Selected Title s i n Thi s Serie s

208 Yuki o M a t s u m o t o , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i U e n o , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l equations an d Secondar y Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d

coding, 200 2

202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r

nonlinear PDE . I , 200 1

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology ,

2001

197 Kenj i U e n o , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes ,

2001

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C* -algebras an d ellipti c

operators i n differentia l topology , 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n geometry, 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces ,

2000

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e

phenomena, 200 0

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f

random variable s an d rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d

applications, 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r

theory 1 : Fermat' s dream , 200 0

185 Kenj i Ueno , Algebrai c Geometr y 1 : From algebrai c varietie s t o schemes ,

1999

184 A . V . Mernikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9 182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d

conservation law s fo r differentia l equation s o f mathematica l physics , 199 9 (Continued in the back of this publication)

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 20 8

An Introductio n t o Morse Theor y

Yukio Matsumot o

Translated b y Kiki Hudso n Masahico Sait o

American Mathematica l Societ y '•? Providence , Rhod e Islan d

10.1090/mmono/208

Editorial Boar d

Shoshichi Kobayash i (Chair ) Masamichi Takesak i

MorsellliOSil

M O R S E R I R O N N O K I S O

( A N I N T R O D U C T I O N T O M O R S E T H E O R Y )

b y Y u k i o M a t s u m o t o

Copyright © 199 7 b y Yuki o M a t s u m o t o Originally publ ishe d i n Japanes e

by Iwanam i Shoten , Publ ishers , Tokyo , 199 7

Trans la ted fro m t h e Japanes e b y Kik i Hudso n an d Masahic o Sait o

2000 Mathematics Subject Classification. P r i m a r y 57-01 ; Secondary 57R19 , 57R65 , 57R70 , 57M25 , 57M99 .

ABSTRACT. Thi s boo k aim s a t introducin g Mors e theor y t o undergraduat e o r ba -sic graduat e students . Th e emphasi s i s o n Mors e theor y o n finite-dimensional manifolds. Th e topic s covere d includ e Mors e functions , handlebodies , handl e de -composition o f variou s manifold s an d Li e groups , slidin g an d cancelin g handles , Poincare duality , intersectio n forms , low-dimensiona l manifolds , an d Ki r by calcu -lus.

Library o f Congres s Cataloging-in-Publicat io n D a t a

Matsumoto, Y . (Yukio) , 1944 -[Morse riro n n o kiso . English ] An introductio n t o Mors e theor y / Yuki o Matsumot o ; translate d b y Kik i

Hudson, Masahic o Saito . p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-928 2 ;

v. 208 ) (Iwanami serie s i n moder n mathematics ) Includes bibliographica l reference s an d index . ISBN 0-8218-1022- 7 (pbk . : acid-fre e paper ) 1. Mors e theory . I . Title . II . Series . III . Series : Iwanam i serie s i n moder n

mathematics.

QA331 .M442713 200 2 514—dc21 200104575 1

© 200 2 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . Visit th e AM S hom e pag e a t URL : ht tp: / /www.ams.org /

10 9 8 7 6 5 4 3 2 1 5 1 4 1 3 1 2 1 1 1 0

Contents

Preface

Preface t o th e Englis h Translatio n

Objectives

Chapter 1 . Mors e Theor y o n Surface s 1.1. Critica l point s o f function s 1.2. Hessia n 1.3. Th e Mors e lemm a 1.4. Mors e function s o n surface s 1.5. Handl e decompositio n

a. Th e cas e when th e inde x o f po i s zer o b. Th e cas e when th e inde x o f po i s one c. Th e cas e when th e inde x o f p$ i s two d. Handl e decomposition s

Summary Exercises

Chapter 2 . Extensio n t o Genera l Dimension s 2.1. Manifold s o f dimension m

a. Function s o n a manifold an d map s betwee n mani -folds

b. Manifold s wit h boundar y c. Function s an d map s o n manifold s wit h boundar y

2.2. Mors e function s a. Mors e function s o n ra-manifolds b. Th e Mors e lemm a fo r dimensio n m c. Existenc e o f Mors e function s

