... Homotopy Propertiesof Topological Constructions ... E x e r c i s e s .............................................

56 60 61 65 66 70 72 78

S P A C E S ...........................................................................................................................

81

CELLULAR SPACES AND THEIR TOPOLOGICAL PROPERTIES...........

81

Fundamental Concepts.................................... Glueing Cellular Spaces from Balls....................

81

Chapter CELLULAR

§1.

SPACES

FUNDAMENTAL CONCEPTS....................................... 1. 2. 3. 4. 5. 6. 7.

§2.

1

1.

2.

2

86

VIII

§2.

3. The Canonical Cellular Decompositions of Spheres, Balls, and Projective Spaces................ 4. More Topological Properties of Cellular Spaces . . . . 5. Cellular Constructions ............................... 6. Exercises..............................................

88 89 94 98

SIMPLICIAL SPACES..........................................

99

1. 2. 3. 4. 5. 6. 7. 8. §3.

Euclidean Simplices................................... Simplicial Spaces and Simplicial Maps................. Simplicial Schemes .................................... Polyhedra.............................................. Simplicial Constructions ............................. Stars. Links. Regular Neighborhoods................... Simplicial Approximation of Continuous Maps........... Exercises..............................................

99 101 105 106 107 114 118 119

HOMOTOPY PROPERTIES OF CELLULAR SPACES ...................

120

1. 2. 3. 4. 5.

Cellular P a i r s ........................................ Cellular Approximation of Continuous Maps............. k-Connected Cellular Pairs ........................... Simplicial Approximation of Cellular Spaces........... Exercises..............................................

120 123 126 130 132

Chapter 3 SMOOTH MANIFOLDS ...............................................

133

§1.

133

FUNDAMENTAL CONCEPTS ...................................... 1. 2. 3. 4. 5. 6. 7.

§2.

§3.

§4.

Topological Manifolds................................. Differentiable Structures............................. Orientations.......................................... The Manifold of Tangent Vectors....................... Embeddings, Immersions, and Submersions............... Complex Structures .................................... Exercises..............................................

STIEFEL AMD GRASSMAN MANIFOLDS ...........................

133 141 150 156 161 166 171 171

1. Stiefel Manifolds...................................... 2. Grassman Manifolds .................................... 3. Some Low-Dimensional Stiefel and Grassman Manifolds ............................... 4. Exercises..............................................

171 177

A DIGRESSION: THREE THEOREMS FROM CALCULUS ...............

187

1. Polynomial Approximation of Functions................. 2. Singular Values........................................ 3. Nondegenerate Critical Points.........................

187 191 194

EMBEDDINGS. IMMERSIONS. SMOOTHINGS. APPROXIMATIONS . . . . 1. 2. 3. 4. 5.

Spaces of Smooth Maps.................................. The Simplest Embedding Theorems....................... Transversalizations and Tubes......................... Smoothing Maps in the Case of Closed Manifolds . . . . Glueing Manifolds Smoothly ...........................

185 186

197 197

200 202 205 208

IX

6. 7. 8. 9. 10.

Smoothing Maps in the Presence ofa Boundary.......... General Position...................................... Maps Transverse to a Submanifold..................... Raising the Smoothness Class of aManifold............ Approximation of Maps by Embeddings and Immersions....................................... 11. E x e r c i s e s ............................................

214 219 225 228

THE SIMPLEST STRUCTURE THEOREMS...........................

239

1. 2. 3. 4.

Morse F u n c t i o n s ...................................... Cobordisms and Surgery................................ Two-dimensional Manifolds ........................... E x e r c i s e s ............................................

239 243 255 263

BUNDLES..........................................................

265

§1 .

BUNDLES WITHOUT GROUP STRUCTURE...........................

265

1. 2. 3. 4. 5.

General Definitions................................... Locally Trivial Bundles............................... Serre Bundles......................................... Bundles With Map Spaces asTotal Spaces................ Exercises.............................................

265 267 269 273 276

A DIGRESSION: TOPOLOGICAL GROUPS AND TRANSFORMATION GROUPS.....................................

276

1. 2. 3. 4.

Topological Groups ................................... Groups of Homeomorphisms............................. Actions............................................... Exercises.............................................

276 282 285 296

BUNDLES WITH A GROUP S T R U C T U R E ...........................

297

§5.

233 237

Chapter 4

§2.

§3.

1. 2. 3. 4. 5. §4.

Spaces With F-Structure............................... Steenrod Bundles ..................................... Associated Bundles ................................... Ehresmann-Feldbau Bundles............................. Exercises..............................................

297 299 304 308 311

THE CLASSIFICATION OF STEENROD BUNDLES ...................

311

1. 2. 3. 4. 5. §5.

Steenrod Bundles and Homotopies....................... Universal Bundles..................................... The Milnor B u n d l e s ................................... Reductions of the StructureGroup...................... Exercises..............................................

VECTOR B U N D L E S ........................................... 1. 2. 3. 4. 5.

General Definitions................................... Constructions......................................... The Classical Universal Vector Bundles ............... The Most Important Reductions of the Structure Group................................... Exercises..............................................

311 316 319 321 323 324 324 331 336 34 3 345

X

§6.

SMOOTH B U N D L E S ........................................... 1. 2. 3. 4. 5. 6.

Fundamental Concepts ................................. Smoothings and Approximations......................... Smooth Vector Bundles................................. Tangent and Normal Bundles ........................... D e g r e e ............................................... Exercises.............................................

346 350 353 359 364 371

H0M0T0PY GROUPS.................................................

373

§1.

THE GENERAL T H E O R Y .......................................

373

1. 2. 3.

37 3 378

Chapter

5

4. 5. 6.

7. 8.

9. 10. 11. 12. § 2.

Absolute Homotopy Groups............................. A Digression: Local Systems ......................... Local Systems of Homotopy Groups of a Topological Space............................... Relative Homotopy Groups............................. A Digression: Sequences of Groups and Homomorphisms, and ^-Sequences....................... The Homotopy Sequence of a P a i r ..................... The Local System of Homotopy Groups of the Fibers of a Serre Bundle......................... The Homotopy Sequence of a SerreB u n d l e ............. The Influence of Other Structures Upon Homotopy Groups................................. Alternative Descriptions of the Homotopy Groups . . . Additional Theorems ................................. E x e r c i s e s ...........................................

THE HOMOTOPY GROUPS OF SPHERES AND OF CLASSICAL MANIFOLDS ................................... 1. 2. 3. 4. 5.

§3.

380 38 4 390 398 402 4 05 4 11 416 420 423 424

Suspension in the Homotopy Groupsof Spheres ......... The Simplest Homotopy Groups ofSpheres................ The Composition Product............................... Information: Homotopy Groups ofSpheres................ The Homotopy Groups of Projective Spaces and Lenses..................................... 6 . The Homotopy Groups of Classical Groups............... 7. The Homotopy Groups of Stiefel Manifolds and S p a c e s ................................. 8 . The Homotopy Groups of Grassman Manifolds and S p a c e s ................................. 9. Exercises..............................................

4 24 429 4 33 43 6

HOMOTOPY GROUPS OF CELLULAR SPACES .......................

445

The Homotopy Groups of One-dimensional Cellular Spaces....................................... 2. The Effect of Attaching Balls......................... 3. The Fundamental Group of a Cellular Space............. 4. Homotopy Groups of Compact Surfaces................... 5. The Homotopy Groups of Bouquets....................... 6 . The Homotopy Groups of a k-Connected Cellular Pair......................................... 7. Spaces With Prescribed Homotopy Groups ............... 8 . Eight Instructive Examples ........................... 9. Exercises..............................................

438 439 44 1 443 444

1.

445 446 448 45 1 45 3 45 5 459 45O

4^2

XI

§4.

WEAK HOMOTOP Y EQUIVALENCE................................. 1. 2. 3. 4.

§5.

THE WHITEHEAD PRODUCT..................................... 1. 2. 3. 4.

§6.

Fundamental Concepts ................................ Weak Homotopy EquivalenceandConstructions............ Cellular Approximations ofTopological Spaces......... Exercises.............................................

The Class w d ( m , n ) .................................. Definition and the SimplestProperties of the Whitehead P r o d u c t ............................. Applications......................................... Exercises.............................................

CONTINUATION OF THE THEORY 1. 2. 3. 4. 5.

463 463 468 472 478 478 478 482 484 486

OF BUNDLES.....................

487

Weak Homotopy Equivalence andSteenrod Bundles . . . . Theory of Coverings.................................. Orientations......................................... Some Bundles Over Spheres............................ Exercises.............................................

487 49 0 500 501 503

BIBLIOGRAPHY ...................................................

5 06

INDEX............................................................

508

GLOSSARY OF SYMBOLS.............................................

517

Set-Theoretical Terms and Notations Used in this Book, but not Generally Adopted

Mathematicians manage with a surprisingly modest collection of set-theoretic terms and notations, which can be roughly divided into three groups.

The first contains terms and notations which have attained

general recognition.

The terms and notations in the second group are

equally well-known, but can be understood differently or have varying connotations. The third group consists of terms and notations used less frequently. There is no need todefine terms from the first example, the notations

X U Y, X fl

Y,

X, x .. . x x I n the notations

and

union, intersection, and product of sets, or Imf

group.

For

for the f: X + Y,

and

fI : A -* Y for a map, its image, and its restriction are i^ understood in the same way by all people. The same is true for the notation

x e X

and the terms one-to-one (injective) map and map onto

(surjective map). For the sake of precision, we must usage set by

of terms and notations from the second group. 0.

i.e., the

We understand the notation equality

term countable set: sets.

say a few words about our

X c Y

X = Y is not excluded. we

The identity map

We denote the empty

in the broadest sense, The same is true for the

use it both for infinite countable and finite of the set

there is no ambiguity about

X

is denoted by

X, simply by

id.

id X ,

or, when

We shall say that a map

is invertible if it has an inverse, i.e., it is simultaneously injective and surjective. the set

X

We let

{x E X | ...}

denote the set of points

x

of

which satisfy the condition appearing instead of the three

dots. A family X^ with y £ M,

-*-s a maP a set M onto a set of objects defined by the formula p ^ X ^ .

Next our main task is to list the

terms and notations

appearing in this book and belonging to the third group.

2

Maps If X

may

denote

it by

eachmap

in: A -> X.

write If A f:

x h- f (x) , to

is a subset of a set

be considered as the map

we simply

f

A

Y

then the inclusion of

defined by the formula

xx .

If there isno ambiguity about

in. is a subset of

X

X,

such that

X and

B

f (A) ^ B

A

is a subset of

induces a map

A

in

We and

Y,

X,

then

ab f : A -* B,

and called here the abridgment (or compression) of the map

A,B.When there is no ambiguity

ab finstead of

ab f : A -> B .

usual restriction of

f

to

If B = Y,

where

(respectively,

Y),

(tp: X such that cp^ =

A and

B, one can write

then ab f

is just

the

A.

By a map of a sequence (Y,B1 ,.../Bn ) ,

about

(X,A ^ ,...,A^) (B^, ...,B^)

into a sequence are subsets of

X

we mean a sequence of maps A 1 -s-

Y , tp-Li ab cp .

tpn : An -►Bn )

Wedenote such a map by

(cp/cp^f •••/cp^):

(X ,A^ ,. ..,A^) ->(Y ,B^ , . ..,B^ ) .

If thesubsets

A, ,...,A and Bn,... ,B are fixed, then the map I n 1 ' n ^ f = (cp,cp^,. ../ X x x

make the

is closed if and only if

Obviously, every product

X

is

(obvious) remark is Hausdorff.

f^ x ... x fn : X^ x

Y. x ... x y of continuous maps f .: X „ Y ., ...,f : X 1 n ^ 1 1 1 n n continuous. Moreover, f^ x ... x f^ is open whenever

... x

x^

Y n

is are

open. 6 . If

X

is a metric space, then clearly

(dist (x^j ,x£) - dist(x 1 ,x2)| £ dist (x jj,x ^ ) + dist(x 2 ,x2>for any points x^,x 2 ,x^,x2 G X. This inequality shows that the function dist: X x X

IR

is continuous.

Properties of Products 7.

Every product of

T^-spaces is a

of Hausdorff spaces is Hausdorff. regular.

T -space.

Every product

Every product of regular spaces is

29

The first and second assertions are immediate. a product of

T3-spaces

X 1 ,...,Xn

T 3 ~space.

is a

We show that

Let

U

be a

(x^,...»x n)

in X^ x ... x x . Pick neighborhoods U ,...,U of the points x.,...,x , such that U„ x ... x u c: u, i n I n 1 n and fix neighborhoods V.),.. .,Vn of the same points with Cl V 1 c U , ■■•/ClVn cUn . Since C1(V 1 x ... x Vn ) = Cl V 1 x ... x cl Vn (see 3), one has Cl (V. x ... x v ) c: U . \ n neighborhood of

INFORMATION. normal; see [14]. 8.

S 1f...,S are dense sets in the spaces I n c x... x sn is obviously densein X^ x ... x x .

S1

then

There are product of normal spaces which are not

If

X „ ,...,X , 1 n Consequently,

a product of separable spaces is separable. If

, . . . , are bases of

xu

U 1 x ...

1 n X 1 x ... x x .

X ^ , . . . then the sets

EF form a base of the space n n ^ Consequently, a product of second countable spaces is

with

U

1

£ T

1

U

second countable. If now

r„,...,r are bases of X„ ,...,X at the points i n I n x.,...,x ,then the sets U. x ... x u with U. £ T U £ F , i n 1 n l i n n form a base of X^ x ... x x^ at the point (x^ ,...,x^) . Consequently, a product of first 9.

countable spaces is first countable.

Every product of metrizable spaces is metrizable.

In fact, we can say more: if then the formula

X ^ ,...,XR

dist ((x^ , ...,x^) , (x^ ,...,x^) ) = [£

defines a canonical metric on the product 10.

are metric spaces, (dist(x^,x|))2]

X^ x ... x x .

Every product of compact spaces is compact.

It suffices to consider a product of two spaces. and

Y

X x y.

be compact topological spaces, and let Consider an arbitrary refinement

sets of the form homeomorphic to x e X, A all

U x V Y

and

(see 1.1.13). Y

a finite collection

which covers

x

i = 1,...,n(x).

x y

of

be an open cover of

T

T,

consisting of open

Since the fibers

Ax = (U^x)

x v^x)}*?^

x x Y

are

of elements of

(see 1.7.2); one may assume that X,

X

is compact, one can find, for each point

Since the sets

cover the compact space

A

So let

Ux =

U± (x)

x £ U^x)

for

are open and

there exists a finite collection

A is a cover U ,...,U covering X. It is clear that A 1 = U™ x. x^ 3 D-l x. 1 m J of X x y. Finally, replacing each set W £ A' by a set of r containing

W,

we produce a finite subcover of

r.

30

11.

Every product of locally compact spaces is locally compact.

PROOF.

Let

ke neighborhoods of the points

E

EX. Then U . x ... x u is a neighborhood of (x.,...,x ) n n 1 n ^ I n X ^ x ... x x^. Furthermore, its closure Cl (U^ x ... x u^) is just ...,x

Cl

x ... x clU

(see 3), and so is compact whenever

Cl

, in

, ...,C1

are compact (see 1 0 ) .

An Application: A Method for Constructing Continuous Maps 12.

Proposition 14 below allows us to establish continuity

a map in some situations similar to those treated by Theorem 1.4.3, but where the latter is not applicable. 13 (LEMMA).

Suppose that the

and transforms the fiber

x^

into

x q

compact, then given any neighborhood is a neighborhood PROOF. Xq

U

of_

Xq

W

f (U x

with

Suppose that

is a subset of X, g: Y

->

for each

X

B

X,Y,Z

is a closed subset of

continuous map

g(B) c A.

: Y \ B + Q,

I_f

Q

themap

q

)

,

there

fix a neighborhood

Q

are continuous maps such that x E A, and

f(x^ x

f(U^. x v^) c=w.

and

Q is

c w.

Q)

compact, one can cover it with a finitecollection s set U = (1. i U . i=1 q± 14.

iscontinuous

If the space

of the point

q E Q,

of q

+ z

f: X x Q

a point.

such that

Given any point

and a neighborhood

map

Since

V ^1

of

1

,...,V ^

Q

is

. Now b

are topological spaces,

A

Y,

and

and

f (x x

f: X

x q

z

reduces to a point

Q)

is compact, then for each h: Y +

Z

given by

f (g (y) ) , (y) ) , for

y E Y \ B,

f (g (y)

y E B,

h (y) = x Q) ,

for

is continuous. PROOF. Y \ B;

The map

h

is clearly continuous at the points of

let us verify its continuity at the points

of Lemma 13, given any neighborhood a neighborhood

U

of

the point

last inclusion shows that is a neighborhood of y.

W

g(y)

y E B.

of the point such that

h(g” 1 (U)) c W,

f(U

By virtue

h(y), x

Q)

there is c:

w.

and finally note that

The g~ 1 (U)

31

Information 15.

The notion of a product of an infinite number of

topological spaces can be defined in a natural way; in

[3 ] for details.

product of compact spaces remains compact; see

3. 1. of

The quotient set

its partitions

X/p

p

thiscase too a

Quotients X/p

of a topological space

Equivalently, a subset of

X/p

pr: X -> X/p

is open.

is closed if its preimage is closed.

This topology is called the quotient topology,

and the set

the quotient topology is the quotient space of

thespace

pr: X -* X/p

X

with

by its

is continuous.

In the special case of a partition of

X/p

p. It is clear that

single

by any

is equipped with a natural topology: asubset of

is open if its preimage under the map

partition

X

set

A

the space X 2.

partitions

and the points of by

A

X \ A,

and

q,

and is denoted by

a

called the quotient

X/A. X

and

and a continuous map

into elements of

whose elements are

X/p is

Given two topological spaces p

p

Y

f: X

Y

which takes the

elements of

p

continuous.

This is a straightforward consequence of the definition of

the quotient topology. f

-1

(pr

-1

(U))

If

Indeed, if

is open in

= pr ^((factf)

^ (U) ) q

q,

the map

with respective

U -* Y / q

factf : X/p

is open, then the set

X, and so the identity

implies that

(factf)

is the partition of

Y

is

Y/q

^(U)

f

-1

-1

(pr

(U)) =

is open in

X/p.

into single points, then

= Y and p r : Y -* Y / q is the identity map. In this case, f h- factf defines a one-to-one correspondence between continuous maps

Y/q

X

Y

which are constant on the elements of the partition

continuous maps 3. continuous map

X/p

p,

Y.

In particular, the discussion above shows that given a f: X -* Y,

its injective factor

factf : X/zer(f) + Y

is continuous too.

The converse is also true: every map f : X Dr fact f be represented as the composition X --- ► X/zer(f) ------- ► Y, f

is continuous whenever 4.

and

factf

Y

can

and so

is continuous.

A continuous map whose injective factor is a homeomorphism

will be referred to as a factorial map (or a quotient map).

32

An equivalent definition: a map

f

of a topological space

X

into a topological space Y is factorial if f(X) = Y , and the preimage _i f (B) of a set B c: Y is open if and only if B is open. If we substitute closed sets for open ones, we obtain another equivalent definition. Obviously, the composition of two factorial maps is factorial, and any injective factorial map is a homeomorphism. plain that if then the map

f g

is factorial and the composition is continuous.

factorial imply

Moreover, it is

g

Also,

f

g ° f

is continuous,

continuous and

g ° f

factorial.

The projections onto quotient spaces form the main class of factorial maps. f (X) = Y

to be factorial is that 5.

f

a map

be an open or a

X

and

(X/p)/p'

p*

is a partition of

X/p,

X

projection

X -> X/p.

with

closed map. p

is a

then the quotient

is canonically homeomorphic to X / q ,

partitions

into the preimages of the elements

where

of

p'

q

under the

This canonical homeomorphism is defined as the

injective factor of the composite map

X

X/p

(X/p)p!,

a homeomorphism, because this composition is factorial 6.

the space

f : X-> Y

Taking quotients is a transitive operation: if

partition of space

A crude necessary condition for

If the sets X , then

A

and

B

and is truly

(see 4).

constitute a fundamental cover of

fact [in:A -> X] : A/AflB -> X/B is a homeomorphism. Given an open subset to show that

V = [pr: X -+ X/B]

U _i

of the quotient (fact in(U))

A/ADB,

is open in

it is enough X.

But this

is a consequence of the equalities V fl A = [pr: A -> A/ADB ] “ 1 (U) and B,

if

pr (A fl B) € U,

',

if

pr (A fl B) t U.