2.3. Gradient-lik e vecto r fields a. Tangen t vector s

V

vi C O N T E N T S

b. Vecto r fields c. Gradient-lik e vecto r fields

2.4. Raisin g an d lowerin g critica l point s Summary Exercises

Chapter 3 . Handlebodie s 3.1. Handl e decomposition s o f manifold s 3.2. Example s 3.3. Slidin g handle s 3.4. Cancelin g handle s

Summary Exercises

Chapter 4 . Homolog y o f Manifold s 4.1. Homolog y group s 4.2. Mors e inequalit y

a. Handlebodie s an d cel l complexe s b. Proo f o f the Mors e inequalit y c. Homolog y group s o f complex projectiv e spac e

CPm

4.3. Poincar e dualit y a. Cohomolog y group s b. Proo f o f Poincare dualit y

4.4. Intersectio n form s a. Intersectio n number s o f submanifold s b. Intersectio n form s c. Intersectio n number s o f submanifold s an d inter -

section form s Summary Exercises

Chapter 5 . Low-dimensiona l Manifold s 5.1. Fundamenta l group s 5.2. Close d surface s an d 3-dimensiona l manifold s

a. Close d surface s b. 3-dimensiona l manifold s

5.3. 4-dimensiona l manifold s a. Heegaar d diagram s fo r 4-dimensiona l manifold s b. Th e cas e N = D 4

c. Kirb y calculu s

CONTENTS

Summary Exercises

A View fro m Curren t Mathematic s

Answers t o Exercise s

Bibliography

Recommended Readin g

Index

197 197

199

203

213

215

217

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Preface

In a very broa d sense , "spaces " ar e object s o f study i n geometry , and "functions " ar e object s o f study i n analysis . Ther e are , however , deep relation s betwee n function s define d o n a space an d th e shap e o f the space .

For example , le t u s conside r a lin e an d a circle . Bot h ar e one -dimensional spaces . Identifyin g a line with the x-axis , ther e ar e func -tions whic h tak e a s arbitraril y larg e values , suc h a s

y = x 2, y = x 3.

On the othe r hand , ther e exist s no such function o n a circle. For , an y continuous functio n o n a circl e mus t tak e a maximu m valu e some -where o n th e circl e (maximu m valu e theorem) . Thi s way , w e ca n distinguish a lin e an d a circl e b y whethe r o r no t ther e ar e function s on the m tha t tak e arbitraril y larg e values .

Morse theory i s the stud y o f the relations betwee n function s o n a space an d th e shap e o f the space . I n particular , it s featur e i s to loo k at th e critica l point s o f a function , an d t o deriv e informatio n o n th e shape o f the spac e fro m th e informatio n abou t th e critica l points .

Morse theor y deal s wit h bot h finite-dimensional an d infinite -dimensional spaces . I n particular , i t i s believe d tha t Mors e theor y on infinite-dimensiona l space s wil l become mor e an d mor e importan t in th e future , a s mathematic s advance s further .

In thi s series , there ar e two books on Morse theory . Thi s volume , "An Introduction t o Morse Theory, " describe s Mors e theory fo r finite dimensions. Th e othe r volume , "Geometri c Variatio n Problems " b y S. Nishikawa, deal s with infinit e dimensiona l aspect s o f Morse theory .

Finite-dimensional Mors e theor y ha s th e advantag e tha t i t i s easier t o presen t th e fundamenta l idea s tha n i n infinite-dimensiona l Morse theory , whic h i s theoretically mor e involved . Therefore , finite-dimensional Mors e theor y woul d b e mor e suitabl e fo r beginner s t o study.

ix

x PREFAC E

On th e othe r hand , finite-dimensional Mors e theor y ha s it s ow n significance, no t just a s a bridge to infinit e dimensions . I t i s an indis -pensable too l i n th e topologica l stud y o f manifolds . Tha t is , w e ca n decompose manifold s int o fundamenta l block s suc h a s cell s an d han -dles b y Mors e theory , an d thereb y comput e a variet y o f topologica l invariants an d discus s th e shape s o f manifolds .

These aspect s o f Mors e theor y wil l continu e t o b e a treasur e i n geometry fo r year s t o come .