V fl B =

Properties of Quotient Spaces 7.

Obviously, a qutient space

and only if the elements of the partition

p

X/p

are closed.

satisfies axiom Also,

X/p

T^

i

33

is Hausdorff if and only if any two distinct elements of

p

disjoint saturated neighborhoods.

T^-space

Similarly,

(T^-space) if and only if for any element closed

subset

B

closed

subsetsA

of X

A

of p

and any saturated

(respectively, for any saturated,

and

B

of

X) such that

disjoint saturated neighborhoods in if and

is a

X/p

have

A fl B = 0,

A

and B

have

X.

Moreover, it is readily seen that X/p is second countable only if there isa countable collection of open saturated sets

in X such that any saturated set can be expressed as the of its subcollections.

union of one

It is immediate from 1.6.6 that a quotient of a separable space is separable. Similarly, 1.7.8 implies that a quotient of a compact space is compact.

Closed Partitions 8.

A partition

is a closed map.

of the space

p

is closed if

X

pr: X

X/p

An equivalent condition: saturations of closed sets

are closed. Obviously, a partition which has only one element that is not reduced to a point is closed if and only if this element is closed. 9. T^-space.

The quotient of a

T^-space by a closed partition is a

The quotient of a normal space by a closed partition is

normal. Since the first assertion is straightforward, all we have to show is that the quotient p

is a

T^-space.

subsets of

X.

neighborhoods.

Let

Since

of a

X/p

F^ X

and

T^-space

F^

by a closed partition

be disjoint, saturated, closed

is normal,

Furthermore, since

X

F^ p

and

F^

have disjoint

is a closed partition, the

saturations of the complements of these neighborhoods are closed, and now it is clear that the complements of these saturations are disjoint saturated neighborhoods of

F^

and

F^•

Open Partitions 10. is an open map. open.

A partition

p

of the space

X

is open if

pr: X

X/p

An equivalent condition: saturations of open sets are

34

If

p

is an open partition and

the saturation of

IntA

A is a saturated set,

is open, and hence

complements, we see thatthe saturation of

equals Int A ;

Cl A

is just

then

passing to

Cl A .

Therefore, in the case of an open partition, the interior and the closure of a saturated set are saturated. As it follows from 1.6.6, the quotient of a first countable (second countable)

space by an open partition is first countable

(respectively, second countable). 11. spaces

X

Let p

and

be open partitions of the respective

q

and Y. The product

morphic to the quotient

(X/p)

x

is canonically homeo-

(Y/q)

(X x Y)/(p x q ) .

The injective factor of the map

pr x p r : X x y

(X/p)

definesthis canonical map, which is a homeomorphism because

x (Y/q)

pr x pr

is open (see 2.5).

4. 1.

Glueing (or pasting) topological spaces is a composite

operation which consists quotient.

of taking a sum andsubsequently passing

More precisely, suppose

topological spaces and

p

hy_ (or according to) p.

xv is termed the setsimmv (Xy ) f: X/p Y, if and

that

{

X/p

to a

isa family of

is a partition of the space

we say that the quotient space Xy

Glueing

X =

J Lxu •

Then

is obtained by glueing the spaces

The composite map

X JE2U X/p v-th immersion and is denoted by yield a fundamental cover

where

Y

imm^ .

of X/p,

Clearly, the

and a map

is an arbitrary topological space, is continuous

only if all the compositions

f 0 imm^

are continuous.

Unions 2. Let that for each pair . In addition,suppose

be a family of topological spaces. (y,y’) £ M x m there is given a subset A that for

each pair(y,y')

an invertible map (i) (ii)

, , = Xy

and

cf>^, (Auy ,fl h v v „)

€ Mx m

there

Suppose c X UP’ y is given

such that:

y = id Xy ,

forany

= Ayly flA y ) y l (

y £M;

and the

diagram

35

is commutative for every ii the subset of

X ^ AVy * J L V

J

[x^

The sets

y,y',p" £ M.

For

x £ | |x , y

consisting of all the points

Bx

in^ (cj)^ (x) ) ,

B x where

are pairwise disjoint and define a partition of

The corresPonding quotient space is called Xy by (or a l o n g ) the maps ^yy»-

spaces

denote by

the union of the

This construction is a special case of glueing, when all the immersions cj)^,

i^y

are injective.

Moreover, assuming that all the maps

are homeomorphisms and that the sets

closed, we see at once that all the maps In the general case, a union of

Au u » i™^

are aH

open or all

are embeddings.

T^-spaces is clearly a

T^-space. 3.

Often the union construction is employed when all the

spaces ^

X„ are subsets of a set X and cover X, while A . and y yy ’ are given by Ayy> = Xy n Xy » and ^yy• = id- In this situation, conditions 2 (i) and 2 (ii) are automatically fulfilled, and one may describe the union of the

X y ’s

the following topology: a set the intersection any

C n X^

simply as the set

C c: X

X

equipped with

is open (closed) if and only if

is open (respectively, closed) in

X^

for

y E M. We devote some special attention to the case where the

topology of each set X.

X

is induced by some topology already given on

Then our construction produces a new topology on

X.

It is clear

that the sets open (closed) in the old topology remain open (respectively, closed) in the new topology. X

n X ,

are open in their sets

then the new topology on set

X ;

y

Moreover, if all the intersections

X

X

y

(endowed with the initial topology),

induces the initial topology back on each

the same holds whenever all the intersections

closed in their sets

X

y

Xy.

Limits and Filtrations 4.

Let

X q ,X ^ ,...

be topological spaces, and let

fl X ,

y

are

36

V

X.j, ^

xo

-> X ^ , ___

be embeddings. t>ki (Xk i) f

k-1

Set if

k ' < k'

if

k’ ^ k,

if

k1 = k,

A.

kk'

V and ab (cj)k - 1 id Xk'

^kk'

o ... o rf)k )] - 1 ,

[ab ((¡)^,

The union of the spaces

if

is well-defined because

conditions 2(i) and 2 (ii) are obviously satisfied. the limit of the sequence

k 1 > k,

and

denoted by

This union is called limiX^.,^)

or

limX, • k A specific property of the limit construction is that the maps imm, : X, + lim X, K

Xk

K

K.

are embeddings: indeed, every closed subset

is the preimage under

imm^

of some closed subset of

A

of

limX^ ,

for

example, of imn^, (Clx (4>k . « 1 ° ••• ° 4>k (A) ) k

U .*=k+1

Obviously, if then all the sets imm^ix^)

°Pen (closed) in + i f°r aH are open (respectively, closed) in

k'

lim(Xk ,(J)k ) . Suppose that ^ Xk + 1^ another sequence of topological spaces and embeddings, and that for each k there is given a continuous map

fk : Xk

X£ ,

so that all the diagrams f,

X,

xk vk k +1

k+1

are commutative. continuous map

Then the rule

f(imm, (x)) = imm, (f, (x)) K K K f: lim(Xk ,k ) -+■ limiX^,^) (see 3.2); f

the limit of the sequence 5. of

limX^

k +1

Ijf

f^,^,...,

X 0 ,Xr ...

are

and is denoted by

defines a is called lim f^ .

T.^-spaces, then every compact subset

is contained in one of the sets

imm^(X^) .

37

This is a consequence of 1.7.6. 6.

in

Xk + 1

If_

are normal spaces and

for every PROOF.

k,

then

X^j

x -| x

=i ^ (x., ) ,cp(Xl ,x2 ,1 ) = in2 (x2) .

that the quotient space of x2) ^ (X^ U X2)

may be alternatively defined

jc

is given by

(pr(x,x n ,t)) = ((1 -t)jc A

x £ (... (X1 * X0) * X,)...)* X I J n— I multiplication of a point of con X^ x . . .

where

A

I

.

n-1

x n

£X, n x conXn_^

(x),pr(x n ,t)),

and by

t £ I; 1 -t

the

is

defined by the rule (1 -t)(pr(x 1 ,t1 ),...,pr(xn- 1 /tn_1)) = = (pr (x1 , (1 -t) t j , ...,pr (xn - 1 , (1 -t) tn-1) ) . Clearly, the image of the embedding

jc 1,‘''

{ (pr(x.,t.) ,...,pr (x ,t )) £ con X. x 1 1

n n

... x con X

I

n

v n

is precisely

I t.+...+t \

n

= 1},

which allows us to identify the iterated join (...((X 1 * x2) * X )...) * Xn 5. the two joins morphic.

with this set.

The ^-operation is associative, meaning, as usual, that (X^ * X2) * X^

* (X2 * X^)

and

The canonical homeomorphism

are canonical homeo-

(X^ * X 2) * X^ -* X^ * (X2 * X^)

is the composition of the canonical homeomorphism x

* (X2 * X^)

(X2 * X^) * x -| with the suitablecompression

canonical homeomorphism + con X 2

X con X^

con X 2

x con X^

of the

x con X^

X con X 1 .

A consequence of the associativity of the ^-operation is that the multiple join spaces

X 1 * ... * X r

is meaningful for any topological

x ^ /.../Xn . 6.

The product

con X^

x ... x con Xn

is canonically

44

homeomorphic to

con(X^ * ... * Xr ).

The canonical homeomorphism con(X, * ... * X ) + con X. x I n i

... x con X

n

is defined as pr(jc“ 1 1 '*

(pr(x1 ,1^) , . . . ,pr(xn ,tn ) ) ,t) ~

n

(pr(x^,tt^ /max ( , ..., 7.

* and

con Sm

and

su Sm

)), ...,pr (x^,tt^/max (t ^ , ...,t^ ) ) ) .

are canonically homeomorphic to

D™

o m+1 S

The canonical homeomorphisms

Dm 1

con Sm

and

su Sm -> Sm

are defined by the formulas pr((X l ,...,Xm + 1 ) ,t) H. (tX l ,...,tXm + 1 )

and pr ( (x f . . . ^x m+1 ) / t ) 8.

The join

»

X * D°

(X1 Sin Tit f . . . /Xm+1 s i n TTt ,COS 771) .

is canonically homeomorphic to

con X .

The join X * S is canonically homeomorphic to su X . The join k k+ 1 X * S is canonically homeomorphic to the iterated suspension su X; m^ m2 m +m2 +1 in particular, S * S is canonically homeomorphic to S The canonical homeomorphism pr(x,0,t) *+ pr(x,t).

con X

The canonical homeomorphism

given by the formulas *+ pr(x,(1-t)/2).

X *

pr(x,1,t)

*+

pr (x, (1 +t )/2) ,

is given by

X *

su X

is

pr(x,-1,t) h-

Finally, the canonical homeomorphism

X * Sk

suk +1 X

is the composite map k V— i v n 1■ sur_1X * su Sk_r

-> sur_1X * Sk_r * S°

- sur"1X * S° * Sk"r - surx * Sk"r . 9.

Combining the canonical homeomorphisms constructed in

6 -8 , we obtain the composite homeomorphisms

m

s

m

* ...

* s

m

s

m

* ...

*s

n

m * s n

+m +1 n -

...

^

“ '1

iun

D

x ... x D

m - —i

x

m -i~1 -> con (S * ... * s and

m D

m1 - 1

0

ml - ^ -*■ con S

m _1

*D * . . . * S n

-* COn(S

m i_1

n * D * ...

0 con D

X

X

...

X

X D

x

...

n»1 Th er e fo re , the jo in S * and the jo in Dml * . . . * m.+ . . . +m +n-1 S 1 n the b a l l respectively.

m -1 * ... * con S n n

* D -*■ i~ 1

* S

n m _1 * D * s n ) ->-

mn - 1 - ^ con S

m. m D x l x . . . x D n m1 1 m D

... x con S n

m 1+. ..+m -1 m. + .-.+m ) ->■ con S n -*■ D n

m -1 -*■ con S

m * ... * D

S

m -

-*■ con S

n m -1 con D x con S n -*

X

m x I x D n -> m

x D n

x D

m,+...+m +n-1

x D n - >- D

n

m m1 m . . . * S , the product D x ... x d n, Dmn are c a n o n i c a l l y homeomorphic t o the sphere m + . . . +m m1+...+ m +n-1 D n , and the b a l l D 1 n ,

The Mapping Cylinder and the Mapping Cone 10. attaching the

Let

f :X ^ ->

product X^ x I

be a continuous map. to X^

(x,1 ) ►>f (x),is called themapping Cyl f .

cylinder of imm2 (X2)

The sets

imm^ (X^ x 0)

and

Cyl f ,

and the sets

imm^(x

bases of

generatrices. and

by the map

X2 #

x

f, and isdenoted by x € X^

are its X^

and they are usually identified with these two spaces; the

a canonical retraction

rtf : Cyl f -> X2 ,

rtf (imm^(x,t)) = imm^(x,1) [ = f (x) ] . map X, ^ 2 —

f.

X^ x 1 4

Clearly, the bases are canonically homeomorphic to

generatrices are canonically homeomorphic to

equals

of

are the lower and upper

with

I)

The result

cyl £

x2

I.

Moreover, there is

defined on

imm^ (X x I)

as

It is evident that the composite

46

If

X2 = X 1

and

f = id X^ ,

homeomorphic to the cylinder over 11. the space conf ,

con X ,

defined in 1).

Let

topological space B^,...,Bn

is canonically

X^ x i.

denoted by

f: X^

X^

(do not confuse it with Conf

= Cyl f/X^ .

Spaces of Continuous Maps

C(X,Y) X

Con f

Equivalent definition:

7.

e C (X,Y )

Cyl f

The mapping cone of the continuous map

X^

1.

X^,

then

be the set of all continuous maps of a

into a topological space

such that

Y.

(p(A^ ) c B ^ , ...,cf)(A^) c B^ ,

are given subsets of

C (X ,A ^,...,A ^ ;Y ,B ^,...,B^) .

X

and

Y,

The set of all maps where

A 1,...,A^

and

respectively, is denoted by

It may be interpreted as the set of all

continuous maps

(X,A. , ... ,A ) -+ (Y,B. , ...,B ) . I n I n We equip C(X,Y) with the compact-open topology: by definition,

this is the topology with the prebase consisting of all sets with

A

compact and

B

open.

C(X,A^ ,...,A^ ?Y ,B ^ ,...,Bn ) If

Y

Together with

C(X,Y),

all the sets

become topological spaces.

is a point, then

C(X,Y)

reduces to a point.

is discrete and consists of the points

x„,...,x , then I n canonically homeomorphic to the product Y x ... x Yof n the space

Y;

this homeomorphism is given by

To each there

pair of continuous maps

corresponds a mapping

(j) h- g o 0 o f .

C(X,A;Y,B)

C(X,Y)

cf>-+ ($

C(X,Y)

X is

copies

of

(x^) , ..., (x )) .

f: X ’ -> X

+ C(X',Y'),

If

and

g: Y -> Y 1

given by therule

This mapping is continuous, and we shall denote it by

C(f,g). 2.

I_f

Y

Indeed, if such that

is a Hausdorff space, (p,ip E C(X,Y)

(J)(x) ? ^(x). Let

and

U and V

$

then sois

C(X,Y).

^ \jj, then

there is

be disjoint neighborhoods of

the points (x) and i|;(x). Then C(X,x;Y,U) and disjoint neighborhoods of the points $ and \p. 3.

If

is metrizable.

X

is compact and

Moreover, if

Y

Given

$ £C(X,Y),

a finite number of balls (see 1.7.11).

is metrizable,

are

then C(X,Y)

defines a metric on

the set

0(X )

C(X,Y),

can be covered

of an arbitrarily small radius

It is clear that

neighborhood of the point

Y

C(X,x;Y,V)

is equipped with a metric, then

dist ((J>,ip) = suPxex dist ((x ) , ip (x) ) compatible with its topology. PROOF.

x £X

$,

0/

= il®= 1 C (X ,“1 (U±) ;Y ,Ui)

is a

contained in the ball of radius

2e

by e

is

47

centered at 4). Therefore, every ball in hood of its center. On the other hand, if with

0(A) cz B,

then

Dist((j>(A),Y \ B)

A c X

C(X,A;Y,B)

centered at

C(X,Y)

contains a neighbor­

is compact and

B c Y

is open,

contains the ball with radius

(J) (see 1 .7.15) .

Therefore, every

neighborhood of Z .

To prove the first assertion, pick a point set

and

[ V (x)] (y) = cj)(x,y)

C(Y,Z) be a continuous mapping, and

is Hausdorff and locally compact.

= [^(x)](y)

and

X q € X,

Then it is enough

a compact

to exhibit a

(pV (U) c: C(Y,B;Z,C).For each point

of

x^

and

ch(U x v ) c C, and then extract a finite cover ^ Y Y from the collection {Vy y } t 0r s0 . It is clear that

y V

such that

,...,V of B s s U = H _L-iy. ._1 U is a 1

48

neighborhood of

xQand that

(f>(U x B) c= U ? *

cj>(U

1

remains to remark that the inclusion

6

(U

x

b)

cz

x v ) C(X,C(Y,Z))

defined by the r

X, Y

and

The continuity of the mapping just

Assume that

Hausdorff and locally compact. and a point A

Q

of

X x Y,

q E Q.

q x Cl Vq ) - t|j is continuous. It is readily see that the mappings V A h- (j) and ip ip are inverses of one another.

49

A Surprising Application 8.

Let

f: X ->• X 1

be a factorial map.

Hausdorff and locally compact, then the map is factorial.

fxidY:XxY-*X'x

One can assume that X' = X/zer(f) and that projection X -> X/zer(f) . Consider the projection p r : X x Y ->• (X x Y)/(zer(f) Pr V : x

x zer (id Y) ) .

C (Y , (X x Y)/(zer(f)

of the partition

zer(f),

fact prV : X 1

If the space

f

Y

Y

is the

The mapping

x zer (id Y) )

is constant on the elements

and hence it induces continuous mappings C (Y , (X x Y) / (zer (f) x zer(idY))

and (fact prv)A : X' x y -+ (X x Y)/(zer(f) x zer(idY)). It it clear that the second of these mappings is the inverse of the injective factor of factor of

f x i d Y : X x Y ^ X ' 9.

and f x

f x i d Y i X x Y + X '

Let

f: X

X'

g :Y ->• Y 1

Yare Hausdorff and locally compact, g:

x x Y -+ X' x y'

Thus the injective

is a homeomorphism.

x y

and

x y.

be factorial

maps.

If

X1

then the map

is factorial»

In fact, one can express

fx g

as the composition

f x id . id x q X x Y — ---- — >X' x Y ------ X' x Y' and recall that a composition of factorial maps is again factorial.

8. 1.

The Case of Pointed Spaces

In the sequel, the class of topological spaces equippe

with a simple additional structure - a distinguished point (i.e., topological pairs

(X,xQ ),

where

xQ

is a point) will play an

important role; we call these spaces pointed spaces, and call the distinguished point a base point.

The constructions described in the

previous subsections must be naturally modified when applied to such spaces.

For some of these construction, the modification entails

merely the addition of a base point to the resulting space: for example, the quotient space of pointed space

(X,xQ)

has the natural base point

50

PIt (Xq ),

the product of the pointed spaces

the natural base point from map

X

(x^,...,x ),

x

yQ ,

has

and the space of continuous maps

into a pointed topological space

const: X + Y,

(X^ ,x^ ),..., (X^x^) (Y,Yq )

contains the constant

and hence has the natural base point

const.

Other constructions such as the sum, suspension, and join need more serious modifications. We shall describe these modified constructions below, and also introduce a new one - the tensor product of pointed spaces.

In every

case, pointed spaces produce pointed spaces, and base point-preserving maps again produce base point-preserving maps. factf ,

C(f,g),

and

f^ * ... x fR

We remark that the maps

preserve base points whenever the

initial maps have this property. We use the symbol

bp

as a general notation for the base

points.

Bouquets and Tensor Products 2.

The construction below replaces the sum construction f

pointed spaces. Let

be a family of topological spaces with base

points . The quotient space of the sum J Lye^ Xy ky the subset consisting of all points in^(x^) is called the bouquet (or the wedge) of the spaces the numbers

,

and is denoted by

1,...,n,

^

^

consists of

we also write

(X. ,x1 ) V ... V (X ,x ) . The point II n n c pr ° in^(x^) E V(X^,x^) does not depend on v; it iscalled the center of the bouquet V(X ,x ), and is taken as its base point. The bouquet

V(X ,x ) y

y

is obviously a union of the spaces ^

(see 4.2), and so there exist the embeddings maps

Prv : V^Xy'xy)

,

imm^:X^

V(X

,x ).