This boo k wa s writte n wit h AMSWFE&, wit h whic h th e autho r was unfamiliar . Th e autho r woul d lik e t o expres s hi s gratitud e t o Professor Toshi o Oshim a a t th e Graduat e Schoo l o f Mathematica l Sciences, Universit y o f Tokyo , fo r installin g th e progra m fo r him , a s well a s othe r hel p th e autho r received .

Finally, th e autho r woul d lik e t o than k th e editor s o f Iwanam i Shoten, wh o were o f grea t hel p i n th e publicatio n o f the book .

Yukio Matsumot o February, 199 7

Preface t o th e Englis h Translatio n

This book was published i n 199 7 in Japanese b y Iwanami Shoten , as a volum e i n th e serie s "Foundation s o f Moder n Mathematics. " I t is a great pleasur e t o me as the autho r tha t th e Englis h translatio n i s published b y th e America n Mathematica l Society . I wis h t o expres s my gratitude by reflecting o n the circumstances leading to the Englis h translation.

It wa s Novembe r 199 7 when I receive d a reques t fo r publicatio n of the Englis h translatio n fro m Professo r Katsum i Nomiz u a t Brow n University. Sinc e then , u p unti l a fe w month s ago , I hav e bee n in -volved in administrative wor k for th e Mathematica l Societ y of Japan , and i t seemed difficul t t o carry ou t th e translation b y myself, so I had to as k someon e else . Dr . Kik i Hudson , wh o ha d bee n a colleagu e fo r over thirt y year s i n th e grou p o f topologist s i n Tokyo , cam e t o m y mind a t once . Sh e wa s a topologis t wh o wa s fluent i n foreig n lan -guages. Sh e alread y ha d experienc e i n translatin g severa l Japanes e books i n mathematic s int o English , an d vic e versa .

Kiki accepte d m y reques t wit h a goo d grace . I recal l tha t he r work starte d i n 1998 . I was looking forward t o seein g he r translatio n completed, bu t I refraine d fro m makin g contac t wit h her , a s I wa s afraid i t migh t distur b he r i n he r bus y schedule . I regre t thi s ver y much now . Fo r I wa s surprise d an d deepl y saddene d b y th e unbe -lievable new s tha t sh e passe d awa y fro m cance r i n Septembe r 1999 . Several month s befor e he r death , Kik i calle d an d tol d m e tha t sh e had gotte n hospitalize d fro m a bad cold , bu t wa s feeling bette r a t th e time. I had neve r imagine d tha t he r illnes s wa s so grave .

After sh e passe d away , he r husband , Dr . Hug h Hudson , tol d m e that sh e ha d finished he r translatio n u p t o th e beginnin g o f Chapte r Three, i n spit e o f he r seriou s illness . I wa s filled wit h sadnes s an d gratefulness.

Hugh contacte d Ms . Chri s Thivierg e a t th e AMS , an d maile d Kiki's manuscrip t t o her . I did no t wan t t o wast e Kiki' s grea t effort ,

xi

xii P R E F A C E T O T H E ENGLIS H TRANSLATIO N

and aske d Dr . Masahic o Sait o a t th e Universit y o f Sout h Florid a t o finish he r work. Masahic o was my student a t th e University o f Tokyo. He, too , gladl y accepte d m y request , an d translate d th e res t o f th e book, payin g attentio n t o consistenc y throughou t th e book .

When the first draf t wa s completed, a n old friend o f mine, Profes -sor Jos e Mari a Montesinos-Amilibi a a t th e Universida d Complutens e de Madrid , kindl y rea d throug h th e manuscript , an d gav e m e nu -merous comments . Furthermore , h e pointe d ou t a fe w error s i n th e original. I believ e tha t Jos e Maria' s comment s greatl y improve d th e contents an d expositio n o f the book .

The Englis h translatio n o f th e boo k woul d hav e neve r bee n ac -complished withou t th e grea t contribution s an d hel p o f th e peopl e I mentioned above . I woul d lik e t o expres s m y deepes t gratitud e t o them.

With th e translatio n i n han d now , I appreciate , deepl y fro m m y heart, th e ol d saying : "Me n d o no t liv e alone. "

Yukio Matsumot o Tokyo, Japa n

June, 200 1

Objectives

The primar y concer n o f Mors e theor y i s th e relatio n betwee n spaces an d functions . Th e cente r o f interes t lie s i n ho w th e critica l points o f a functio n define d o n a spac e affec t th e topologica l shap e of th e space , an d conversely , ho w th e shap e o f a spac e control s th e distribution o f the critica l point s o f a function .

Morse theory of finite-dimensional manifold s i s a powerful too l for the topolog y o f manifolds , an d offer s a unifie d metho d t o "visualize " manifolds wit h theoretica l eyes . O n the othe r hand , Mors e theory fo r infinite-dimensional space s clarifie s th e dee p relation s betwee n vari -ational problem s an d geometry , an d i s on e o f th e basi c principle s o f modern mathematics .

In this book, which deals with finite dimensions , we first introduc e fundamental concept s such a s critical points , the Hessian , an d handl e decompositions, wit h surface s a s examples .

These ar e generalize d t o highe r dimension s i n Chapte r 2 . Th e existence o f enough Mors e function s wil l be proved i n thi s chapte r a s well.

In Chape r 3 , handle decomposition s associate d wit h Mors e func -tions ar e discusse d i n genera l dimensions , an d th e theor y o f handle -bodies i s developed. Whe n w e said tha t Mors e theory offer s a unifie d method t o visualiz e manifolds , w e ha d handlebodie s i n mind . Fur -thermore, w e explicitly construc t Mors e function s o n classica l space s such as projective spaces and Lie groups, and compute indices and th e numbers o f critica l points . Moreover , th e fundamenta l tool s fo r deal -ing wit h handlebodies , suc h a s slidin g handle s an d cancelin g pairs , will be explained . Thi s chapte r i s mos t essential .

In Chapte r 4 , w e discus s th e relatio n betwee n handl e decompo -sitions an d cel l decompositions , an d w e se e tha t homolog y theor y i s made easie r t o visualize by using handlebody structure s o f manifolds . For example , Poincar e dualit y i s nothing bu t a n algebrai c expressio n

xii i

xiv OBJECTIVE S

of "turnin g a handlebod y upsid e down. " Also , i t seem s tha t discus -sions o n intersectio n form s o n manifold s ca n b e mad e well-balance d between intuitio n an d theoretica l rigo r b y usin g handlebod y struc -tures.

Chapter 5 i s devote d t o low-dimensiona l (dimensio n 4 an d less ) manifolds. I n lo w dimensions , handlebodie s ca n b e visualize d explic -itly b y Heegaar d diagram s an d Kirb y diagrams , whic h ca n b e draw n on a piec e o f pape r a s frame d links . Suc h concretenes s make s u s fee l familiar wit h low-dimensiona l manifol d theory , an d a t th e same time , the relatio n t o kno t theor y become s eviden t immediately . I t ca n b e said that kno t theory and low-dimensional manifol d theor y are almos t the sam e subject . Bot h ar e ver y concret e an d mak e u s fee l familiar , but the y ar e no t eas y subjects . W e realize th e difficult y righ t awa y if we try to untie a tangled thread. (Imagin e a closed circle C in 3-space , which i s knotted i n a very very complicated manner . Togethe r wit h a specified intege r n , a closed 3-manifol d M(c,n) i s assigned t o the pai r (C, n) vi a a Kirb y diagram . Suc h a correspondenc e wil l convince th e reader o f the incredibl e complexit y o f the subject. )

Low-dimensional manifol d theor y i s an area which is very activel y studied today . Th e origina l pla n wa s t o mak e Chapte r 3 an d thi s Chapter 5 th e mai n topic s o f th e book , bu t unfortunately , du e t o page limitations , onl y fundamenta l notion s ar e covere d i n Chapte r 5 .

In my opinion, th e mos t interestin g aspec t o f handlebody theory , either i n low or high dimensions , i s its geometric entity , which we feel as i f we can touch an d se e with ou r eyes . I feel that , i n Morse theory , we are no t muc h differen t fro m childre n playin g wit h woo d blocks .

The purpos e o f thi s boo k woul d b e mor e tha n hal f achieve d i f such a feelin g i s passed alon g t o th e reader .