X

y

The

defined by xv ,

if

v ’ / v,

x,

if

v 1 = V,

prv (immv ,(x)

are specific to the bouquet construction. and pr^ ° imm^, = const if v 1 f v. If

M

Clearly,

pr

V

° imm

also indexes another family of pointed spaces

and a family of continuous maps

f^: X^

Y^

such that

then the map fact y ) : V(Xy ,xy) + V(Yy ,y ) continuous; we denote it by V f y

V

= id X

v

(Y ,v ) y

y

f (x ) = y ,

is well-defined and^

51

3.

Let

x »(x,x2,- ..,xr )

(x^

,x^

( x ^ ,x^)

[x £ X^] , ...

definecanonical embeddings in^,...,in^.

be pointed spaces.

, x » (X l ,... /xR_1,x )

The

rules

[x € X ],

X„ -► X„ x ... x x , .... X -*X x ... x x ! 1 1 n' n 1 n1 Moreover, the rule x (pr^ (x) , ...,prn (x))

denoted

by

defines

a canonical embedding

(XwX. ) V . . .V (X ,x )-> X. x ... x x , 1 1 n n 1 n which allows us to regard the bouquet (X„,x ) V ... V (X ,x ) as a I t n n subspace of X x ... x x . Clearly, in .: X . -► X. x ... x x is the I n 1 1 1 n composition of the embedding imm.: X. -+ (X.,x.) V ... V (X ,x ) with 1 1 1 1 n n the inclusion (X ,x.) V ... V (X ,x ) -+■ X.x ... x x , while the 1 1 n n 1 n projection pr : (X ,x. ) V ... V (X ,x ) -v x . is the restriction of i l l n n 1 p r .: X . x ... x x ■> X .. I 1 n 1 The quotient space is called the by

(X. x ... x x )/[(X.,x.) V ... V (X ,x )] I n i l n n tensor product of the spaces X^,...,X , and is denoted

( X . , x . ) ® ... ®

II £ (X^/x^) ® ...

(X ,x ) . The pointpr[ (X.,x ) V ... V (X ,x )] £ n n II n n ® (X^/X^) is called the center of the tensorproduct

(X ,x ) ® ... ® (X ,x ), and is taken as its base point. I I n n The tensor product is a commutative and associative operation: there are obvious canonical homeomorphisms ■> (X2/x 2) ® (X^x^)

and

(X^,x^) 0 (X2 ,x2) -*

(X^ ,x^ ) ® [ (X2 ,x2) ® (X3 ,x3),bp] ->

t(X^/X^) ® (X2 ,x2),bp] ® (X^,x^); the more general equality

this is also the way we understand

[(X^,x^) ® ... ® (X^_^,x^_^),bp] ® (xn 'xR ) =

- (X1 ,x1 ) 0 . .. ® (Xn ,xn ) . If

(Y.,y.),..., (Y ,y ) are other pointed spaces and 1 1 n n f : X ^ -> Y ^ , ...,f^ : X^ -* Y^ are continuous, base point-preserving maps, then the map

fact(f. x ... x f ): (X.,x.) ® ... ® (X ,x ) -> 1 n 1 1 n n (Y^,y^) 0 ... ® (Y^^y^) is well defined and continuous; we denote it c

by

f . 0 ... 0 f . 1 n

Cones, Suspensions, and Joins 4.

The cone over the pointed space

quotient of the usual cone is denoted by

con(X,xQ).

con X -> con(X,xQ) base point.

con X

by its generatrix

The image of

is the vertex of

con(X,xQ),

The image of the base of

con X -* con(X,xQ)

is the base of

pr(xQ x I) con X

con(X,xQ);

(X,x^)

is defined as

pr(xQ x I),

and

under the projection

and is taken as its

under the projection this projection carries

the first base onto the second one, and thus allows us to identify the base of

con(X,x^) with X. If (Y ,y q ) is another pointed space and

f: X

Y

is

52

continuous, with f (xQ ) = y 0 ' then the maP fact con f : con(X,xQ) con(Y,yQ) is well defined and continuous, and we denote it simply by con f . Equivalently, one may describe space of the cylinder 5.

X x I

by

con(X,xQ)

(X x 0) U (x^ x I).

The suspension of the pointed space

as the quotient of the usual suspension pr(xQ x I),

and is denoted by

under the projection

suX

as the quotient

su X

su(X,xQ) .

su(X,xQ)

(X,x^)

is defined

by its generatrix

The image of this generatrix

is the vertex of

su(X,xQ )

and

is taken as its base point. If

(Y, y^)

continuous, with s u (Y,Yq )

is another pointed space and

f(Xg)

= yQ , then the map

f: X

Y

is

fact su f : su(X,xQ ) -+

is well defined and continuous, and we denote it simply by

su f . Equivalently, we may decribe of the cylinder

X x I

by

su

(X x (0 U 1)) U (xQ x I),

(X,Xq ) 0 (1/(0 U 1) ,bp) = (X,Xq ) ® (s\ort^). description:

as the quotient space

(X,X q )

i.e., as

Another equivalent

su(X,Xg) = con(X,XQ)/X.

6.

The join

of the pointed spaces

(X^ ,x^ )

and (X^fX^)

defined as the quotient space of the

usual join X^ * X^by its

generatrix

denoted by (X^,x^)

image of

pr(x^ x x^

xxi) , and is

pr (x^ x x^ x

* (X2 ,x 2 ) as its base point. -+ (X^,x^)

If f^ : X^ -* Y^

is the center of

(Y^,y^) and

(Y 2 ,y 2)

and

f2 : X 2

(X^x^)

and is taken

The canonical homeomorphism (su (X1 ,x1) ,bp) V (su (X2 fX2) ,bp)

(X^x^)

=

(X2 ,x2),

the

su((X^,x^) V (X2 ,x2 ),bp)

is given by [x € X± ,

m su(S ,ort^) ,

are canonically homeomorphic to respectively.

and

is canonically homeomorphic

immi (pr (x,t) )

m con(S ,ort^) ,

f ^ (x^ )

the maP fact(f 1 * f2> : ( X ^ x ^ * (X2 ,x2) -> d^fined and continuous, and we denote it

bouquet (su(X^,x^) ,bp) V (su(X2 , ^ 2 ) 'bp) to su((X1 /x 1) V (X2 /X2 )/bp).

8.

* (X2 ,x2),

are another pointed spaces, and

For any two pointed spaces

pr (immi (x) ,t)

->

Y 2 are continuous maps such that

= y 1 and f 2 (X 2 ^ = Y 2 'then (Y-j/Y-j) * simply by f^ * f2 . 7.

* (X^,x^ ) • The

under the projection X^ * X 2

I)

is

Dm+1,

and

s m+1,

i = 1,2].

(Sm ,ort1) * (Sn ,ort ) and

s m+n+1f

The canonical homeomorphism su(Sm ,ort

) -+■ Dm+1

is defined

pr ((x1 ,. ..,xm+1) ,t) h- (tXl + (1 -t) ,tx2 ,. ..,txm+1) .

as

Sm+1

The canonical homeomorphism su (Sm ,ort.j ) the generatrix passing through the point XT\^

S

if

with center x = ort^) ;

pr(x,t)

as

(ort^ + x)/2 t

x G Sm onto the circle on

(which degenerates to the point

varies from

0

1,

to

xm+>| ^ 0

ort^

the image of the point

moves uniformly on this circle, starting from

continuing into the half space

transforms

ort^ ,

and

(see Fig. 1). First base

s m+n+1

Finally, the canonical homeomorphism

(Sm ,ort^) * (Sn ,ort^) +

j_s defined on the bases

by the formulas

/

»

(x1 '* * *,Xm+ 1 ) and

and

and maps

/

/-v

a

^

, U , . . . , U ,

x /3 x 2 t• • • r

the generatrix joining the points

the arc of the great circle on x 1,

sn

3x1+1 x /3 x /3 /3(1-x ) 1 ^ m*t" l 1 ( 4 ' ~2 '**•' 2 /0 /---/0 , 4 ) 3X.+1

( x^/ » » * / X^ +^ ) H ' (

and

Sm

sm+n+

x E S™

2

\

/1 /J(xr 1) 9 4 and

'

x 1 E Sn onto

which joins the images of

x

in such a way that the lengths are linearly transformed (see

Fig. 2). 9.

Since

con(Sm ,ort^) = (Sm ,ort^) ® (1,0) (see 4), and A su(Sm ,ort1) = (S (Ort^ 0 (S jort.j) (see 5), the homeomorphisms con(Sm ,ort1) -> Dm+1 n ^ 1

and

su(Sm ,ort.|) -> Sm+1 ,

to the canonical homeomorphisms

defined in 8 , lead for

54

and Dn = (S1 ,ort.) 0 ... 0 (S^ort.) 0 (1,0). v------ ]-----v-----------n- 1 Now one can define the maps id ® ... ® id ® pr: Dn = (S ,ort^) ® ... 0 (S ,ort^) ® (I/O)

->

(s\ort^) ® ... (s\ort^) ® (1/(0 U 1) ,bp) = Sn , and (pr ® ... ® pr ® id I) ° pr:

In = I x ... x i -v

(1/(0 U 1) ,bp) 0 ... 0 (1/(0 U 1) ,bp) 0 (1,0) We denote them by Int Dn whileID

homeomorphically onto

and

fact DS : Dn /Sn ^

Sn

the quotient space

The Mappings

DS

Dn /Sn ^

base point-preserving mapping

Z

in ^

and

preserving mapping, and suppose that ^

n ^ 1 Sn .

C(Y,y0 ;Z,zQ)

X

and

(Z,Zq )

be pointed topologica

is continuous and preserves base = C (X,xQ ;C (Y,y0 ;Z/z0) ,const) given

bythe formula

$ »

(see 10)

^

is continuous.

If

X

and

Y

are Hausdorff and compact, then this mapping is a homeomorphism and its inverse is given by the formula The preimage of under

the mapping

which

shows that

$ h- C(X

(Y,yQ),bp;Z ,zQ)

X

Y , Z)

ip w*

and ip »

ipA ,

and the vertical arrows the composite mappings C(X,xQ ;C(Y,y0 ;Z,z0),const)

in

« C (X,C(Y,yQ ;Z,zQ)) C(ld X,in) ->

---- > C (X,C(Y,Z)) and C ((X,xQ) 0 (Y,y0) ,bp;Z,z0 )

in

> C((X,xQ) ® (Y,yQ> ,Z) ---- ►

C (pr,id_Z) ^ c ^x x Y ,Z). Since this diagram is commutative, the fact that is an embedding implies the continuity of h- » y x E X.

Every

(or connecting f.

f

A map homotopic

to a constant map is also said to be null homotopic. Often a homotopy continuous maps (0 £ t £ 1).

ft : X -* Y,

F:

X x I

related to

Y is interpreted as F

via

ft (x ) = F(x,t)

Acoording to 2.7.6, the continuity of

this family is continuous as a map of thesegment

a family of

I

F

implies that

into

C(X,Y).

Moreover, if X is Hausdorff, then the continuity of the family is equivalent to that of the map F. Obviously, the constant homotopy f: X + Y, given by

F'(x,t) = f (x) ,

connects

F of a continuous map f

to

f;

if the

f

57

homotopy

F

connects

f

to

f',

F' (x,t) = F(x, 1 -t) ,

defined by

F connects f to then their product

then the inverse homotopy, F', connects

f'

f' and the homotopy F ", defined as

to

F'

f;

if the homotopy

connects

F ( x ,2t ),

for

t £ 1/2,

F'(x,2t-1),

for

t ^ 1/2,

f'

to

f",

F M (x, t)

is a homotopy connecting

f

to

f".

Thus homotopy is an equivalence

relation, which yields a partition of called homotopy classes. 2. and

f'

We denote the set of these classes by

be continuous maps of a space x £ X

contained in f

into equivalence classes,

Anexample is the rectilinear homotopy.

If for each from

C(X,Y)

f',

Namely, let

into a subspace

the segment joining

Y, then

to

X

f(x)

F(x,t) = (1-t)f(x)

to

f'(x)

+ tf'(x)

Y

of

f IRn .

is entirely

defines a homotopy

referred to as rectilinear.

Obviously, any two maps of an arbitrary space into Dn

tt(X,Y).

IRn

or

are rectilinearly homotopic. 3.

Let the maps

continuous maps g o f' o h

g: Y -> Y'

be homotopic.

and

X,

h: X'

Then given any

the maps

g o fo h

and

are homotopic. In fact, let

Then

f ,f ': X ■+ Y

g ° F 0 4.

F: X x I -* Y

(hx id I)

be a homotopy from

is a homotopy from

As3 shows, the mapping

induced by two continuous maps

X

homotopy classes into homotopy classes. fact C (h,g) : tt(X,Y) + tt (X',Y ')

g ° f ° h

C(h,g)

h: X 1

f to

: C(X,Y)

and

g: Y

to

f*.

g ° f' ° h.

CiX'/Y1) Y'

transforms

The resulting mapping

is denoted by

ir(h,g),

that it depends only on the homotopy classes of

h

and

and

3

implies

g.

Stationary Homotopies 5. F : X x I ■+ Y A-homotopy if

Let

A

be a subset of the space

is said to be stationary on F(x,t)

= F(x,0)

for all

A

X.

A homotopy

or, simply, to be an

x € A

and

t € I.

Two maps

which can be connected by an A-homotopy are A-homotopic. As with usual homotopy, A-homotopy defines an equivalence relation, dividing the set of continuous maps A

with a given map

f: A + Y,

X

Y

which coincide on

into equivalence classes.

The latter

are called A-homotopy classes or, in full, homotopy classes of continuous

58

extensions of the map

f

to

X.

We denote the set of these classes by

7T (X, A; f ) . Notice that a rectilinear homotopy from is stationary on the set of points where

f

and

f g

to

f*

(see 2)

agree.

If one wants to specify that a certain homotopy is ordinary, i.e.,

not stationary, then one says that it is free.

Homotopy Equivalence of Spaces 6.

continuous map id X

A continuous map f: X -* Y

f ° g

is a homotopy inverse of the g ° f

is homotopic to

is homotopic to id Y .

Acontinuous

has a homotopy inverse is called a homotopyequivalence.

there is a homotopy equivalence Y

X

if the composition

and the composition

map which

g: Y

X

Y,

is homotopy equivalent to the space

If

then one says that the space X.

The following are obviously homotopy equivalences: the identity map of any space, a map which is a homotopy inverse of a homotopy equivalence, and the composition of two homotopy equivalences. Thus, homotopy equivalence among topological spaces is an equivalence relation.

It divides the topological spaces into classes called

homotopy types (instead of saying that one says

also that

X and

Y

Y

is homotopy equivalent to

X,

have the same homotopy type) .

Every homeomorphism is clearly a homotopy equivalence. 7.

If one of the continuous maps g o f: x

and their composition

Z

f: X -> Y

and

g: Y ^ Z,

are homotopy equivalences, then

the other map is also a homotopy equivalence. Indeed, let that of

f g.

h

be a homotopy inverse of

is a homotopy equivalence. Similarly, if

homotopy inverse of 8.

g: Y + Y'

and

7T(X,Y)

g

f o h

and suppose

is a homotopy inverse

is a homotopy equivalence, then

h ° g

is a

f. is a homotopy invariant.

f : X -* X 1

Evidently, if

That is to say, if

are homotopy equivalences, then

7r(f,g): tt(X,Y) -> tt(X',Y')

inverse of inverse of

Then

g ° f,

is invertible. f'

(respectively,

f (respectively, of Tr(f,g).

g) ,

g')

is a homotopy

then the map

tt(f 1 ,g ')

is the

59

Contractible Spaces 9. to a constant map. ]R

and

A space Dn

X

is contractible if the map

id X

is homoto

are examples of contractible spaces (see 2).

10-. A space is contractible if and only if it is homotopy equivalent to a point. If f: D

-> X

id X

is homotopic to a constant map

taking the value

(p(X) ,

inverses of one another: indeed, If now

f:

one another, then 11 .

X

id X

_If

X

and the map

f ° g = (J) and

and

g: X

then the map

g: X

are homotopy

g ° f = id

.

are homotopy inverses of

is homotopic to the constant map

fog.

is contractible, then any two continuous maps of

an arbitrary topological space into id X

cf),

X

are homotopic.

is homotopic to any constant map

X

In particular,

X.

This is a straightforward consequence of 10 and 8 .

Deformation Retractions 12. its subspaces

Aretraction A

p

of a topological space

onto one of

(see 1.4.10) is called a deformation (strong

deformation) retraction if the composition homotopic

X

(respectively, A-homotopic) to

X ■ p-> A id X .

— ►X

If the space

is X

admits

a deformation retraction (a strong deformation retraction) onto A, then A is called a deformation retract (respectively, a strong deformation retract) Obviously, p

of X. if p : X

and the inclusion

A

X

A

is a deformation retraction, then

are homotopy equivalences, each being a

homotopy inverse of the other.

It is clear also that any space which

admits a deformation retraction onto one of its points is contractible, and that every point of a contractible space is a deformation retract of the ambient space.

Relative Homotopies 13.

Let

sequence of subsets

X

(respectively

Y)

be a space with a distinguished

(respectively,

B 1 ,...,Bn >.

A map

60

F: (X x I ,A 1 x I,...,An x I) -> (Y ,B ^ ,. .., connecting the continuous maps if

abs F

is

is evidentthat

connecting the maps

B^.

abs f 1 .

and

In

is a homotopy

Moreover, it is

(X x I,A^ x I,...,An x I) ->

yield an equivalence relation.

This relation divides

into homotopy classes forming a set denoted

tt(X ,A ^ ,...,A ^ ;Y ,B ^ ,...,B^) .

the map

abs f

ab abs f , ab abs f 1 : A_^

C (X ,A ,...,A^ ;Y ,B ^ ,...,B^) by

the maps

ababsF: A^ x I -+ B^

readily seen that the homotopies (Y,B,j,...,B )

is called a homotopy

f,f! : (X ,A^ ,. ..,A^) -* (Y ,B ^ , .. .,B^)

a homotopy connecting

this case, it

)

We may give an analogous definition of

7i(h,g)

from 4. A continuous map

g:

(Y,B^ , .. .,B^) ->

(X ,A^ ,.. .,A^)

to be ahomotopy inverse of the continuous map (Y,B^ ,..., B ^ )

if

g ° f

homotopic torel id Y.

is homotopic to

A continuous map

is called a homotopy equivalence. ( Y , B^,...,B^)

f:

is said

(X,A ^ ,♦..,Ar ) ■>

rel id X

and

fog

is

possessing a homotopy inverse

Two sequences

(X,A^ ,.. .,A ^ )

are said to be homotopy equivalent, or to have

and

the same

homotopy type, if they are related by a homotopy equivalence. Propositions 7 and 8 , as they stand, apply to the case of relative homotopy. 14. X

and Y

The situation discussed in 13 encompasses the case whe

are pointed spaces (in this case

A^and

B

are points,

n = 1 , and the homotopies defined in 13 are just the homotopies stationary at

A^) .

Moreover, the definition of a contractible space

given in 9 extends to pointed spaces (however, the homotopy from to a constant map must be stationary at the base point).

The

true for theorems 1 0 an 1 1 , as well as for the definitions of

id X

same is a

(x and must have the same base point, and the homotopy from the composition deformation retraction and deformation retract, given in 12

X —

A —— — > X

to

id X

must be stationary at this point) .

A

Also,

the remarks in 1 2 remain valid, while the definition of strong deformation retraction is entirely unaffected by the presence of a base point.

2. 1.

Paths

A path in a topological space

of the closed unit interval

I

into

X.

The points

are called the origin and the end of the path termed loops. Given a path

s,

the formula

s.

t k s(1-t)

X s(0)

is any continuous ma and

s(1)

Closed path are also defines a new path,

61

called the inverse of with

s

and denoted by

s 1 (1 ) = s 2 (0 ),

s

_1

.

s^ (2 t ) ,

for

t £ 1 /2 ,

s2 (2 t- 1 ),

for

t ^ 1 /2 ,

s1s2 * Since

of a map

D

Obviously,

I =x i f

-+ X.

and

the formula

defines a path, called the product of the paths denoted by

s1

Given two paths

(s~ 1 ) ” 1 = s

s^

and

s2 ,

and

( s , ^ ) " ^ s2 1 s”1.

and

any path can be considered as a

homotopy

If one adopts such an interpretation, then the

inverse path becomes the inverse homotopy, while the product of paths becomes the product of homotopies. On the other hand, every homotopy between two continuous maps f,f'

: X -> Y

defines a path in

C(X,Y),

joining

f

and

f*

(see

2.7.6), and again the inverse path corresponds to the inverse homotopy, and the product of paths to the product of homotopies.