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Bibliography

J. Cerf , Su r le s diffeomorphisme s d e l a spher e d e dimensio n troi s (T 4 = 0) , Lecture Note s i n Math . 53 , Springer-Verlag , 1968 . M. H . Preedman , Th e topolog y o f four-dimensional manifolds , J.Diff. Geom., 17 (1982) , 337-453 . R. E . Gomp f an d A . I . Stipsicz , J^-manifolds and Kirby calculus, Graduat e Studies i n Mathematic s 20 , Amer . Math . Soc , 1999 . V. Guillemi n an d A . Pollack , Differential topology, Prentice-Hall , 1974 . M. A . Kervair e an d J . W . Milnor , Group s o f homotop y sphere s I , Ann. of Math., 7 7 (1963) , 504-537 . R. C . Kirby , A calculu s fo r frame d link s i n S 3, Invent. Math., 4 5 (1978) , 35-56. S. Lang, Differential and Riemannian manifolds, GT M 160 , Springer-Ve r lag, 1995. F. Laudenbac h an d V . Poenaru , A note o n 4-dimensiona l handlebodies , Bull. Soc. Math. France, 10 0 (1972) , 337-347 . W. B . R. Lickorish , A representation o f orientable combinatoria l 3-manifolds , Ann. of Math., 7 6 (1962) , 531-540 . W. S . Massey , A basic course in algebraic topology, GT M 127 , Springer -Verlag, 1991 . Y. Matsushima , Differentiable manifolds, Shouka-bou , Tokyo , 196 5 (i n Japanese). Englis h translation : Differentiable manifolds, Pur e an d Applie d Mathematics, 9 . Marce l Dekker , 1972 . B. Mazur , Mors e theory , Differential and combinatorial topology, A sym-posium in honor of Marston Morse (ed . b y S . S . Cairns) , Princeto n Univ . Press, 1965 , 145-165 . J. Milnor , O n manifold s homeomorphi c t o th e 7-sphere , Ann. of Math., 6 4 (1956), 399-405 . J. Milnor , Lectures on the h~cobordism theorem, Princeto n Univ . Press , 1965 . J. Milnor , Topology from the differentiable viewpoint, U.P . o f Virginia , 1965 . J. M . Montesinos , Heegaar d diagram s fo r close d 4-manifolds , Geometric topology (ed . b y J . C . Cantrell ) Academi c Press , 1979 , 219-237 . J. R . Munkres , Elementary differential topology, Annal s o f Mathematic s Studies 54 , Princeto n Univ . Press , 1963 . R. Palais , Extendin g diffeomorphisms , Proc. Amer. Math. Soc, 1 1 (1960) , 274-277. W. Rudin , Principles of mathematical analysis, 3r d ed. , McGraw-Hill , 1976 .

213

214 BIBLIOGRAPHY

[20] I . Satake , Linear algebra, Shouka-bou , Tokyo , 195 8 (i n Japanese) . Englis h translation: Linear algebra, Pur e an d Applie d Mathematics , 29 . Marce l Dekker, 1975 .

[21] H . Sato , Algebraic topology, "Iwanam i Kohza , Genda i suugak u n o kiso"(Iwanami serie s i n moder n mathematics) , 199 6 (i n Japanese) . Englis h translation: Algebraic topology: An intuitive approach, Translation s o f Math -ematical Monograph s 183 , Amer . Math . Soc , 1999 .

[22] S . Smale , Diffeomorphism s o f th e 2-sphere , Proc. Amer. Math. Soc, 1 0 (1959), 621-626 .

[23] S . Smale , Generalize d Poincare' s conjectur e i n dimension s greate r tha n four , Ann. of Math., 7 4 (1961) , 391-406 .

Recommended Readin g

1. J . Milnor , Morse theory, Ann . Math . Studies , 5 1 (1963) , Princeton U.P .

The boo k start s fro m Mors e theor y o f finite dimensions , and reache s th e Mors e theor y fo r energ y functional s o n loo p spaces o f Li e groups . An d a t th e end , a beautifu l theore m called th e Bot t periodicit y i s proved , whic h i s abou t th e sta -ble homotop y group s o f Li e groups . Th e expositio n i s simpl e and eas y t o understand . Thi s i s a classica l boo k wit h a well -deserved grea t reputatio n o n Mors e theory .

2. J . Milnor , Lectures on the h-cobordism theorem, Princeto n U.P., 1965 .

The boo k explain s th e /i-cobordis m theorem , whic h i s im-portant i n differentia l topolog y fo r higher-dimensiona l mani -folds. Th e handlebod y theor y fo r manifold s i s given i n detail . This boo k i s als o eas y t o rea d an d understand . Muc h o f th e treatment o f handlebodies i n the curren t boo k i s based o n thi s book o f Milnor's .

3. R . C . Kirby , The topology of 4-manifolds, Lect . Note s i n Math., 137 4 (1989) , Springer-Verlag .

4-manifold theor y i s develope d base d o n th e commo n grounds betwee n frame d link s an d low-dimensiona l manifol d theory. Th e expositio n i s sometime s intuitive , s o firs t read -ers ma y hav e a har d tim e gettin g int o it , althoug h plent y o f pictures ar e provided .

4. R . E . Gomp f an d A . I . Stipsicz , ^-manifolds and Kirby calcu-lus, Graduate Studie s in Mathematics 2 0 (1999) , Amer. Math . Soc.

This boo k contain s material s o n 4-dimensiona l topology , from fundamenta l technique s t o mos t recen t results . Kirb y calculus i s emphasize d significantly . Man y exercise s ar e als o

215

216 R E C O M M E N D E D READIN G

provided. Thi s i s quit e a thic k book , s o i t ma y b e toug h t o read i t through , bu t i t coul d b e use d a s a dictionar y a s well .

5. D . Rolfsen , Knots and links, Math . Lectur e Series , 7 (1976) , Publish o r Perish , Inc .

An introductio n t o kno t theory , wit h a lo t o f pictures . 6. A . Kawauchi, A survey of knot theory. Translate d an d revise d

from th e 199 0 Japanes e origina l b y th e author . Birkhause r Verlag, Basel , 1996 .

This give s a n overvie w o n moder n aspect s o f knot theory . 7. K . Fukaya, Gauge theory and topology, Springer-Verlag Tokyo ,

1995 (i n Japanese) . This i s mainly o n the Mors e theory o f infinite dimensions ,

but a beautifu l expositio n o n Mors e theor y i s foun d i n th e introduction.

CP2, 19 0 CP™, 89 , 17 1 (C2 ,^-approximation, 51 , 52 C°°-function, 3 3 z-cell, 13 3 i-skeleton, 13 4 A;-dimensional submanifold , 3 6 m-dimensional dis k (ra-disk) , 3 4 m-dimensional torus , 7 2 m-dimensional uppe r half-space , 3 5 m-handle, 7 5 (m-l)-dimensional spher e (( m —1)-

sphere), 3 5 ? m , 8 4 SO(m), 9 1 SU(m), 98 , 102 , 103 , 17 1 S2 x S 2, 19 1 5 m x S n, 8 4 S m , 8 3 T m , 7 2 U(m), 96 , 10 3 A-handle, 7 6 0-handle, 30 , 7 5 1-dimensional disk , 2 7 1-handle, 2 8 2-dimensional disk , 1 9 2-handle, 2 9

abelianization, 17 0 annulus, 2 1 array o f vecto r fields, 13 4 attaching map , 79 , 13 4 attaching sphere , 7 9

base point , 16 5 belt sphere , 12 6 Betti number , 14 0 bicollar neighborhood , 6 8 boundary group , 13 8

boundary homomorphism , 13 6 bouquet, 16 7

canceling handles , 12 0 cell, 13 3 cell complex , 13 3 cell decomposition , 14 1 center, 10 4 centralizer, 10 4 chain, 13 5 chain complex , 13 8 chain group , 13 5 closed z-cell , 13 3 closed surface , 1 4 coboundary, 14 9 coboundary group , 14 9 coboundary homomorphism , 14 9 cochain, 14 8 cochain complex , 14 9 cochain group , 14 8 co-core, 7 7 cocycle, 14 9 cocycle group , 14 9 cohomology class , 14 9 cohomology group , 14 9 collar neighborhood , 38 , 6 8 commutator, 17 6 compact, 5 1 compact surface , 1 8 complex projectiv e line , 9 0 complex projectiv e plane , 9 0 complex projectiv e space , 8 9 connected, 2 2 connected sum , 171 , 18 5 coordinate neighborhood , 5 1 core, 7 7 critical point , 1 , 4 , 16 , 41 , 48 critical value , 23 , 42, 4 8 cycle group , 13 8