If

X

is

Hausdorff and locally compact, then a homotopy connecting two maps f,f": X

Y

may be even defined as a path in

C(X,Y)

joining

f

and

f '. 3.

Since any path is a continuous map, it can be also

subjected to homotopies.

Unfortunately, the generally accepted

terminology for such homotopies is not in complete agreement with our definitions in subsection 1 (which are also generally accepted). precisely, when we consider paths,

More

the homotopies and the homotopy

relation are understood always as (0 U 1 )-homotopies (i.e., homotopies stationary at the extremities of the interval relation, respectively.

I) and (0 U 1 )-homotopy

Moreover, a free homotopy of a loop is

understood always as a usual free homotopy whereby the path remains a loop all the time (i.e., as a continuous map F (0,t ) = F (1,t)

for all

3. 1.

F: I x I

X

such that

t £ I).

Connectedness and k-Connectedness The properties of topological spaces we study in this

subsection represent weaker versions of the contractibility in the absolute case, and of deformation retractability in the relative case.

62

Connectedness 2.

A topological space is connected (see the Preface) if

each pair of its points can be joined by a path. connected if the set Since

tt(D°,X)

tt(D°,X)

every

X

is

contains just an element; see 2.2.

is a homotopy invariant, connectedness is

a homotopically invariant property. spaces are connected.

Equivalently,

In particular, all contractible

For example,

3Rn

and

Dn

are connected for

n. For

n > 0,

Sn

is also connected: any two points of

can be joined by a path, which in fact is contained in

S

Sn \ p,

where

p

is a third point (recall that the punctured sphere Sn ^ p is n 0 homeomorphic to 1 ). S is not connected: a path joining -1 and [0 ,1 ],

would be a continuous function on

1

taking two distinct values

but no intermediate ones. The only connected subsets of the real line

are the empty

JR

set, the finite or infinite intervals, the finite or infinite semi­ intervals, and the closed intervals.

Indeed, if

a

exact lower and upper bounds of a connected subset contains the interval 3.

A

of

are the

IR,

then

A

(a, 6).

Given an arbitrary topological space

being joined by a

6

and

X,

the property of

path defines a relation between its points, which

obviously satisfies all the requirements for an equivalence relation . This relation defines a partition of maximal connected subsets of

X,

X

into subsets which are the

and are called the components of

Clearly, the set of components may be identified with denote it by

We

comp X .

Every continuous map

f: X

fact f

= 7T(idD^,f) : comp X -+ comp

replace

f

whenever

(D ,X) .

tt

X.

Y

induces the map

Y .

This map does not change

by an arbitrary homotopic map, and

f

factf

is invertible

is a homotopy equivalence (see 1.4 and 1.8).

plain that if

f(X) = Y,

then

factf (comp X)

= compY .

It is also In particular,

the image of a connected space under a continuous map is connected. 4. subsets

X A2

A^ and Indeed,

contains

also 5.

can be written as the union of two connected with

A^ fl A^ ^ 0,

a component of

A^

and

A^r

X

i.e.,

Consider a partition of

connected subset of

X

then

X

is connected.

which contains a point contains X

x Q € A^n A 2

X.

into open sets.

Then every

is contained in one of the elements of this

whenwe

63

partition.

In particular, every subset of a connected space which is

both open and closed is either empty or the whole space PROOF.

Let

A

be a connected subset of

an element of the partition, such that f: X

which takes

continuous,

f(A)

U

into

1

X,

U n A f 0.

and

X \ U

and let

U

be

Consider the map

into

f(A) = 1

is connected, whence

X.

-1.

Since

f

is

A c u.

and

k-Connectedness r The following properties of a continuous map

6.

with

r ^ 0

are equivalent:

(i)

f

(ii)

is homotopic to a constant map; f

(iii) S

r— 1

f: S

extends to a continuous map the compositions

-homotopic,

where

DS+

and

D

r +1

X;

f ° DS+ , f ° DS_: Dr DS_

X

are the embeddings of

are D

r

in

r S ,

defined by 2

2 1/2

DS+ (x1,...,x )

= (x1 ,...,xr , (1—x 1 — ..>-xr )

DS_ (x ,...,xr)

= (x1 ,... ,xr ,— ( 1 - x ^ ..*“X r)

)

and

(iv)

f

2

2

1/2

);

i_s ort^-homotopic to a constant map.

The proof follows the following scheme: (i) ! l V (iv)

^ (ii)

(iii)

^>"7

(i) =*> (ii) .

A homotopy F: Sr x I

map takes the upper base

of the cylinder

Consequently,

F

Sr x I ■+ Dr + \

defined by

S x I

from

f

to a constant

into one point.

may be expressed as the composition of the map

and a continuous map clear that

X

((x^ ,...,x^ +^) ,t ) h- (x^ (1 -t) ,...,x^ +^ (1 -t)), Dr 1

g:

X

(see 2.3.4 and 1.7.9), and it is

= f.

g

sr (ii) => (iii) continuous extension of ((x^ /• •• and

and f.

) ,t)

(ii) => (iv) .

Suppose

g: D

r +1

X

Then the formulas g(x^,...fxr f(1 ” 2 t)( 1 -x^-... -xr )

)

is a

64

((x1 ,. -./Xr+1) ,1 ) ^ g(t+( 1 -t)x1 , (1 -t)x2, ..., (1 -t)xr + 1 ) define an Sr ^-homotopy an ort^-homotopy

Dr x I -> X

Sr x I -+ X

(iii) =* (ii) . to

from

f

f ° DS +

to

f ° DS_,

and

to a constant map.

f : Dr x I -> X from f ° DS + r takes every generatrix of the cylinder S x i into one

f ° DS_

point. Dr x I

from

An Sr ^-homotopy

Consequently, F can be expressed as the composition of the map Dr + 1 , defined by ((x1 , ...,xr) ,t) n- (x1 , ...,xr , (2 1 — 1 ) (1 -x^— •• •-x^) 1 /' 2 ) ,

and some continuous map clear that

r+ 1 g: D -* X

(see 2.3.4 and 1.7.9), and it is

= f. sr The implication (iv) => (i) is trivial. g

7.

A nonempty space

(0 £ k £ °°) ,

if any continuous map

S

X

is said to be k-connected

X

with

to a constant map, i.e., satisfies condition 6 (i).

r £ k

is homotopic

Theorem 6 shows that

this definition has three more equivalent formulations, based on conditions 6 (ii) , 6 (iii) and 6 (iv). r maps D X which agree on f: Sr

X

such that

f ° DS+ = f

Moreover, since for any continuous r~ 1 S there is a continuous map and

f ° DS_ = f ^ r

we conclude that

a nonempty space X is k-connected if and only if any continuous maps r r” 1 r~ 1 f^,f2: D -* X, r ^ k, which agree on S are S -homotopic. Obviously, for nonempty spaces O-connectedness is nothing else but connectedness. connected.

The

1-connected spaces are usually called simply

Note that a 0-connected space is simply connected if and

only if any two paths with common extremities are homotopic. The homotopy invariance of the sets

tt (S

,X)

implies that a

space which is homotopy equivalent to a k-connected space is itself k-connected.

In particular, every contractible space is ^-connected.

The Relative Case 8.

The following properties of a continuous map

f: (Dr ,Sr ^) -> (X,A) (i)

f

with

r > 0

are equivalent:

is homotopic to a constant map;

(ii) abs f into a subset of A ;

is S

r— 1

-homotopic to a map which carries

D

r*

65

PROOF. homotopy from

f

(i) =* (ii).

If

F: (Dr

l,sr

x

x

I ) -> (X,A)

is a

to a constant map, then the formula

F(x/dist(0 ,x),2 (1 -dist(0 ,x))),

if

dist(0 ,x) 5 (2 -t)/2 ,

F(2x/(2-t) ,t) ,

if

dist(0,x) s; (2-t)/2,

(x,t) H-

r~ 1 2T defines an S -homotopy D 'x 1 -> carries Dr into a subset of A. (ii) => (i) . S

r—1

from

abs f

consider the map

If

G: Dr

X

from

I -> x

x

to a map which

is a homotopy stationary on 2T D into a subset of A,

to a map which carries F: Dr x I -> x

abs f

given by G ((x1 ,...,xr ),2 t),

if

t i

G( (2x1 (1-t) ,...,2x (1-t) ,1) ,

if

t 5 1/2

Then rel F : (Dr x I,Sr ^ x i) constant map.

(X,A)

is a homotopy from

f

1 /2 ,

to a

9. A pair (X,A ) is k-connected (0 £ k £ °°) if for any map r r— 1 r— 1 f: (D ,S ) -* (X,A) with r £ k, abs f is S -homotopic to a map whose image is contained in It is clear

A.

that the pair

if each component of the space

X

(X,A)

is 0-connected if and only

intersects

A.

If

k > 0,

then

(X,A) f:

is k-connected if and only if every continuous map r r— 1 (D ,S ) -* (X,A) with r £ k is homotopic to a constant map; see 8 . A pair which

k-connected.

As a consequence,

deformation retract of (X,A) X,

X,

the

we see that when A pair

is homotopy equivalent to

the pair of

is homotopy equivalent to a k-connected pair is

(X,A)

(X,A) is “-connected;indeed,

(X,X).

It will be clear later that

is already ^-connected if

or even when the inclusion

is a strong

A + X

A

is a deformation retract

is a homotopy equivalence;

see 5.1.6.5.

4. 1. point

A topological space

Xq € X

neighborhood

Local Properties

if each neighborhood V

of

to the constant map

x^

U

of

X x^

such that the inclusion

V -> x^.

is locally contractible at the contains.another V

U

is homotopic

A topological space is locally contractible

if it is locally contractible at any of its points.

66

If we replace in these definitions the homotopies by XQ-homotopies, the we get the definitions of a space strongly locally contractible at the point locally contractible space IRn , Dn

and

x^,

X

which is

and of a strongly

X.

Sn

are examples of strongly locally contractible

spaces. 2. Xq E X V

of

U.

A topological space

if each neighborhood Xq ,

U

X

is locally connected at the point

of

Xq

contains another neighborhood

such that any two points in

V

can be joined by a path in

A topological space is locally connected if it is locally connected

at any of its points. It is clear that a locally contractible space is locally connected.

As an example of a connected space which is not locally 2

connected we may take the subset of m^x^ + m 2 x 2 ' 3. of

with

m -j

an(^

m2

]R

consisting of the lines

integers.

A space is locally connected

its open sets are open.

if and only if the components

In particular, in a locally connected space

every neighborhood of an arbitrary point contains a connected neighbor­ hood of this point. PROOF. subset of Then in

U V

X,

A

Suppose that

X

a component of

is locally connected, U,

contains a neighborhood

V

can be joined by a path in

and of

U.

Xq E A Xq

Hence

U

is an open

an arbitrary point.

such that any two points V c A

and

x Q E IntA .

This proves that in a locally connected space the components of the open sets are open.

5. 1.

A topological pair

topological space

Y,

(A,B)

f |^'

Let

is a Borsuk pair if given any f: X -+ Y,

there is a homotopy

and any homotopy X x i -* y

is a topological triple such that

are Borsuk pairs, then 2.

(X,A)

(X,A,B)

(X,A)

any continuous map

F: A x I Y of the map which extends F. If

Borsuk Pairs

(X,A)

(X,B)

be a topological pair. X x i.

(X,A)

f

and

is obviously a Borsuk pair.

to be a Borsuk pair it is necessary that

a retract of the cylinder is also sufficient.

of

When

A

Then in order for (X x 0) U ( A x i)

be

is closed, this condition

67

PROOF OF THE NECESSITY. in: X = X x 0 -> ( X x O )

u

Any homotopy of the map

(Ax I )

in: A x I (X x 0) U (A x I) onto ( X x O ) U (A x I).

which extends the homotopy is a retraction of the cylinder X x I

PROOF OF THE SUFFICIENCY. be a retraction. map f: X -* Y, composition

G

p:XxI+(XxO)U

Then given any topological space and any homotopy

X x I -P-* ( X x O ) where

Let

F: A x I

y

Y,

(A x I)

any continuous

of the map

^[a *

the

U (A x I) -iL, y,

is defined by f(x) ,

if

t = 0,

F (x ,t) ,

if

x G A,

G(x,t)

is a homotopy of 3. way.

I_f

f

which extends

The following statement completes Theorem 2 in an essential

X

is Hausdorff, then the assumption that

is a retract of the cylinder closed. that

X x I

(X x 0) U

(A x I)

implies automatically that

A

is

Indeed, it suffices to note that the above hypothesis implies

(XxO)

U ( A x I)

isclosed in

is the preimage of this 4. Xand

F.

setunder the

If the sets

(A,A

fl B)

A

and

B

X x I (see1.5.5), map X

X x

i,

and that

x *+ (x,1).

form a closed cover of the space

is a Borsuk pair, then

(X,B)

is also a Borsuk pair.

This is a consequence of 2; in fact, any retraction p:

I -> [ A

A x

x 0] U

[

(Afl B) x

x i

(x x 0)

'p(x, t) ,

if

x e A,

(x,t),

if

x e b.

defines a retraction

X

I]

U (B x

i) by

(X,t)

5. (Z x x ,Z x a )

If

(X,A)

is a Borsuk pair and

A

is closed, then

is a Borsuk pair for every topological space

PROOF.

If

p

is a retraction of the cylinder

Z.

X x I

onto

(X x 0) U (A x i), then id Z x p is a retraction of the cylinder (Z x X) x I = Z x (Xxl) onto [(Z x X) x 0] U [ (Z x A) x I] = = Z x [ (X x 0) U (A x I) ] .

A

68

Borsuk Pairs and Deformation Retractions 6.

If

(X,A ) is

a Borsuk pair and theinclusion

ahomotopy equivalence, then

A

is a deformation

7T: X

A

be a homotopy inverse of

PROOF.

Let

A

retract

X is

of X . the inclusion

A X. Extend the homotopyfrom 71 |^ “ 71 ° in: A A to id A to a homotopy of the map tt ; this yields a homotopy from tt to a retraction of X

X

onto

A,

which we denote by

A —— — > Xis homotopic to

is also homotopic 7. (X

to A

lf_

xI , (x x 0) U

id X ,

f: X x i X

x

(x x 1 ))

p

X —

A— ^ — >X

is adeformation retraction. X

and

is a Borsukpair,

then

A

is a

X.

p : X -> A

be a homotopy from

A — — — > X.

, thecomposition

and thus

strong deformation retract of Let

id X

Since the composition

is a deformation retract of

(A x i) u

PROOF.

p.

be a deformationretraction, id X

and let

to the composite map

Define a homotopy

g: [(X x 0) U (A x i) u (X x 1 )] x i

x

by f x, g((x,t1 ),t2) =

|f(x,(1-t2 )t1), I f(P(X),1-t2),

and extend it to some homotopy G :

if

t1 = 0 ,

if

x E A,

if

t 1 = 1,

(X x I) * I -> X

f: X x i ->■ x. It is clear that (x,t) h- G((x,t),1) A-homotopy Xx I ->• x fromid X to in ° p. 8.

I_f

(X,A) is

retract of the space equivalence. PROOF.

a Borsuk pair and B

A, then the map

rel: (X,B)

Consider a B-homotopy from

of a strong deformation retractionA -> B

id A

of the map yields

is a strong deformation (X,A)

Now extend it to a homotopy

G of

(X,A) -+ (X/B),

is a homotopy inverse of

x » G(x,1),

is a homotopy

to the composition

and the inclusion

id X .

an

B

-»■A.

It is clear that the map rel.

69

Local Characteristics of Borsuk Pairs 9. Y

Suppose that

is any topological space, and

given

anyhomotopy

there is an

is a Borsuk pair with f: X

Y

Let G

f

(x,t) *-> G(x,tct>(x))

be a homotopy

of

U

to a map which

(X,A)

of_

A

takes

is distinguishable

U

Then

of

A,

F. f

extending

X\ U, A.

F, and let

Then the formula

defines an (X^U)-homotopy of

I_f

neighborhood

normal,

is any continuous map.

extending

be anyUrysohn function for the pair

10.

X

F of the mapand any neighborhood

(X^ U )-homotopy of

PROOF.

(J)

(X,A)

f

extending

F.

is a Borsuk pair, then there exists a such that the inclusion

U

into a subset of

(in particular, if

X

U

A.

X

.is A-homotopic

Iif Xis normal

is metrizable and

and A

A

is

closed), then this condition is also sufficient, i.e., the converse of the above statement is valid. PROOF

OF THE NECESSITY.

a retraction, then the set a(x,1) € A x (0,1]

U

If

of

all

a: X x i

(X x 0) U (A x I)

points x E X

is

with

is open, and the composition pr

U

x I —

— > X X I

— £-* (x X 0) U

is an A-homotopy from the inclusion into a subset of PROOF let

(J): X

I

distinguishes

U X

I)

---- -— > X

— » X X I

to a map which takes

U

A. OF THE SUFFICIENCY.

A-homotopy such

(AX

that

F(x,0) = x

Let and

:

F

f(x,1) E A

be a Urysohn function for the A

(see 1.5.9).

Ux I

x

be an

for all x E U, pair

A,X

^ U,

and which

The formula

fF (x ,min (t/c{) (x) ,1 )) ,

if

x E U ^ A,

if

x E A,

G (x ,t) = < x, defines amap continuous. H: X x I

G:

U x I -> X ,

This in X x I

and Theorem

2.2.14 shows that

G

is

turn implies the continuity of the map

defined by (G (x ,max (0 ,t-(|> (x) )) ,max (0 ,t-2(J)(x) )) ,

if

x E U,

H (x ,t ) =

(x ,0) , It is readily seen that

if

x E X \ U.

H(X x I) = (X x 0) U (A x I)

and that

70

(x x0) U (A x i)

ab H :X x i

11. Let

(X,A )

is a retraction.

be a topological pair such that

A

is a

strong deformation retract of one of its neighborhoods.

If

normal and

is metrizable

and

A

is distinguishable (in particular, if

Ais closed), then

(X,A)

X

X

is

is a Borsuk pair.

This is a corollary of 10. 12. If V

of A,

(X,A)

there is another neighborhood

and the inclusion a subset of

W -> V

A-homotopy W

F:

x E U. in

x

W c= V, the

U

W

of

A,

W c= v

such that

is A-homotopic to a map which takes

W

into

A.

PROOF. By all

is a Borsuk pair, then given anyneighborhood

10, there exists a neighborhood U

U x I -+ X

such that

F(x,0)

= x

and F(x,1)

Now 2.2.13 shows that every point with

F(W x i)

F(W x I) c V. x c v, and that

inclusion W -> I 13.

I_f

X

Set

of

x E U

Aand an EA

for

has a neighborhood

W =U

W . It is clear that xEA x ab F : W x I -> v is an A-homotopy from

to a map which takes

W

into A.

isa topological space and

EX

issuch

that

(X,x)

is a Borsuk pair, then

If

is normal and locally contractible at a distinguishablepoint

X

then

(X,x)

X

x

is strogly locallycontractible

at

x. x,

is a Borsuk pair.

This is a consequence of 12 and 10.

6 . CNRS-spaces

1.

A subset

A

neighborhood retract of hoods in X.

of atopological space X

if

A

X is said

tobe a

is a retract of one of its neighbor­

The retracts and the open sets are trivial examples of neighborhood retracts. 2.

rf

A

is aneighborhood

neighborhood retract of Indeed, let in

Xt

A,

then

p: U -> A

and let a: V -* B

X and

B

is a

is a neighborhood retract of

be a neighborhood retraction for

be a neighborhood retraction for

Then a° (p |w): W B, where retraction for B in X. 3.

B

retract of

w = p_1 (V) ,

B

in

X. A A.

is a neighborhood

A topological space is a CNRS-space or, simply, a CNRS,

if itis compact

and can be embedded in a Euclidean space (of a certain

71

dimension) as a neighborhood retract; CNRS is the abbreviation of compact neighborhood retract of a sphere. Dn

and

Sn

4.

A compact neighborhood retract of a CNRS is a CNRS.

This is

are obvious examples of

CNRS’s .

a resultof 2.

5. The image of any embedding of a CNRS in a normal space is a neighborhood retract. PROOF. normal space g(X)

Let

Y,

f: X +

Y

and let g: X + TRn

is a neighborhood retract of

f ^ = [abf: X

f (X) ]

and

g ° f .^: f(X) ^ nRn is clear that

f^

° g^

U= h

-1

(V)

f(X)

into

X

in the

such that p : V -> g (X)

is closed (see 1.7.9), h: Y + JRn

is a neighborhood of f(X)

> 0 such that any two maps,

e

X

and let

U

f

(see 1.5.12).

,

and that

onto

Given anycompact neighborhood retract

a number Y

Since

CNRSX

Further, consider

o p o [abh : U -> V ] is a retraction of 6.

space

JRn .

extends to a continuous map

1

It

be an embedding of

g^ = [abg: X + g (X) ] ,

be a neighborhood retraction.

is

be an embedding of the

and

f (X) .

X

of

JRn ,

there

g,

of an arbitrary

which satisfy

sup^EY dist(f(y),g(y)) < e are homotopic. set where

f

Moreover, one may choose a homotopy stationary on the and g

PROOF. that one

agree.

Let

may take

e

a: U + X

be a neighborhood retraction. We show

to be the

distance between

X

and

HRn ^ U

(which is positive by 1.7.15). Let f,g: Y X be continuous and satisfy SUPyEY dist extremities maps

Y —

(y) '9 (y) ) < f(y) and g(y) X —^ — > U

and

Y —2-» x —— — > U

F: Y x I -* u,

rectilinear homotopy a ° F : Y x i + x

For anY Point y C Y, the segment with the lies in U. Consequently, thecomposite

7.

if

and it is plain that

is a homotopy from

is stationary on the set where A

f

can be connected by a

f

and

to g

g.

Furthermore,

a ° F

is

agree.

is a neighborhood retract of

a CNRS

X,

then

(X,A)

is a Borsuk pair. PROOF.

Let

o:

U+ A

X

as a neighborhood retract of

V

the neighborhood of

for which

A

in

dist (x,a (x) ) < e,

be a neighborhood retraction. Consider ]Rn X

and pick

e

as in 6 .

Denote by

consisting of all the point

and let

be the composition

x E U

72

CM tt V Then

>A

— ► X.

dist ((J)(x) ,x) < c

the inclusion

V

X

for

x £ V,

and

is A-homotopic to

G(x,1),

id X

Let us check that

Consider the homotopies

rel fact G : ( (X/A)

x l,pr(A) x I ) -*

Thefirst connects the maps

rel id X, rel g : (X,A) + (X,A) while the second connects the maps rel id (X/A) , rel fact g: (X/A,pr(A)) -+(X/A,pr(A)). It is clear that = [relfactg:

[rel g : (X,A)

(X/A,pr(A))

(X,A)]

[ rel fact g : (X/A,pr (A) ) = [relpr:

(X,A)

(X,A) ] = ° [relpr : (X,A)

(X/A,pr(A))]

and

(X/A,pr (A) ) ] =

(X/A,pr(A))] o [relfactg:

(X/A,pr(A)) -> (X,A)].

Attachings 8.

I_f

homotopic, then the spaces equivalent. f: X 2

(X^ ,C) X^

is a Borsuk pair and X^

and

X^ U , X^

such that the following diagram is commutative:

imm. X,

2

PROOF. maps

are homotopy

Moreover, there is a homotopy equivalence

X^ ■+ X^ U , X ^

and let

(a2 (t1 ,t2)) = G (f o F(y,t2 ),t1);

2

p2

with

a^(0 ,1 ) = A 2,

Further, for

y E Y

Y

76

J

dime. dime, |_(D 1 X D 2)

= (2 )

dim e dime, , 1 v ^ 2 = (J |_D 1) x (J [D ) ~Cha • * Cha- > Xl x x2 , where the first map is the sum of the canonical homeomorphisms dim(e^xe2)

dime^

D

D

dim e 2 x d

1

(W),

.

Since

p^

and

P2

satisfy condition

2

the maps cha and cha are factorial (see 1.3). Furthermore, dim e 2 since | |D and X are locally compact (see 4.3), the map cha

1

2

x cha

is factorial too (see 1 .2 .7.9), which in turn implies that

the composite map (2) is factorial. x P2

Therefore, the decomposition

has property (W ) . 4 (INFORMATION). X2

X^

and

of

cells, then

If every point of each of the cellular spaces

has a neighborhood which intersects only a countable family X. x 1

c

X, = X. x Z

x, ; see z

1

[6 ] for

aproof.

Attaching 5.

Consider two cellular spaces X^

and

X2 ,

a subspace C

of X ^ , and a cellular map tp: C + X^- According to 1 .2 .4 .8 , X2 U(p X 1 is a well-defined topological space, while 2.3 and 1.2.4.9 imply that X 2 Ucp X 1

and

imm2 e 2 ,

norma-*-where e^

respectively, and put

Now decompose X 2 X^ into the sets imm^e^ and e 2 run over the cells in X^ \ C and X2 ,

dim(imm^e^)

= dim e^

and

dimiimir^^)

= dim e 2 •

This is a cellular decomposition: as a characteristic map for

imnue^

one may take the composition of an arbitrary characteristic map

chae i

with imm^e^

imitK.

Clearly, the only cells that the closure of the cell

intersects are either

intersecting

Cl e^ ,

or

imm^e^,

imir^^'

where

where e^

is a cell in is a cell in

X2

X^

96

intersecting

(p(Cle^

fl C) .

Moreover, we see that

Cl imm2 e 2

intersects only the cells imm2 f:2 , where is X 2 “ a ~ cell in “ “2 Consequently, our decomposition has property (C). intersecting Cl e 2 To see that it has property (W) too, let F be a subset of X,2 Utp X,1 The having closed intersections with all the cells in X U X,

- X^ , (j) : X^ limit

lim(Xk ,(f)k )

Xq,X^,...

X2 ,...

X^ \

into the sets ),

are cellular embeddings.

By 1.2.4.4, the

is a well-defined topological space, which is also

normal (see 2.3 and 1.2.4.6 ). lim (X^, (f)^)

are cellular spaces and

Now consider the decomposition of

imm^e^,

k = 0 ,1 ,...,

where

e^

is a cell in

dimiimm^e^ ) = dim e, . If we k take the composition of an arbitrary characteristic map cha with ek imm^ as a characteristic map for the cell imm^e^, we see that this

decomposition is cellular. and (W) ,

lim(X^,c()k )

and put

Since it obviously satisfies conditions

becomes a cellular space, and

imm^

(C)

become

cellular embeddings. Notice that this definition of the limit includes as a special case the inductive process of glueing a cellular space from balls that we discussed in subsection 2 .

97

More Special Constructions 7. and

Inti

Since decomposing the segment makes

I

0, 1 ,

X;

see 2 and 3.

X x I

The bases of

are cellular subspaces (in the sense of 1.9); hence when we pass

to the quotient space su X

into the cells

into a finite cellular space, the cylinder

is cellular for any cellular space X x I

I

of

con X ,

con X

of

X x I,

and then to the quotient space

we find ourselves in the situation covered by the

construction 5. Therefore, the cone and the suspension over a cellular space are also cellular spaces. If

f: X^

which transform

X^

is a cellular map, then the attaching processes

X^ x I

into

Cyl f , and

again into the category described in 5.

con X^

into

Con f

fall

Therefore, the mapping cylinder

and the mapping cone of a cellular map are cellular spaces. 8 .The

and

X2

X^

X^

of

[(X. X

X,)

two cellular spaces

is defined as X- * 1

where

cellular join

(p:[ (X1

c

X = (X.I 2

|X_) U

1 -— L 2

(p

C

Z

cp(x1 ,x2 ,1) = in2 (x2);

cellular, the space

*c X 2

According to 3, when topologically the same as

x I],

xc x2> x 1 ] ^ x -\ I Ix 2

xc X 2) x 0] U [ (X^

(p(x1 ,x2 ,0) =

1

is g iven by

cf. 1.2.6.3.

Since

ip is

is cellular. is locally finite,

X^

X^ * X2 -

X 1 *c X 2

is

In general, the cellular

decomposition of X^ *c X 2 is cellular for X 1 * X 2 too, and so the cellular weakening of the topology of X^ * X 2 yields X^ *c X 2> However, this process does not affect the topology of the compact subsets of

X^ * X2 ;

cf. 2 .

The Case of Pointed Spaces 9.

Suppose that

X

that we take as a base point. su(X,Xq)

are quotients of

they are cellular spaces.

is a cellular space and The cone

con X

and

c o n ( X , x Q)

su X

xQ

is a 0-cell

and the suspension

by subspaces, and as such

Similarly, the bouquet of a family of

cellular spaces with 0 -cells as base points is the quotient of the sum of this family by a subspace, and hence is a cellular space.

98

Finally, we define the cellular cellular join of the cellular spaces and

x^taken as

base points,as

( X 1 Xc X 2 ) / f

respectively.

(X1

tensor product and the and

X 2 withthe 0 -cells

x^

the quotient spaces

x x2)

u (x1 X x2)]

and

(X1 * c

X2 ) / / ( x 1 *

X2 ) f

(X^,x^) ®c (X2 ,x2>

These are cellular spaces, denoted by

and

(XwX.) * (X„ ,x ) . If X. is locally finite, then they are 1 1 C 2. 2. 1 identical with (X1 ,x1) ® ° * Thus ' ®c (X2 'x 2> and (X ^ ,x ^)*c (X2 ,x2) arise from the cellular

weakening of the topologies of (X^,x^) * (X2 ,x2),

(X^,x^) 0

respectively,

and it

(X2 ,x2) and is clear that

this process

does not affect the topology of the compact subsets of (X1 ,x^)

® (X2 /X2) and (X^,x ^)

6.

1.

Show

arbitrary point

x

Exercises

that given an arbitrary cellular space

X

£ X,there exists a cellular space

together with

a cellular homeomorphism 2.

* (X2 ,x2>.

f: X + Y

Show that the sphere

such that S°°

Y

and an

f(x) E ske^Y.

and the ball

D°°

are homeo-

morphic to cellular spaces. 3.

Show that every connected, locally finite cellular space

can be topologically embedded in 4.

CO

IR .

Show that every connected, finite dimensional, locally

finite cellular space can be embedded in 3R^, 5.

for large enough

Show that every finite cellular space admits a cellular

embedding in a cellular space homeomorphic to

,

for large enough

INFORMATION. Every finite celular space of dimension be embedded in a cellular space homeomorphic to D 2n+\ 6.

q.

Consider the bouquet

space (see 5.9) homeomorphism.

B = Vt ^

and show that the map

(I =i,0)

id: B x^b

n

as a cellular

B x b

is not a

q. can

99

§2.

1.

(r ^ 0)

SIMPLICIAL SPACES

Euclidean Simplices

1 . Let A be a subset of ]Rn consisting of r +1 points which are not contained in any (r-1)-dimensional plane. The

convex hull of

A

(i.e., the smallest convex set containing

called the Euclidean simplex spanned by The points of number

r

A

Esi A

Obviously, a point of Esi A

A, and is denoted by

are the vertices of the simplex

is its dimension.

Esi A ,

is

EsiA .

and the

is also called a Euclidean r-simplex.

Esi A

is a vertex if and only if

contains no nondegenerate segment whose midpoint falls on the

given point.

Therefore, the set

A

Every simplex spanned by the simplex for any

EsiA .

is uniquely determined by a subset of

It is clear that

A

EsiA .

is called a face of

Esi A^ DEsi A^

= Esi(A^ D A^) ,

A^ ,A^ cz A. Two faces spanned by complementary subsets

A

A)

are said to be opposite. pr (x1 ,x2 ,t)

A^

and

A^

of

In this case, the formula

(1-t)x^ + tx 2

x 2 £ Esi A^ ,

(x^ € Esi A^ ,

Esi A^ * Esi A^

t € I)

defines a homeomorphism of

the join

onto Esi A .

Thus,

every Euclidean simplex is

canonically homeomorphic to the join of any

of the pairs of its opposite faces. Since with

p +q = r - 1

the spaces

Esi A

Dr

is canonically homeomorphic to any join

(see 1 .2 .6 .9), a trivial induction proves that both and

Drare homeomorphic to a join of

r+1

We conclude that every Euclidean r-simplex is homeomorphic to It is clear that the boundary of the simplex

Esi A

points. D . in the

r-plane that it determines is precisely the union of its (r-1 )-faces. Usually, this boundary and its complement in

EsiA

to as the boundary and the interior of the simplex 2.

are simply referred EsiA .

We may equivalently describe the simplex

set of all sums

£a£A t&a ,

where

t& ^ 0

there is no (r-1)-plane containing uniquely for any point

x =

A,

^aa?

and

as the

l a £ A t& = 1. Since

the numbers ^a

EsiA

ta are determined ca-*-^ec^ ^he a-th bary -

100

centric coordinates of x and is denoted by bar a (x). — . —.— -- -—. . Obviously, a face Esi B of the simplex Esi A in the barycentric coordinates of for

a E A \ B.

Moreover, if

EsiA

by the equations

EsiB EsiA

i.e., equal to

is the center of the simplex

and takes

3.

A map

A

into

Esi A B.

-+ Esi B

EsiA .

is called simplicial if it is affine

It is clear that such a map takes each face of

simplicially into a face of

Esi A

onto the interior of the simplex which is its image. Obviously, every map Esi A

Esi B .

bar a (x)

coincide for all a E B. having all barycentric coordinates equal,

Esi A

map

bar^ (x) = 0

x E Esi B , then the coordinates

computed in EsiA and The point of 1/(r+1),

is defined

Esi B ,

A -+ B

If the given map

then its simplicial extension

Esi A

and takes the interior of

extends uniquely to a simplicial A

B

Esi B

is injective (invertible), is an embedding

(respectively, a homeomorphism). 4. ordered.

EsiA issaid to be an ordered simplex

Since the subsets of

if the set

A

is

an ordered set inherit a natural order,

all the faces of an ordered simplex are ordered simplices. If orders of

A

EsiA and

and B

EsiB

are ordered r-simplices, then the

define an invertible map

simplicial homeomorphism

EsiA

-* EsiB .

A + B,

and hence a

Consequently, all ordered

Euclidean simplices of the same dimension are canonically simplicial homeomorphic. 5.

The simplex spanned by the points

is called the unit r-simplex and is denoted by

ort ,...fOrt^ Tr .

of

r +1 3R

This simplex is

notable due to the fact that its barycentric coordinates are the usual r +1 T" coordinates in IR . The given order of its vertices transforms T into an ordered simplex, and thus every ordered Euclidean r-simplex is canonically simplicial homeomorphic to

T .

Mote that given an ordered simplex Esi A , the homeomorphism r EsiA -+ D discussed in 1 is now canonical. The canonical homeor r morphism T -* D and its inverse are denoted by TD and DT, respectively. That TD maps the boundary (the interior) of r— 1 r S (respectively, onto IntD ) is plain.

Tr

onto

Topological Simplices 6.

of dimension

A topological space r

X

is an ordered topological simpl

(or an ordered topological r-simplex) if there exists

101

a homeomorphism of the simplex

Tr -> X; X,

of the simplex. ball

D

jc

this is called a characteristic homeomorphism

while

X

is sometimes referred to as the support

For example, all ordered Euclidean r-simplices and the

are ordered topological r-simplices; see 5. The standard way to detroy an order is to introduce

simultaneously all possible orders. topological space

X

Accordingly, we say that the

is a topological simplex of dimension

r

(or a

topological r-simplex) if there are given (r+1 )! homeomorphisms r r T -> X , which can be transformed into each other by simplicial homeor r morphisms T -+ T . The terms characteristic homeomorphism and support are employed in this situation too; however, now we have at our disposal (r+1 )!

equally rightful characteristic homeomorphisms. If

X

is a topological r-simplex (an ordered topological

r-simplex), and X

Y

Y

transforms

is a topological space, then every homeomorphism Y

into a topological r-simplex (respectively, into

an ordered topological r-simplex). image of a Euclidean r-simplex

Consequently, every homeomorphic

(ordered Euclidean r-simplex) is a

topological r-simplex (respectively, an ordered topological r-simplex). The vertices, faces, boundary, interior, barycentric coordinates, center, and simplicial maps are defined in an obvious fashion for topological simplices.

The faces of a topological simplex

(ordered topological simplex) are topological simplices (respectively, ordered topological simplices).

As with a Euclidean simplex, a

topological simplex becomes an ordered one as soon as we fix an order of its vertices.

2.

Simplicial Spaces and Simplicial Maps

1 .

A triangulation of a set

X

topological simplices such that: (i) every face of an arbitrary simplex in simplex in

A

of

is again a

A

A;

(ii) A,

is a cover

if a simplex in

A

is contained in another simplex of

then the first is a face of the second; (iii) the intersection of the supports of two overlaping

simplices of

A

A set

is again the support of a simplex in X

A.

endowed with a triangulation is known as a

simplicial space; the simplices of the triangulation are called simplices of the space, and the O-simplices are its vertices.

The

smallest simplex in the triangulation which contains a given point

X

by

102

x E X

is denoted by si x . According to 1.2.4.3, a triangulation transforms the given set

into a topological space, and 1 .2 .4.1 shows that the supports of the simplices of the triangulation yield a fundamental cover of this space. Since the intersection of two simplices in the triangulation is closed in each of them, the simplices in the triangulation keep the same topology when considered as subspaces of this topological space (see 1 .2.4.2) . Let

a

be a vertex of the simplicial space

X.

Then the

a-th barycentric coordinate

bar (x) is well defined for any point x a belonging to any simplex which has a as one of its vertices (see 1 . 2 and 1.6), and we obtain a continuous function bar (x) = 0 a not have a

for those points

x G X

bar : X -> IR if we set a contained in simplices which do

bar_ is called the a-th barycentric function. a Given two arbitrary distinct points x,y E X, there obviously is a

vertex

a

as a vertex.

such that

bar_ (x) f bar_ (y). a a

Consequently, every simplicial

space is Hausdorff. When a set

X

endowed with a triangulation already has a

topology, it is useful to find conditions ensuring that the topology defined by the triangulation is identical with the initial one. have an immediate necessary and sufficient condition:

We

the topology of

each simplex in the triangulation coincides with the topology induced by the initial topology of

X,

and the cover of

X

by the supports of

these simplices is fundamental in the initial topology.

If this

condition is satisfied, then the given triangulation is said to be a triangulation of the initial topological space

X.

Example: the cover

of a topological simplex by all its faces is a triangulation of this simplex. A simplicial space is ordered if its simplices are ordered in such a way that the orders of the faces of any simplex agree with the order of the simplex itself.

In particular, this holds whenever the

order of the simplices is induced by some order on the set of all vertices of the given space, which incidentally shows that a simplicial space can be always ordered. 2.

We shall presently describe a fundamental class of

simplicial spaces.

Given an arbitrary nonempty set

A,

we let

Si A

denote the set of all nonnegative, finitely supported functions $: A IR such that with the subset of Si A

^ = ^ ^ B c: a , then we identify consisting of all functions £ Si A

such that

x £ A \ B.

(x) = 0

elements, then

Si A

for

If

A

is finite and has

Si B

r +1

is obviously a topological simplex: indeed,

Si A

103

is a subset of the (r+1)-dimensional Euclidean space of all functions A + JR.

Moreover, corresponding to the

(r+ 1) i

homeomorphisms

(x^ ,.. ./xr + ■ Y

Every injective simplicial map is a

ske^Y Y.

f: X ^ Y

is uniquely defined

from the set of vertices of

A map

ske^X ->• ske^Y

X

into

extends to a

if and only if it carries the vertices of each

into the vertices of a simplex of

Y.

is injective (invertible) if and only if

A simplicial map ab f : ske^X

ske^Y

is injective (respectively, invertible). Two simplicial spaces which can be transformed one into another by a simplicial homeomorphism are said to be simplicial homeomorphic. 6.

A simplicial map

f: X

simplicial spaces, is monotone if a

and

b,

of X

Y,

where

f(a) < f(b)

X

and

Y

are ordered

for any pair of vertices,

which belong to the same simplex and satisfy

a < b.

Every simplicial map between simplicial spaces can be made monotone by suitably ordering the spaces.

Moreover, if

X

and

Y

are

105

simplicial spaces and simplicial map

Y

is ordered, then one can transform a given

f: X -+ Y

into a monotone one my suitably ordering

X;

indeed, it suffices to order arbitrarily the preimage of each vertex of Y,

and then order the simplices of

f(a) < f(b)

or

f(a) = f(b)

3. 1. M

A

is a set and

and

contains, along with each set scheme that map

(M',S ')

(,0)

by $ .

of

Obviously,

the scheme d> and

into

(p: M

( M ^ S 1)

$

($,$)

all the parts of

M'

(M,S)

(A) £ S'

for all

into

AGS.

is invertible and

is called an isomorphism.

(M,S)

If

0(S) = S',

Two simplicial schemes

which can be related by an isomorphism are isomorphic. A simplicial scheme (M ',S 1 )

scheme

complete if

if

A G S ’

2. simplices of A

and

X

X

A c= M

S c=s'. imply

and the cover of

For example, the scheme of The map X*

A

(M,S) is

AGS.

ske^X

Si A

ske^X

of a

by the 0-skeletons of the X

(see 2.2)

and is denoted by consists of the set

by all its finite subsets.

ab f : skeQX + skeQX'

induced by a simplicial map

takes the 0-skeleton of each simplex of

0-skeleton of a simplex of sch X * ,

The subscheme

is termed the scheme of the space

and of the cover of

f: X

and

is a subscheme of the simplicial

The simplicial scheme given by the skeleton

simplicial space sch X .

M c M'

(M,S)

X*.

X

into the

Hence it defines a map of

called the scheme of the map

f

and denoted by

schX schf .

into The

discussion in 2 . 5 implies that a simplicial map is uniquely determined by its scheme, that every map of some simplicial map

X -* X',

sch X

into

sch X '

is the scheme of

for any simplicial spaces

X

and

X*,

and that a simplicial map is invertible if and only if its scheme is an isomorphism.

In particular, two simplicial spaces

simplicial homeomorphic if and only if their schemes

X

and schX

X’ and

are sch X '

are isomorphic. 3.

If

X

is a subspace of the simplicial space

X',

then

106

sch X

is a subscheme of

if

is complete.

X

sch X 1

s c h X 1 ,and

sch X

is complete if and only

Moreover, it is clear that every subscheme of

is the scheme of a subspace of X*. In particular, let (M,S) bean arbitrary simplicial

and consider the simplex schSiM,

and so

(M,S)

SiM .

Obviously,

(M,S)

scheme,

is a subscheme of

is the scheme of a subspace of

Si M .

Thus,

every simplicial scheme is the scheme of a simplicial space.Moreover, given an arbitrary simplicial space scheme of

X

A simplicial scheme

orders of the subsets of a < b.

(M,S)

and

X

to be the can be

Si ske^X . (M,S)

are ordered and the order of each set schemes

we may take (M,S)

and conclude that every simplicial space

simplicially embedded in 4.

X,

A.

(M',S!)

is ordered if the sets of is compatible with the

A f. S

A map

(,$)

S

between ordered simplicial

is monotone if

by

■ { ij=! i S ' i

" 1

•

All that remains is to verify that if x = (x„,...,x ) and cf 1 q +1 X' = (x^,...,x’+1) belong toskenT4 and f(x) =f (x'), then x = x ’. Since each of the points x and x' lies in an n-dimensional face of q T , at most n+1 ofthe numbersx. ,... ,x ., and n+1 ofthe numbers 1 q+ 1 xi x'+ r are different from zero. Consequently, no more that 2n+2 numbers

x^-x^,...,x^+^-x^+^

j-,'••• »j2n +2

positive integers x-j = X'

for

j f

j,

i =

= °' Jr

The determinant of the matrix x. = x * . Dr

for

i|:]

i 1 «}

{

1 = 0 ,...,2 n+ 1 .

L 1 —U

r = 1 f...f2 n +2 f

3 (INFORMATION).

f 10 —

r- 2n+2

I

and finally

For any

n

does not vanish, and x = x*.

there are n-dimensional

polyhedra which cannot be topologically embedded in is

and

1 ,..., 2 n + 1 , we have

j r (xi Jr

so

such that

j2nt2.

¡? :i

for

are different from zero, i.e., there are

ske T2n+2; n

An example

see [10] for a proof.

5. 1.

IR2n.

Simplicial Constructions

Many of the topological and cellular constructions

described in§ 1.2

and Subsection 1.5

can be replaced by parallel

constructions which produce simplicial spaces out of simplicial ones. The simplest examples are the

J

and

V

operations: a sum of

simplicial spaces and a bouquet of pointed simplicial spaces with vertices as base points are obviously simplicial spaces.

There are also

more elaborate constructions, the more important ones being discussed below.

The main one is the barycentric subdivision construction, which

refines triangulations and has no analogs in § 1.2 and Subsection 1.5. 2(LEMMA). space

X by

Let

T

be a fundamental cover of the topological

triangulated subspaces.

the intersection

A 0 B

Suppose that for any

is a complete subspace of both

(considered as simplicial spaces) and inherits from

A

A,B €

r

A

and

B

and

B

the

108

same triangulation.

Then there exists a unique triangulation of

relative to which the elements of This triangulation of

r X

X

become simplicial subspaces.

is simply the union of the

triangulations of the elements of

r.

One may check directly that this

union satisfies conditions (i), (ii), and (iii) in 2 . 1

[the completeness

of the intersections

Uniqueness is

A n B

is necessary for (iii)].

also evident.

Barycentric Subdivision 3. baX , X

The construction below produces a new simplicial space,

from any simplicial space

X,

such that

b aX

is identical to

as a topological space, but has a finer triangulation, called the

barycentric subdivision of the initial triangulation. Consider first a Euclidean simplex numeration

a,*,...,a O r

of the vertices of

{x E X I bar

a_

(x) £ bar

0

a„

X,

X.

For an arbitrary

form the set

(x) X .

a correct definition, i.e., the triangulation of

X

This is clearly

thus obtained does

not depend on the choice of the simplicial homeomorphism the (r+1 )! available ones. Finally, let consider the cover of

X X

Tr ,

Tr

X

among

be an arbitrary simplicial space, and by its simplices, each subdivided as above.

It is easy to verify that this cover satisfies the conditions of Lemma 2, and hence we obtain a new triangulation of

X,

which is precisely

the barycentric subdivision of the initial triangulation of

X.

109

We note that the barycentric subdivision transforms a finite (locally finite) simplicial space into a finite (respectively, locally finite) one.

Moreover, if

4.

X

is a polyhedron, then so is

The set of vertices of the space

the set of centers of

the simplices of

X.

ba X

baX .

equals exactly

The centers of the simplices

S1'**‘'Sm °f X are the vertices of a simplex of ba X if and only if S1'*‘*'Sm Can be reindexed to form an increasing sequence. This observation enables us to give a concise description of the barycentric subdivision in the language of schemes: sch ba X = (S,ba S) , finite parts of

S

where

ba S

if

sch X = (M,S),

is precisely the collection of those

that can be ordered by inclusion.

we obtain a canonical

order of

whenever the simplex (of with center a ’.

ba X : if

X) with center

X*.

ba X 1

is a complete subspace of Indeed,

sch ba X

a

sch ba X

whenever

X

f: X

ba X'

X* = t \

f (ort^ ) = ort^ , f (ort^) = f(ort^) = ort^) . sch X'

naturally induces a map ba X

ba f

monotone.

and is clearly always rf

X

of the simplices of

ba X *

X

times

ba X

the map

.

take

2

X = T ,

However, the sch ba X

Thelatter is

sch ba X ', denoted by

does not exceed the maximal diameter

n/(n + 1) ,

It is enough to show that if bigger than

X',

sch ba X ’.

is a polyhedron, then the maximal diameter of the

simplices of the polyhedron

with vertices

is a subspace of

is not simplicial (the simplest example:

and hence a simplicial map

5.

shows that

is clearly a complete subscheme of

f: ba X

sch f : sch X

a < a1

then

is contained in the simplex

In general, given a simplicial map

map

At the same time,

a,a’ € skeQba X ,

In particular, the above description of baX

then

where X

n = dim X .

is the Euclidean simplex

a«,...,a , then the diameter of the simplex (2 ) is no O r [r/ (r+1)]diamX . Consider the part X ’ of X defined by

the inequality

bar

(x) ^ r/(r+1). X' is the Euclidean simplex ar obtained by contracting X towards the vertex a^ by a factor of r/(r+1).

Consequently,

that

contains the simplex (2).

X'

6

(COROLLARY).

is a positive integer bamX

diamX'

has diameter

m

< e.

s?

[r/(r + 1 )]diam X ,

For any polyhedron

X

and we finally note

and any

e

> 0

there

such that every simplex of the polyhedron

110

Simplicial Products 7. and X^

dim X^ X2

If

X^

and

are

>0 , then it is

simplicial spaces with

readily seen that their cellular product

does not admit a triangulation such that the interiors of its

simplices are products of interiors of simplices of However, we shall presently show that

X2

X^

X^

X^.

X^

and

X^

This construction produces a simplicial space out of

called the simplicial product of To begin with, let

X^

X^

and

an ,...,a ,

and let

X,

vertices

bg,...,br .

Set for

x^ G X^

U

q

= ^ =0 bara ' k

“i^l*

nondecreasing sequence

(x^ ,x2> G X^ x x 2

and

JRn

qr

,

q

in a

Further, let {1 ,...,q+r},

elements of

denote the set of all points

such that each of the numbers

^p^x i'x 2 ^'

yQ (x^) ,...,Yq _ 1 (x1) .

is equal to one of the numbers

with

x^ G X^:

Y-j (x^ ,x2) , .. .,y^ +r (x^ ,x2) .

y G M

with

6 j (x2 ) = ^ 1 =0 barb 1 (X2 )' J 1

denote the collection of subsets with where

IRm

a 0 (x1 ),...,a^_^(x^),6 0 (x2 ),...,6 r_^(x2)

and arrange the numbers

s(y),

X^ x^ x ^ -

be the Euclidean simplex in

Z

are

X^ xc X2 ,

and denoted by

X^,

be the Euclidean simplex in

vertices

and let

and

admits triangulations,

and we shall construct a canonical triangulation when ordered.

dim X^ > 0

P £ U/

One may check

directly that there is no (q+r- 1 )-dimensional plane containing the q+r +1 points (aQ /bQ) /• ••t (an ,b • — -j) 7 J1

(a1 ,b. 1

(a.,b. _n)?

Ji

1

1

J9

z

:3>

(aq-i ,bj q- ( q- i ) ] ' ' ' ' ' (V i ' bj q- q); (aq'br) '

(aq'b j - q *

where

G y, ^

j-j < •••< j

Also,one may verify that

jk + r (k+1)

k =0 l =i -k[Yk+l + 1 (X1 ,X2 ) Jk for

.

(x1 fx2) € s(p),

Tq+r +i (x i/x2^ =

‘

jQ = 0 ,j^ +1 Moreover,

" Yk + 1 (x1 ,x2 )] (ak'bl ) = (xi'x 2 ] = q +r + 1 ,

Yq (x^,x ^ )

=

and

and

Yp+ 1 ,$).

We defineScyl f

sch f

sch Scyl f = (M1 I |M ,S) ,

where

i

sets

A c M 1J [M2 such that:

(ii)

(Min” 1 (A)) U in2 1 (A)

in” 1 (A) E $ (S 1) ; a2
Scyl f .

sch

Scylf

in2 ° (a),

induces

1)

M 1J_[_M2

acertain map

sch (X„ x I) -* sch Scyl f, and hence a simplicial map X 1 * I -> Scyl f . 1 S I S Clearly, together with the inclusion X 2 -+■ Scyl f , this simplicial map

114

yields a continuous map a continuous map

csc f : Cyl f

csc f (Cyl f ) = Scyl f , rtf : Cyl f -► X^ partition

(X^ x I )J Scyl f .

zer(csc f) .

Moreover, we see that

and that the canonical retraction Also, the canonical

Cyl f

elements of the partition

zer(csc f)* zer(id I)

composition of the inclusion obviously equals

f.

Scylf

X^ -> Scylf

homotopy equivalence if and only if

Stars.

Links.

s

which

st s

s.

in a simplicial space

contain s.

which

Clearly,

(x) > 0 ,...,bar

sis the union of

a q

Sts

or

sts

or

the interiors

st(s,X).

It is

> 0,

(x)

are the vertices of

do not intersect

lk s

is the

X.

Notation:

The link of the simplex Sts

Notation:

X

is the open set defined by the inequalities

0

aQ ,...,a^

is a

Regular Neighborhoods

The open star of the simplex of all simplices containing

s.

s

Moreover,

is the union of all simplices in

s.Notation:

is a subspace of

C i s t s = Sts .

Iks or

the spaces X

lk(s,X).

and

Sts .

The following are obvious facts. If

s'

is a face of

s,

then

St s 1 c s t s ,

st s' c st s ,

If

are vertices which do not sit in the same

a^,...,a^

simplex,then the intersection aQ,...,a u ^

lk s 1 c Iks .

sta,

are vertices of a simplex

s,

is empty. then

q n r n st a . = st s . i=0

If

l

X

is

is a homotopy equivalence.

is a subspace of

where

with this retraction ->Scylf

St(s,X).

Sts

rtf

and the

X^f

the inclusion X^

X

an

Consequently,

Scyl f

union of all simplices of

bar

.

X^

f

1. The star of a simplex Clearly,

id(Cylf)

is constant on the

We conclude that the inclusion

always a homotopy equivalence, whereas

6.

X^-homotopy from

^ ^ > X^ — — — ► Cyl f

strong deformation retraction

readily seen that

which in turn induces

(see 1.2.6.10) is constant on the elements of the

to the composite map defines a

Scyl f ,

is a subspace of

s t (s,X) = st (s,X *) n X

X*,

then

However,

if

115

for any simplex

s

of

X.

Moreover, if

St(s,X) = St (s ,X 1) n X

and

X

is complete, then

lk(s,X) = lk(s,X') fl X.

Finally, if s 1is a simplex of

lk(s,X),

then

lk(s' ,1k(s,X)) = lk(s",X ) , where

s"

is

the smallest simplex containing s

2.

s f.

We can extend the definition of the star, open star, and

link to points of a simplicial space St(x,X ) , the open star are defined as St x

stx

= St six,

Obviously,

for x £ X,

= st(x,X),

st x

stx

X:

= stsix,

and lk x

is a neighborhood of

pr (y ,t) ^ A

h: U x I

U

U.

of a

with

st(a,X)

with

is a deformation

In fact, there is even a

from

id U

with the inclusion

1 - t

A

A

to the composition of

A -> U .

This homotopy is given

barb (x) bar (x) I a bEske^A -, bar, (x) b bEske^A

if a £ ske^A,

bar (h(x,t)) = < a t bar (x) , a

if

a £ skenX \ ske A. u u

In particular, this shows that every subspace of a simplicial space

X

is a deformation retract of its regular neighborhood in

ba X

(see 5.4).

Barycentric Stars and Barycentric Links 6 . The barycentric star of the simplex

space X

s

is the union of all simplices of ba X

first vertex the center

of

s. Notation:

equivalent description:

bsts

bsts

of a simplicial

which have as their or

bst(s,X).

is the set of all points

An

x £ X such

that bar (x) = bar, (x), a b

if

a,b £ s D skenX, U

bar (x) 5 bar, (x) , a b

if

a G ske_X, u

and b G (X \ s) D skenX. u

It is clear that the barycentric stars of the simplices of cover

X

whenever

and are subspaces of s f

s',

and

bsts

baX .

Moreover, bst s f

c bsts'

whenever s o s'.

7. The union of those simplices bst s

bst s'

which do not contain the center of

of the barycentric star s

is the barycentric link

X

118

of the simplex

s,

and is denoted by

blk s .

The star

bsts

is

clearly simplicial homeomorphic to the cone over

blk s .

the rectilinear projection from the center of

induces a homeomorphism

of

blk s

onto the link

lk s

barycentric subdivision of simplicial one). (con Iks,Iks)

of the simplex

lk s

(bst s,blk s)

Let

f: X

Y

A simplicial map f

if

g: X

g(x) £ sif(x)

canonically homotopic to

f:

X

joining

f(x)

x

I

2.

and

g(x).

a

of

PROOF. f,

and let

x £ X

x £

x £ X; st a^ ,

g: X Y

x £ st a .

if

g(x)

Y

sif(x) ,

3.

from

of the

is a simplicial approximation

if and only if

g

f(sta)

c= stg(a)

for

is a simplicial approximation

Recalling that

g(x) £ si f (x) ,

that

g

is

vertex

lies in the interior of a simplex with Thus,

g(a)

f(sta)

aQ ,...,a

whence

Therefore, the points g (aQ),...,g(a ) ,

x x I

Y

is

g(x) = f(x).

u

is a vertex of

c stg(a)

for every vertex

are the vertices of

g(aQ),...,g(a )

and since

si f (x) ,

si f (x) ,

and a

of

then

4

f (x) e f (ni=o s t a ±) c ni=0 flsta.)

simplex

I

x

lies in the interior of a simplex with

Now suppose that q

where

Assume first that

we conclude that

Pick

Y

X.

vertex g(a) (see 1.3). hence f (x) c: stg(a). X.

X

f: X

It is clear that this homotopy is stationary

f: X

simplicial, and that x a,

Y are

x £ X.

of the map

the canonical homotopy

A simplicial map

of the continuous map every vertex

and

is a simplicial

for any point g

X

onto the (possibly degenerate) rectilinear segment

on the set of the points

of

Y

is an affine mapping from each generatrix

cylinder

(and the

and

be a continuous map, where

Every simplicial approximation g

X

Simplicial Approximation of Continuous Maps

approximation of

to

in

transforms this homeomorphism into s

Therefore, the pairs

simplicial spaces.

f

s

are homeomorphic.

7. 1.

s

Moreover,

g(x)

■ Y

of simplicial spaces has a

simplicial approximation if and only if for each vertex

a

of

X

there

119

is a vertex

b

of_

Y

such that f(sta)

cr st b .

The necessity of this condition is an immediate consequence of

2.

To prove

f(sta)

its sufficiency, fix a map

cp: skeQX -+ skeQY

c st (p (a )for every vertex

vertices of a simplex of

demonstrate that

0 such that, given any subset with

diamA

< e,

stars (see 1.1.7.16). bamx

and,

of a finite simplicial spac

Without loss of generality, we may assume that polyhedron constitute

that

(see 6.1),

applying 2 , this extension is a simplicial approximation of

X

such that

have

Let

f(A) m

e /2

(see 5.6).

Then

the diameter of its star is less than

f: bamX -> Y

e,

given any vertex and 3 shows that

has a simplicial approximation.

8.

1.

open

belarge enough so that the simplices of

diameters less than

bamX,

is contained in one of these

Let

X

Exercises

be a simplicial space.

Show that the formula

dist(x,y) = [ I (bar (y) - bar (x) ) 2 ] 1 ^ 2 a€skeQX defines a metric on

X,

and verify that the resulting metric topology

coincides with the initial topology if and only if 2.

Show that for every polyhedron

triangulation of

IRn

X c IRn

is locally finite. there is a

by Euclidean simplices, relative to which

becomes a simplicial subspace of 3.

X

IRn .

Show that every connected, locally finite, .n-dimensional

simplicial space can be simplicially embedded in

IR

triangulated

by Euclidean simplices. 4.

X

Let

f: X

Y

be continuous, where

X

and

Y

are

120

simplicial spaces.

Produce a new triangulation of

following two properties:

a)

X

with the

each of its simplices is contained in

one of the simplices of the original triangulation;

b)

f

has a

simplicial approximation relative to the new triangulation.

§3.

HOMOTOPY PROPERTIES OF CELLULAR SPACES

1. 1.

Suppose that

subspace of zero on

X.

A

Cellular Pairs

Let

X

is a rigged cellular space and

h ^ : A U ske^X -+ I

and equal to

1

a sequence of functions

on

is

denote the function equal to

(A U skeQX) \ A ,

h^_: A U ske^X

hr-1 ^

A

I

and define inductively

(r = 1,2,...),

'

such that

if

x E A U sker_^X,

if

x = chae u,

by the formula

if

x € U

if

x = cha (ty ),

Fr (x,t) cha (((1 —t)i + t)y), where

e G cell^X ^ cell^A,

homotopies

Gr :

x I

u,

i

E (0,1],

r

>. Î ,

and

Fr (x,2 rt- 1 ) ,

if

Gr- 1 (Fr (x,1),tl'

i£

Each homotopy

Gr

an A-homotopy

U x i -> u

r- 1 '

y G Sr ^ .

Now construct

by if

Gr (x,t) =

A -*■ X,

0 g t £ 2 r, t < 2~r+1 ,

2 rt1

i t c 1-

extends the preceding one, and together they yield from

id U

to a map which takes

The compression of the last map to a map

U

A

U

into

A.

is the desired strong

deformation retraction. 3.

Every cellular pair is a Borsuk pair.

This is a consequence of 2 combined with 1.3.5.11, because cellular spaces are normal, and their subspaces are distinguishable. 4.

If_

(X,A)

is a cellular pair and the inclusion

is a homotopy equivalence, then of

A

A

X

is a strong deformation retract

X. In order toprove this,

to the

pair (X,A),then apply

(X x I , to the

3 and1.3.5 . 6

firstapply theorems

Theorem

(X x0) U (A x i) u (X x 1 ))

3 to the pair

and, finally, apply Theorem1.3.5.7

pair (X,A).

Cellular Pairs and k-Connectedness 5. (X,A)

Let

k

be a nonnegative integer or

is a cellular pair such that all the cells in

dimension at most

k,

topological pair.

Then every continuous map

f(A)

B

and let

(Y,B)

X ^ A

°°.

Suppose that

have

be an arbitrary k-connected

is A-homotopic to a map which takes

f: X X

Y

such that

into a subset of

B.

In particular, every continuous map of a k-dimensional cellular space

122

into a k-connected topological space is homotopic to a constant map. PROOF.

We exhibit a sequence of A-homotopies

{f ^ : (A U ske^X) x I -*

each extending the preceding one, and

satisfying the conditions: (i) F^(x,0) = f(x) (ii)

for all

x £ A U ske^X;

F ((A U skerX) x (1-2~r-1)) c: B; —

(iii)

F (x,t)

Then the map (A U ske^X) x i

does not depend upon F : X x I ->Y

will be

homotopy of

f|A ,

for t

which equals

ahomotopy fromf

into a subset of B. We proceed by induction.

t

to a

Define

—

F^ on map which takes

F_^

and assume that homotopies

1

^ 1- 2

X

as the constant F_^, . . . '

each

extending its predecessor and satisfying (i)-(iii), are already constructed. Fg =

•

If

A U ske X = X and we simply take q So suppose now that q < k. Since the pair

({A U ske^X) x homotopy

q > k,

then

I,(AU ske^_^X) x I)

G of the

map f|AUsk

Using the fact that

x,

B,

for each cell

he

there is a

suchthat

G|

^ x) xl = Fq_., .

a map

e £ cell

k-connectedness of the pair e £ cell^X \ cell^A,

(see 3),

q (Y,B)

h^:

->Y

the formula S^ 1

which takes

X \ cell A. Now take advantage of the q to deduce that, given any cell

there is an S^ 1-homotopy

to a map whose image is a subset of

B.

x i -y y

from

We put

F .(x,t), q- 1

F (x,t) = M.

(AU

F . ( (A U ske .X) x (1-2 q )) c B, q- 1 q- 1

he (y) = G (cha^ (y) ,1-2defines into

isBorsuk

if

x £ A U ske

.X, q- 1

G (x,t) ,

if 0 sc t £1-2-q,

He ba a

is simplicial and h (t) c f (k ) c h (a). Since B => Sr , f. is 2~ y I y y \ S -homotopic to f, and to complete the proof of our lemma it suffices to examine (I) and (II) for f r a t h e r than f. Consider an arbitrary ball 6 X

According to

f: ske^X x i -> x

to a map which takes ske^X

into

A.

from

Define

127

F:skekX x I

i + xby

x

C=

Ffx,^,^)

=

ffx,^),

and set

(skekX x i x 0) U (skekX x (0 U 1) x I) u (skekA

x Ix i)

and D = ske. X x i x 1 . k Obviously,

C

skekXX 1 X

I 'and the map Y=

and

D

F

is cellular.

Y

Y

and

B

is a cellular space, and it is clear that

containing

skekY.

To verify that imm2 (A)

have the same homotopy type, note that retract of B and fact, the formula

Now define Y

imm2 (X)

by

b = imm2 (A) U imm1 (D).

X Up j (skekX x i x i),

By 1 .5.5, a subspace of

are subspaces of the cellular space

(X,A)

and

is

(Y,B)

is a strong deformation

is a strong deformation retract of

(imm1 (x,t1 , 1 ) ,t) h- imm1 (x,tt1 , 1 )

B

Y.

In

t,t1 £ I],

[x €skekX,

-

(imm2 (x),t) h- imm2 (x) defines a homotopy

t £ I],

B * I -»■ B,stationary on

B ->- imm2 (A) .

a retraction

[x € A,

imm2 (A) ,

from

id B

to

Similarly, the formula

(imm1 (x, t^ ,t2) ,t) h- imm1 (x,t1 ,tt2)

t,t 1 ,t2 £I],

[x £ skekX,

•
Y ,stationary on

Y -> imm2 (X).

a retraction

[x £ X,

(Y,B)

from

id Y

(Y,imm2 (A))

(see 1.3.5.8 )

to

is

and

and it remains to observe that the pairs

(imm2 (X),imm2 (A)) 2.

imm2 (Y),

Consequently, the pair

homotopy equivalent to both the pairs (imm2 (X),imm2 (A)),

t £ I],

and

(X,A)

are homeomorphic.

Every k-connected cellular space

(0 £ k £

°°) is homotopy

equivalent to a cellular space whose k-skeleton reduces to a point. PROOF. 0-cell

xQ

in

lular pair A

X.

(X,A)

is contractible,

and 1.3), and 3. Y

Let

X

be a k-connected cellular space and choose a

The pair such that Y

(X,xQ)

is homotopy equivalent to a cel­

A => ske^X

(see 1).

Set

has the same homotopy type as

it is clear that

ske^Y

Y = X/A. Since X(see 1.3.7.7

is just a point.

Theorem 2 says nothing about the dimension of the space

which replaces the given space

that one can always choose

Y

X.

However, its proof demonstrates

to satisfy

dimY

£ max(dim X ,k +2).

Our

128

k = 0

next task is to prove that for sharpened to

dimY

(see 6 ).

- dim X

4 (LEMMA).

r

: Y

If id Yv

Vi nV then the formula

Y

00

be a topological space, and let iY^^^ =q 0 whenever k - 1 > 1 . be a fundamental cover of Y such that yk n y i If Y, „ D Y, is a stronq deformation retract of Y, for all k > 1 , — k- 1 k ^----------------------k — — — — then Yn is a strong deformation retract of Y . PROOF. from

Let

the last equality may be

x I

is a homotopy, stationary on k

to a map which takes

y G Y, J k

if

into

Yk - 1

n Yk ,

t £ 2

and

(y /1 ) F i ( F 1 +1 ( ... F k (x,1 ) ...,1),2 t-1 ) ,

if

y € Y. J k

2 1 a t < 2 1+1

defines an Y^-homotopy into

Y x i

y

from

id Y

and

(1

S k),

to a map which takes

Y

Yq . 5.

Given any connected cellular space

contractible one-dimensional subspace of PROOF.

X

Fix an arbitrary 0-cell k

1-cells.

x^

Since

there is a

containing all the 0-cells. in

the set of all 0 -cells that can be joined to which touches at most

X,

^0

ske^X

X

and let

by a path

Now given any 0-cell

closed 1 -cell

c (x)

joining

x E A^ \ Ak-1 x

k ^ 1,

to some cell in

O'

if

k = 0,

if

k > 0,

I

be t ske^X

is connected (see

1.4.7), and a path can touch only a finite number of cells, = ske^X.

A^

k- 1

U k =0 Ak pick a

k- 2 '

and set

Yk = U

c (y) ,

y e A k'Ak-i and

uu

Y = Uk=Q Yk .

containing

skeQX,

Obviously,

and the cover

of Lemma 4. Therefore, YQ i.e., Y is contractible. 6.

Y

is a one-dimensional subspace of {Yk }

of

Y

X

satisfies the conditions

is a strong deformation retract of

Y,

Every connected n-dimensional cellular space is homotopy

equivalent to a cellular space of dimension at most only one 0-cell.

n,

and having

In particular, every connected one-dimensional

cellular space is homotopy equivalent to a bouquet of circles. This results from 5, 1.3, and 1.3.7.7.

129

Applications to Cellular Constructions 7. su(X,Xq),

If the cellular space

where

xQ

X

is k-connected, then

su X

and

is a 0 -cell, are (k+1 )-connected.

The proof reduces to three remarks.

First, since

su X

and

have the same homotopy type (see 1.4.5, 1.3.6 .8 , and 1.3. 7. 7),

su(X,xQ)

the (k+1 )-connectedness of one is equivalent to the (k+1 )-connectedness of the other. verify that if

Secondly, according to 2 and 1.3.7.12, it is enough to su(X,xQ)

ske^X = X q ,

is (k+1)-connected when

ske^X = x Q .

then the (k+1)-connectedness of

s u

is a

(X,X q )

corollary of Theorem 2, because under this assumption

And thirdly, 1 su(X,x )

ske,

k+f

u

also reduces to a point. 8.

a 0-cell of and

Suppose X^,

X^

is a k^-connected cellular space and

i = 1,2.

(X.,x ) 0 (Xn ,xn) 1 1 c 2 2

Then the tensor products

is

(X^,x^) 0 (X^^x^)

are (k +k +1)-connected. ---------1 2

Again, the proof reduces to three remarks. (X^,x^) ® (X^fX^)

x^

First, since

induces on its compact subsets topologies which are

identical to those induced by the topology of

(X^,x^)

(X^fX^,

the

(k1 +k2 + 1 )-connectedness of one of these spaces implies the (ki+k2 + 1 )-connectedness of the other. it is enough to verify that

(X^,x^)

Secondly, using 1 and 1.3.7.12, (X^/X^)

is (k^+k2 + 1 )-connected

when

skev X = x and ske, X = x 9 . Thirdly, under these k^ 1 1 .K2 circumstances, the (k1 +k2 +1 )-connectedness of (X^,x^) ©c (X2 ,x2) follows from Theorem 2, because

skek

reduces to a point. 9 (LEMMA). 0 -cells (XwX„) 1 1

x^ * C

®c (X2 'x 2>)

also

^ and X 2

For any cellular spaces

with

and x 2 taken as base points, the cellular join (X0 ,x„) is homotopy equivalent tosu((X„,x„) ®_ (X0 ,x0 ),bp). z

PROOF. su((X 1 ,x1) ©c

z ------------ — 1 1

I I

By definition, the spaces

(X2 ,x2 ),bp)

(X^x^)

are obtained from

C

z

*c (X2 ,x2>

z

and

(X1 x^ X2> x I by taking

quotients two times, and the projection (X1 xc x2^ * 1 -> su ( (x^ ,x ^) ©c (X2 ,x2 ),bp) is constant on the elements of the partition

zer (pr:(X., xc x2 ) x I -> ^

, x ^ ) *c

(X2 ,x2)). The resulting

map f = fact [pr: (X1 : su((X

xc x2 ) x I -v(X1 ,Xl) *c (X2 ,x2)]: ®c (X2 ,x2 ),bp) -> (X1 ,x1 ) *c (X2 ,x2 )

130

is factorial (see 1.2.3.4). zer(f)

Since the only element of the partition

which does not reduce to a point is

-1

(bp), we see that

(X2 ,x2 ),bp) = [ (X1 /X1 ) *c (X2 ,x2)]/f 1 (bp) .

sudx^x^ note that

f

f_ 1 (bp) = [(X ,x.j) *c (x2 ,x2 >] U [ ( x ^ x ^

Finally,

*c (X2 ,x2 )],

and

since this union is contractible, the quotient space [(X.jjX.j) *c (X2 ,x2)]/f 1 (bp) (X1 'x 1 > *c (X2 'X 2 )* 10. Letthe

cellularspaces

respectively k^-connected. (X1 ,x^ ) * (X2 ,x2), and 0 -cells, are

is homotopy equivalent to

X^

X 2 be

and

Then the joins

X^ * X2 ,

(X1 /x 1) *c (X2 ,x2 >,

and

X^ *c X2 , x1

where

k^and

x 2 are

(k^ +k2 +2 )-connected.

The proof reduces to four remarks.

First, since

(X1 fX1) * (X9 ,x9) i s a q u o t i e n t of X * X 9 by a c o n t r a c t i b l e s p a c e 1 1 C Z Z I c z (the closed 1 -cell x^ * x 2), (X^,x^) *c (X2 ,x2 ) and X^ *c X 2 have the same homotopy type.

Secondly,

subsets the same topologies as does

X^ *c X 2

induces on its compact

X^ * x 2 ,

and hence the (k^+k2 +2 )-

connectedness of one of these spaces implies the (k^ +k2 +2 )-connectedness of the other.

(X ,x^) * (X2 ,x2)

Thirdly, and from the same reason,

(ki+k 2 +2 )-connected if and only

if (X^,x^)*c (X2 ,x2)

is (k^+k2 +2 )-

Fourthly, the (k^+k2 +2)-connectedness of

connected.

is

(X^,x^) *c (X2 ,x2)

is an immediate consequence of 9, 8 , and 7.

4. 1

Simplicial Approximation of Cellular Spaces (LEMMA).

00

Suppose that

X

and

Y

are cellular spaces and

OO

r r=0 and Let f : X Y

r r=0 are filtrations of X and Y be a cellular map such thatf (X^) c: Y^

If all the maps

ab f : xr ^ Yr

PROOF.

Since the

by subspaces. (0 £ r £ «>) .

are homotopy equivalence, then so is = Cyl(abf : X^ + Y )

f.

are cellular sub­

spaces of Z = Cyl f and satisfy the conditions Z cz Z „ and i r r +1 U n Z = Z, they yield a filtration of Z (see 1.1.9). Thus, the k image of any continuous map D Z is contained in one of the sets Z^

(see 1.2.4.5),

and so the pair

(Z,X)

is «-connected provided that

all the pairs

(Z ,X ) are ^-connected. Now note that (Z ,X ) is r r r r »-connected if and only if ab f : X^ Y^ is a homotopy equivalence; similarly,

(Z,X)

is “-connected if and only if

equivalence (see 1.3, 1.6, 1.3.3.9, and 1.3.7.13).

f

is a homotopy

131

2. Given any cellular space

X,

there is a simplicialspace

which has the same homotopy type and the same dimension as X ,and finite or countable together with X . The proof

consists of producing three sequences:

simplicial spaces r•

{Y }

1 00

Yr + 1 J

11 :

{fr * . skerX

,

one of

one of simplicial embeddings

and one of cellular homotopy equivalences

Yr }r_Q, (l)

,

is

with the following four properties:

fr|ske X = lr-1 ° fr-1; ' r- 1

(ii)

dim Yr = dim ske^X ;

(iii)

if

ske^X

is finite (countable), then

Y^

is finite

(respectively, countable); (iv)

if

ske X = ske .X, r r- 1

then

Y

r

= Y

. r- 1

and

i . r- 1

* l dYr - 1 • This will enable us to define the simplicial space having dimension

dim X ,

well as a cellular map

and finite or countable together with

f : X -> lim(Y ,i ) r r

Finally, we shall use Lemma 1 to show that Define

Yq

and

that simplicial spaces simplicial embeddings constructed for ske

„X U A, cr-1 cp

r < where

limfY^i^)

f^

Y , ir_'| '

as

ske^X

X,

as

fI , = imm 0 f. sKe x. r 1 r is a homotopy equivalence.

such that f and

id ske^X ,

and assume

cellular homotopy equivalences

f^,

and

satisfying (i)-(iv), are already

q. By 1.2.1, we may represent A = II n v (D = Dq ) J— Letcell qX e

and

ske^X as cp

is a continuous

map of

Z = I I , -- v (S = Sq“1) into ske X. Next triangulate - — Le€cell X e q-1 q A so that Z becomes a complete subspace and the map ° ^ : ^ ^ Yq-1 admits a simplicial approximation g: E -> Y .. Further, order Ya _ 1 q i Si 1 and £ in such a manner that the map g becomes monotone. Applying c

successively Theorems 1.3 .7.10, 1.3.7. 8 , and then again 1.3.7.10, we obtain three homotopy equivalences: a homotopy equivalence aqrees with ^

f . q- 1

on

ske

A ske X -* Y . Uf q q 1 q - 1 >■ Scyl g ]

inclusion

which

Yq_i ;

and a homotopy equivalence where

which

Yq_i Ug ^

(Scyl g)

(see 2.5.11), whichagrees with

Y _1 Scyl g on Y . q i vi At last, we may define Y

M

as

(Scyl g)

U. A,

f

A, the

^1

as

the

132

composition of the three homotopy equivalences above, and

as the

composite embedding

A

of the cylinder

i q * Scyl g -> Y ^ . The triangulations of

Scylgyield together a

2.5.2).

It is plain that

Y , f , q q

and 3.

countable.

(see q -*-s a simplicial embedding and that

PROOF. X

Y

i „ satisfy conditions (i)-(iv) for r = q. q-1 Let X and Y be cellular spaces with X finite and Then the settt(X,Y)

case when

triangulation of

and

and

Y

is countable.

By theorems 2 and 1.3. 1.8, we need only consider the Y

are simplicial spaces.

Theorem 2.4.7 shows that the cardinal of

Under this assumption,

tt (X,Y )

cardinal of the set of all simplicial mappings

does not exceed the

bamX

Y

(m = 0 ,1 , . ..) ,

and the latter is obviously countable.

5. 1. equivalent,

Exercises

Suppose that the cellular spaces i = 1,2.

Show that

X^ x^ x 2

and

X^

and

Xjj ^

equivalent, and that the same is true for the spaces

X^ x^

are homotopy are homotopy

X^ *c X 2

and

X i *c X 2 " 2. Show that every cellular space is homotopy equivalent to a locally finite cellular space. 3.

Show that every cellular pair is homotopy equivalent to

a simplicial pair, and that every finite cellular pair is homotopy equivalent to a finite simplicial pair. 4.

Show that there is no cellular space having the same

homotopy type as the subspace of the real line consisting of the points 0 and 1 /n, n = 1 ,2 ,... .

Chapter 3. Smooth Manifolds

§1 .

1. 1.

Topological Manifolds

This chapter comprises an elementary introduction to

differential topology. manifolds.

FUNDAMENTAL CONCEPTS

The basic objects of this theory are the smooth

They are defined in the next subsection and represent (as do

cellular and simplicial spaces) topological spaces with an additional structure.

The present subsection is devoted to topological manifolds,

which occupy

an

intermediate position between smooth manifolds and

topological spaces, and do not carry an additional structure.

Locally Euclidean Spaces 2.

A topological space is said to be a an n-dimensional

locally Euclidean space if each of its points has a neighborhood homeomorphic to the space

]Rn

or to the half space

]R^,

where

]R^

is the

set of all points (x„ ,...,x ) G ]Rn with x>, £ 0. The half space ]Rn " I n I is defined for n ^ 1 ; we do not define it for n = 0 and, accordingly, a 0-dimensional locally Euclidean space is simply a topological space such that each of its points has a neighborhood

homeomorphic to

]R^,

i.e., a discrete space. In a locally Euclidean space neighborhood homeomorphic to

IRn

X,

the points having a

are called interior points,

remaining ones are called boundary points.

while the

The interior (boundary)

points form the interior (respectively, the boundary) of the locally Euclidean space

X,

denoted by

ference between the notations

intX int,9

(respectively and

Int,Fr

3X).

(The dif­

should prevent us,

134

in each context, from confusing the interior and boundary points, and the interior part and boundary defined here with the interior and boundary points and the interior part and boundary of a set in a topological space.)

Clearly, the interior of

whereas the boundary of

X

X

is a dense open set,

is closed.

If each point of a topological space has a neighborhood homeomorphic to an open subset of

IRn

or

,

then obviously it already is

an n-dimensional locally Euclidean space.

Consequently, every open

subset of an n-dimensional locally Euclidean space is also an n-dimensional locally Euclidean space.

In particular, the interior of

an n-dimensional Euclidean space is an n-dimensional locally Euclidean space without boundary. subset

U

Moreover, the interior and boundary of an open

of a locally Euclidean space

= U n int X

and

X

are given by

int U

=

d\J = U fl 3X.

Since a locally Euclidean space is locally connected, its components are open (see 1 .3.4.3), and hence also closed. Obvious examples of n-dimensional locally Euclidean spaces are

3Rn , 3R^,

Sn ,

and

Dn .

It is clear that

3nRn = 0

and

Furthermore, all the boundary points of the half space limiting hyperplane that

x^ = 0 ,

^,

consisting of the points

we see that the product

X^ x

and

n^

of

D

such

lie in the

Sn 1.

3. Since the product X^

lie in the (x ,.../XR )

and all the boundary points of the ball

limiting sphere

3Sn = 0.

dimensions

HR

n

x 3R

n2

is homeomorphic to

X

n i +n2

of two locally Euclidean spaces

and

,

,

X^

and without boundary, is an

(n i+n2)-dimensional locally Euclidean space. i.e., a product

3R

This is true in general,

of boundaryless locally Euclidean spaces s X i,...,X of dimensions n i ,...,n , is an (n I+...+n )-dimensional s s s boundaryless locally Euclidean space. Turning to locally Euclidean 1

x ... x x

spaces with boundary, note that the formula (x ,. ..,x

fyw...fy 1

n1

1R

for

x hr

n2

1

), 2

n l+ n 2 ]R r

n 1 > 0.

((x ,...,x ),(y ,...,y )) 1 nl 1 n2 which gives the canonical homeomorphism

also defines a homeomorphism

Similarly, the formula

((x , .. .,x

ni n2 3R x ]R ) , (y , ...,y

n1

** (y-i /•••fY

,...,x 2

for

n^ > 0 , 2

)

defines a homeomorphism

n2

x ]R_

ni n2 nR x ]R

((x^,...,x^ ) , (y^,...fy

2

n l+ n 2 -+ nR_

for

n i+n? -* 1R

"

(-2 x^y^,x^ - y^,X2 ,...rxn ,y2 '#’’'yn ^ R1

))

n2

1

and the formula

n i+n? 3R

n 1 > 0,

)) ^

defines a homeomorphism

n 2 > 0.

Thus, each of the products

135

n 1R_

x

IR

^1

,

n2

JR

x

3R_ ,

^ IR

and

x

IR_

conclude that given locally Euclidean spaces n^

and

n2 ,

the product

Euclidean space.

X^ x X^

n. +n^ HR.

is homeomorphic to and

X^

We

of dimensions

is an (n^+n2)-dimensional locally

In general, the product

locally Euclidean soaces of dimensions "

X^ x ... x X g

n., ...,n 1

of arbitrary

is an (n.+...+n )1 s

s

dimensional locally Euclidean space. 4.

The discussion in 2 raises two nontrivial questions.

The first one is whether a nonempty topological space can be a locally Euclidean space of dimension

n

and, simultaneously, a

locally Euclidean space of a different dimension

n' :

n1

n f

?

Chapter 4 this question is answered negatively (see 4.6.5.10). answer is obvious when

n1 > 1

n = 1,

or

n ’ =1,

n > 1.

In

The

In fact,

any connected subset

of a one-dimensional locally Euclidean space

becomes disconnected

after one removes two suitably choosen points (for

example, two points belonging to an open subset which is homeomorphic to

IR );

in contrast, every nonempty locally Euclidean space of n' > 1

dimension

contains a nonempty open subset which cannot be disconnected by removing two points (any open subset homeomorphic to IRn ^ has this property). n 1 =0.

The picture is cristal clear when

However, when

n > 1,

n ’ > 1,

n = 0

or

the proof requires a technique

which we will develop only later. The second question is whether we can formulate more efficient definitions ofthe interior and

boundary points, which would permit us

to actually recognize

them.

At this point we can

show only the trivial inclusion

2 ) , and we are

9IR^ = IR1? 1

For example, consider the half space

IR^.

3IR^ c ir1^

(see

forced to settle for one of the extreme equalities

or

3IR^ = 0

(which obviously are the only possible ones) . n -n— 1 We shall prove in Chapter 4 that 3lR_ = IR^ (see 4.6.5.12). This n = 1

equality is plain for neighborhood in 0

1

IR_

(assuming that the point

which is homeomorphic to

we would disconnect this neighborhood;

1

IR ,

has a

then by removing

this is absurd, because the

latter cannot happen to a connected neighborhood of for

0

0

in

1

IR_).

But

n > 1,

equality

we again need techniques which are to be developed. The ^ •>! 3IR_ = IR^ settles satisfactorily the general problem of

recognizing the interior and boundary points too. that a point

x

of the n-dimensional locally Euclidean space

has a

neighborhood

point

of

U,

Indeed, it follows

U

with a homeomorphism U

and hence, of

X,

if and only

X

which

IR^, is. a boundary ifthishomeomorphism

into a point of the hyperplane IR1? 1 . For Dn this theorem n n—1 asserts that 3D = S . Finally, we note that the alternative takes

x

equality

3IR^ = 0

would obviously imply that

9X = 0

for any

136

n-dimensional locally Euclidean space. 5.

In general, it would be more prudent not to use the

theorems formulated in 4, equality

= IR^ \

i.e., the theorem on dimensions and the

as long as they have not been proven.

This

indeed is the way we shall deal with the theorem on dimensions - the only exception is a harmless remark used the

equality

33R^ = 3R^ \

in 2.3.

However, we have already

and we will take advantage of it again,

before its proof, in 7 and in 2.6, 2.7.

But these are the only

instances where these theorems and their corollaries will be used before their proofs. 6.

The boundary of an n-dimensional locally Euclidean space

is an (n-1)-dimensional locally Euclidean space without boundary. Let and let

U

boundary

x

be a boundary point of the locally Euclidean space

be a neighborhood of 9U

is a neighborhood of

and is homeomorphic to

HRn ^ ,

to Chapter 4 for the equality alternative

x,

= 0

dlR

7.

homeomorphic to x

in

3X,

since

IR^.

X

Then the

3U = U fl 3X,

since 8 3R^ = 3R^? ^ (here the reference n n 1 8nR_ = nR^ is unnecessary: the

is excluded, because

3X ^ 0).

For any locally Euclidean spaces

X ^ ,.

'x s

int(X. x ... x X ) = int X. x ... x intX 1 s 1 s and 9 (X, x ... x x 1

) = OX, x x o x. ..x x ) U . . . U (X x . ..x X x3X ) . s 1 2 s 1 s- 1 s

It is enough to prove the statement for

s =

2. Let

x. E X . i

i

and let cp^ be a homeomorphism of a neighborhood LL of onto n. n. JR or HR_ , i = 1,2. Then cp^ x ^ is a homeomorphism of the x U2

neighborhood

of the point

(x^x^

n1 n2 n1 n2 nl n2 3R x IR , HR x HR_ , IR_ x IR or

3R_

onto one

n1 n? x JR_ ,and composing it

withone of the homeomorphisms exhibited in 3, we x u2

of

onto

n 1+n 2

!IR

or

n 1+n2

3R_

.

cp and analyse the four possible cases. n2

- IR

,

then

n 1+n2

cp(U^ x U2> - 3R

interior points.

If cp1 (U1 ) = 3R

n l+ n 2 cp(U^ x U2> = 3R_ , and n2 _1 ^ 2 ^x 2^ ^ ^1 “

Thus

nl

obtain a homeomorphism

We denote this composition by If

nl cp^ (U^ ) = IR

and

x^ , x 2 ,

and

n? )) = ip~^ ((-00,b^)) . 17.

Therefore, in case (ii)

X

is homeomorphic to

S1 .

Every compact, connected, one-dimensional manifold is S1

homeomorphic to either PROOF.

or

D1 .

For a start, assume that the given manifold is closed.

Then it can be covered by a finite number of open subsets homeomorphic 1

to 1R ,

and we may arrange these in a sequence V, = U 1 U ... U U,

each union

K

I

-rC

is connected.

U^,...,Us

such that

According to Lemma 16, -j

the first of the sets 1 1

morphic to

S ,

V 1,...,V not homeomorphic to IR is homeoi s and being both open and closed, it is the entire

manifold, which is thus homeomorphic to

S1 .

Assume now that the manifold has a boundary.

Then its double

is a closed, connected, one-dimensional manifold, and as such is homeo1

morphic to subset of

S . 1

S .

Therefore, the original manifold is homeomorphic to a Since this subset is connected, closed, nonempty, 1

different from

S ,

and not reduced to a point, it is homeomorphic to

D1 . 18 (LEMMA).

If a topological space

X

can be represented

as the union of a nondecreasing sequence of open subsets, all homeo­ morphic to

IR1 ,

then

PROOF.

Let

X

X = U

any homeomorphism of homeomorphism of (a,b+1),

or

cp2 : V2 ^ A2 ,..., interval U Ai ,

IR1 .

be the given representation.

onto some interval

(a,b)

V j_+-| onto one of the intervals

(a-1,b+1).

of intervals

is homeomorphic to

Hence one

Clearly,

extends to a (a,b),(a- 1 ,b) ,

can construct inductively a sequence

and a sequence of homeomorphisms suchthat ^ = abcpi+ 1 . which agrees with on

A ^,

The map of X onto the v i , is obviously a homeo-

141

morphism. 19. Every noncompact, connected, one-dimensional manifold is homeomoiphic to either IR or IR . PROOF. boundary.

Then

First, assume that the given manifold X

X

has no

can be covered by a countable family of open subsets,

all homeomorphic to

IR ,

and we can arrange these in a sequence

^1'^2'#,‘' such that all unions these unions are homeomorphic to 1

not homeomorphic to

IR

U ... U are connected. Then all HR1 . Indeed, if not, the first of them

is, according to Lemma 16, homeomorphic to

and being open and closed must coincide with

X;

1

S ,

contradiction.

Therefore, one can apply Lemma 18 to our manifold and deduce that it is homeomorphic to

IR .

Now assume that

X

has a boundary.

Then

dopp X

is a non­

compact, connected, one-dimensional, boundaryless manifold, and must be homeomorphic to

1

IR .

It follows that

closed, noncompact subset of homeomorphic to

1

IR ,

X

is homeomorphic to a connected,

different from

IR1 ;

as such, it is

1

IR_ .

2.

Differentiable Structures

1.

n IR

Recall that a real function defined on an open subset of r r is of class C (or a C -function) if it has continuous partial

derivatives of all orders up to and including that

0
supp cp n s u p p \p -— which are inverses of one another, are of class morphisms for

r > 1

and homeomorphisms for

is trivially satisfied whenever

implies the equality of their dimensions. also from the C^-compatibility of

r C

r = 0).

supp cp n supp iii = 0 .

nsupp if;^ 0,then the C -compatibility

supp cp

cpfsupptp n supp ij),

cp

and

r (i.e., C -diffeoThis condition If

of the charts

tp and

ip

In fact, this equality results \p,

as shown by 1.4.

A collection of charts is an n-dimensional Cr-atlas of the set

X

if these charts cover X, are n-dimensional, and each two of r r r them are C -compatible. Two C -atlases of X are C -equivalent if their union is again a C -atlas.

This is clearly an equivalence

relation, and the equivalence classes of n-dimensional Cr-atlases of the set with

X

r > 0

are called n-dimensional Cr-structures.

The Cr-structures

are called differentiable structures.

Clearly, if

0

£ q £ r,

then each n-dimensional Cr-atlas is

143

also an n-dimensional C^-atlas, and two equivalent Cr-atlases are also Cq-equivalent.

Thus, when

0 £ q £ r

every n-dimensional

- structure

uniquely extends to a C^-structure. r Every C -structure contains a maximal atlas, namely the union of all its atlases.

The latter is called the complete atlas of the

structure, and its charts are called the charts of the structure. When we pasr. from a Cr“Structure to its C^-extension, the complete atlas extends too. 4.

A set endowed with an n-dimensional Cr-structure is called

an n-dimensional Cr-space.

The charts and atlases of the structure are

refered to as the charts and atlases of the space. 27 of a C -space X is denoted by AtlX . The coordinate functions of a chart are called coordinates on in X.

supp cp

We shall denote by Cr-space

X

cp

The complete atlas of the Cr-space

or, alternatively, local coordinates

C^X

the C^-space obtained from the

by extending its Cr-structure to a C^-structure,

The Cr-spaces with

X

r ^ q

are also termed

0 £ q £ r.

spaces.

For examples of Cr-spaces we may look at all the open subsets X

of

IRn

or

1R^,

with the Cr-structure defined by the atlas reduced

to the single chart id IRn

and

id 3R^ 5.

U ',

IRn

3Rn

and

3R^

r,

the charts

into n-dimensional Cr-spaces.

its complete atlas consists of all possible homeomorphisms

where

subset of

transform

In particular, for any

Every n-dimensional locally Euclidean space has an obvious

C^-structure: U

id: X -> X.

U or

is an open subset of the space and JR^.

U1

is an open

On the other hand, applying the "union of

topological spaces" construction (see 1 .2 .4.3) to the complete atlas of a given n-dimensional C^-space, we obtain an n-dimensional locally Euclidean space, and this transition is the inverse of the previous one. Therefore, C^-spaces are just locally Euclidean spaces. Since any differentiable structure extends uniquely to a C -structure, every Cr -space with r > 0 is also a locally Euclidean 0

space.

Its topology may be described in a more direct fashion as the

topology of the union constructed from any atlas of the structure. 6.

Obviously, every point of an n-dimensional C -space

can be covered by a chart points with

Im cp = IRn

cp

of

X

such that

Im cp = IRn

or

IR^.

X The

are called interior points and form an open

dense set, called the interior of the space

X,

denoted by

intX .

The remaining points are called boundary points and they form a closed set, called the boundary of

X,

denoted by

9X.

These notations are

144

in agreement with those introduced at 1.2.

In fact, when

r - 0, the

previous and present definitions of the interior and boundary points coincide. When

r > 0,

we use Proposition 2 (ii) in order to recognize

the interior and boundary points. r > 0

a point

and only if space with with for

ofan n-dimensional C -space is

X

Imp c

and

x £ supp cp .

r > 0, r = 0

According to this proposition, when

cp(x) £ 1R^ 1, where cp

In particular, if we regard

3 ]R^ = ]R^

then

a boundary point if is a chart on this 1R^

as a Cr-space

Recall that the corresponding statement

appeared in 1.4 and its proof was postponed until Chapter 4.

The above characterization of the boundary points shows that the interior and the boundary of a its Cr-structure to a

- space

do not change when we extend

C^-structure, for any q £ r.

In other words, for

0 £ q £ r,

int(C^X) = intX

and 3(C^X)

=3X.

We emphasize that the

equalities

int(C^X) = intX

and 3 (C^X)

= 3X

were proved by a refe­

rence to 1.4, i.e., they depended upon results from Chapter 4, whereas the equalities

int(C^X) = intX

9 (C^X)

and

= 9X

for

q > 0

need no

such reference. Using the relation valid for Cr-spaces with

3X = 3(C°X),

r > 0

too.

we see that Theorem 1.8 is

That is to say, a Cr-space is

connected if and only if its interior is connected.

However, this

27

C -variant of Theorem 1.8 can be proved by merely repeating the proof of the original theorem, and therefore we can eliminate the reference to Chapter 4. 7. Suppose

A

is an open subset of an n-dimensional

27

C -space

X. Then thecharts of X whose supports are included in 27 yield a C -atlas of the set A, and define an n-dimensional

A

Cr-structure on A. In this way, any open subset of an n-dimensional r r C -space is an n-dimensional C -space. In particular, the interior of r r any n-dimensional C -space is an n-dimensional C -space without boundary. Moreover,

the interior

Cr-space

X are obviously given by Suppose

tp

and the boundary of an open subset

of the

3U = U n 3 X.

and

is a chart on an n-dimensional Cr-space

ab tp : 3X fl supp tp ->• cp(3X fl supp tp) 3X.

intU = U n intX

U

X.

Then

is an (n-1 )-dimensional chart on 9X,

In this way we may construct a Cr-atlas of the set

and so

define a Cr-structure on 3 X. Thus, the boundary of an n-dimensional r 2T C -space is an (n-1)-dimensional C -space without boundary. If (P1 such that

(tp2) n1

Imcp^ = IR

then the composition of

is a chart on the C or

n1

]R_

x tp2

r 1

-space

(respectively,

X1

(C n?

Im ip^ = ]R

r? -space or

H

with one of the homeomorphisms

no

yi

),

)

145

]R

1

x

2

]R

n l+ n 2 ]R

n1

and

3R_

x

n2

n i+n? -> HR_ ,

nR

defined in 1.3, provides

an (n^+n2 )“dimensional chart on x X 2 - If constructed as above form a Cr-atlas of the set r = min(r 1 ,r2),

and hence define

a C -structure on

rl product of the n 1-dimensional C -space C r

2

X2

-space

3X 2 = 0

with

= Xx X~,

X1

then the charts with

X

x X„.

Thus the

and the n 2-dimensional

is an (n^ +n2>-dimensional Cr-space, where

is the smallest of

the numbers r. r of the n^-dimensional C -spaces X^

and

r„.In general, theproduct

.

one of them has a boundary, is an

i = 1,...,s,

such that at most

(n^+...+ng)-dimensional Cr-space,

where

r = min(r , .. .,r ). Moreover, int (X x ... x x ) = intX i s I s x int X g , and if X^ is the only space having a boundary, then

x ...

X ) = X, x ... x X. 1 x 9X. x x . , x ... x x . For s 1 1-1 i i+1 s both formulas can be found in 1.7; for r > 0 , they are plain.

r = 0

3(X1 1

...

x

x

It is clear that when we extend the d °-structure of the cr r X to a C -structure, the C -structures induced on the open

r C -space subsets of

X

and on its boundary

X

also extend to C^-structures,

and that C^(X 1 x ... x X s ) = C^X.1 x ... x C^X s . In particular, the r r 27 topology defined by the above induced C -structures coincide with the r-L relative topology, and the product of the C -spaces X^, i = 1,...,s, considered as a topological space, is just the product of the topological spaces

X

,...,X . s

I

Smooth Maps 8 . A continuous map

isof

chart

n f

is of class

Y,

-

-1 ,

,x , ab cp (supp ip ))

Cr

Obviously, a map

X

into a C

cpon

X

^r -space

and any

the composite map

(see 1). f: X

1

^ j-— 1 ,

supp cp n f

lXabf , (suppiji) >supp ip

ib

-

.

Im ijj

Such a composite map is called a local

representative of the map

f; Y

we use the notation

loc(cp,^)f. r -spaces is of class C if and only

of C

if the local representatives of of

>r -space

r r class C , or a C -map, if for any chart on

cp (supp cp

of a C

-

Y

f

constructed for all the charts of 27 someatlases of X and Y are of class C . Note that this general definition of C-maps contains the

definition given in 1. map.

The maps of class

f

Now, as before, a C _map is just a continuous C

1

are called smooth, and the maps of class

Ca - (real) analytic. r r The composition of two C -maps is obviously a C -map.

If

A

146

is an open subset or the boundary of a A ->■ X

is a Cr-map.

in

or

Y

XT C-space

• • then the inclusion

X,

If A is open in X or A = 9x, and B is open then the compression A B of any C -map X -+ Y

B = 9Y,

nr

is a C -map. 9.

A map

^1

f

of a C " -space

X

into a C

diffeomorphism if it is invertible and both The space

Y

f

^1

and

-space f ^

is said to be diffeomorphic to the space

a diffeomorphism

X -* Y,

C -diffeomorphism

and Cr-diffeomorphic to

is a

are smooth.

X

X,

Y

if there is

if there is a

X -> Y.

r Of course, the identity map of a C " -space with r ^ 1 is a Cr-diffeomorphism. Also, the composition of two Cr -diffeomorphisms is r r a C -diffeomorphism, and the inverse of a C -diffeomorphism is a Cr-diffeomorphism itself, as we may easily see from 2 (iii).

Therefore,

the property of being C -diffeomorphic is an equivalence relation. Using 2 (i), we conclude that nonempty diffeomorphic spaces have the same dimension. 10.

f . : X. -+ Y. ,. ..,f : X Y be Cr-maps with r ^ 1, 1 1 1 m m m ^ where no more than one of the spaces X^,...,Xm , andno more than one of the spaces

Let

Y^,...,Y ,

has a boundary.

Then

-- ► Y. x ... x Y f A x ... x f : X. x ... x X 1 m 1 m 1 m ■e

is obviously a C -map. If f^,...,f^ f x ... x f . 1 m The canonical homeomorphism

are diffeomorphisms, then so is

X^ x x^

feomorphism for any two Cr-spaces

and

X^

X2

X^ x x^

is a Cr-dif-

r ^ 1,

with

such that

one of them has no boundary. The canonical homeomorphisms (X- x ... x x m . ) x Xm -+ X. x ... x X and X, x (Xn x . . . x x ) -+ 1 m- 1 m_ 1 m 1 „ m r r 2 X^ x ... x x^ are C -diffeomorphisms for any C -spaces X^ , .. .,X with

r ^ 1,

such that no more that one of them has a boundary.

Subspaces 11.

A subset

is a k-dimensional subspace of a chart

0,

then this condition is obviously equivalent to the following one: point of A is covered by a chart cp of the space k k (P(A D supp cp) = Im