217

218 INDEX

degenerate critica l point , 5 , 4 3 diffeomorphic, 1 7 diffeomorphism, 17 , 3 9 differential topology , 1 7 disjoint union , 2 6 dual basis , 15 1

Euler number , 14 0 Euler-Poincare characteristic , 14 0 existence o f Mors e functions , 4 7 exotic sphere , 8 3

frame, 18 6 frame o f vecto r fields, 15 8 framed close d curve , 18 6 framed link , 18 8 free group , 16 7 free part , 14 0 function o f clas s C°° , 3 3 fundamental class , 15 8 fundamental group , 16 6

genus, 1 4 gluing, 3 9 gradient vecto r field, 6 1 gradient-like vecto r field, 6 3

handle, 28-30 , 75 , 7 6 handle decomposition , 8 1 handle slide , 10 6 handlebody, 79 , 17 9 Heegaard diagram , 180 , 18 8 Heegaard splitting , 18 0 hermitian product , 9 6 Hessian, 5 , 4 2 homeomorphic, 1 7 homeomorphism, 1 7 homology class , 13 8 homology group , 13 8 nomotopic, 13 9 homotopy, 13 9 homotopy equivalence , 14 0 homotopy invariance , 14 0 homotopy inverse , 14 0

identity, 2 5 implicit functio n theorem , 3 6 index, 4 4 integral curve , 6 5 interior, 20 , 3 8

intersection form , 16 2 intersection number , 15 9 isotopy, 10 5

Jacobian (matrix) , 4 8

Kirby calculus , 19 4 Kirby diagram , 19 4 Kirby moves , 19 4 Klein bottle , 17 3 knot, 18 8

left-hand disk , 11 2 lens space , 18 3 Lie group , 9 1 link, 18 8 linking number , 18 8 local components , 3 4 longitude, 18 2 loop, 16 5 lower disk , 11 2

manifold wit h boundary , 3 4 map o f clas s C°° , 3 4 mapping cylinder , 14 2 maximum-value theorem , 1 8 meridian, 18 0 Morse function , 43 , 4 7 Morse inequality , 14 1 Morse lemma , 8 , 4 3

non-degenerate, 16 2 non-degenerate critica l point , 5 , 4 3

orient able, 13 5 orientation, 134 , 13 5 orthogonal matrix , 9 0

Poincare duality , 15 0 presentation, 17 0 projective plane , 8 7 projective space , 8 4

real projectiv e space , 8 9 right-hand disk , 11 2 rotation group , 9 1 rotation matrix , 9 1

Sard's theorem , 4 9 sign o f intersection , 15 9 simply connected , 16 7

INDEX

sliding handles , 10 6 smoothing corners , 28 , 7 7 solid torus , 18 2 special unitar y group , 9 8 sphere, 1 4 step function , 5 4 submanifold, 3 6 surface, 1 4 surface wit h boundary , 1 9

tangent vector , 5 6 tangent vecto r space , 5 6 torsion, 16 2 torsion part , 14 0 torus, 14 , 7 2 transversely intersect , 12 0 trivial knot , 19 0

unitary group , 9 6 unitary matrix , 9 6 universal coefficien t theorem , 15 0 unknot, 19 0 upper disk , 11 2 upper half-space , 3 5

van Kampen' s theorem , 16 8 vector field , 6 1 vertex, 13 4

well-definedness o f the index , 13 , 44

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Selected Title s i n Thi s Serie s (Continued from the front of this publication)

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups . Part 2 , 199 9

180 A . A . Mi lyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d

optimal control , 199 8

179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants ,

1998

178 Mi t su o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sa t or u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e

Hamiltonian system s wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d

elliptic systems , 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l

properties o f distribution s o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups .

Part 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c

integrals, 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t

coefficient, 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s

of fractals , 199 7

166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g

problem i n Galoi s theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis ,

1997

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I .

Yadrenko, Probabilit y theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d

methods i n linea r statistica l models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c

dynamics, 199 7

160 V . G . Osmolovskii , Linea r an d nonlinea r perturbation s o f th e operato r div, 199 7

For a complet e lis t o f t i t le s i n th i s series , visi t t h e A M S Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .