The Geometric Topology of 3-Manifolds

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AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 40

The Geometric Topology of 3-Man if olds R. H. Bing

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/coll/040

2000 Mathematics Subject Classification. Primar y 57-XX .

Library o f Congres s Catalogin g i n Publicatio n Dat a

Bing, R . H., 1914 -The geometric topology o f 3-manifolds . (Colloquium publications; ISSN 0065-9258; v. 40) Bibliography: p . Includes index.

1. Geometric topology . 2.3-manifolds . I. Title. II . Series: Colloquium publications (America n Mathematica l Society) ; v.40. QA612.S94 198 3 514. 2 83-1496 2 ISBN 0-8218-1040- 5

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TABLE OF CONTENTS

Preface i x Chapter I . Plana r Complexe s

1. Triangulation s 2 2. Extendin g triangulations 4 3. Specia l triangulations 6

Chapter II . P L Planar Map s 1. Linea r maps 9 2. P L maps 1 1 3. Pushe s 1 3 4. Isotopie s 1 5 5. Meshin g triangulations 1 7

Chapter III. The Schoenflie s Theore m 1. P L Schoenflies theorem 2 0 2. Triangulatin g PL disks 2 2 3. Ske w sets 2 3 4. N o arc separates R2 2 6 5. Jordan-Brouwe r theorem 2 7 6. Schoenflie s theore m 2 9

Chapter IV. Wil d 2-Sphere s 1. Tam e and wild 2-spheres 3 3 2. 3-sphere s 3 4 3. Alexande r horned sphere 3 8 4. Simpl e connectivity 4 1 5. Soli d Alexander horned sphere 4 2 6. Peculia r involutions 4 3 7. Antoine's necklace 4 4 8. A n Antoine wild sphere 4 7 9. Fox-Arti n spheres 4 7

10. Bing' s hooked rug 5 3 Chapter V. Th e Generalized Schoenflie s Theore m

1. Schoenflie s theorem for a 2-sphere 5 7 2. Generalize d Schoenflies theore m 5 7 3. Canonica l collared Schoenflies theore m 6 2

v

VI TABLE O F CONTENT S

4. Loca l flatness 6 4 5. Weaknes s of th e generalized Schoenflies theore m 6 4

Chapter VI . The Fundamenta l Grou p 1. Path s and loops 6 9 2. Th e fundamental grou p 7 2 3. Graph s 7 3 4. Associatin g words with loops 7 4 5. Relation s 7 8 6. Shellin g 8 0 7. Changin g words 8 0 8. Presentation s of groups 8 3 9. Shor t cuts 8 4

10. Wh y compute fundamental group s 8 8 11. A homotopy cube 9 4 12. Othe r treatments 9 5

Chapter VII . Mappin g onto Sphere s 1. Retraction s onto boundaries 9 7 2. AR' s and ANR's 9 8 3. Extendin g mappings onto spheres 9 8 4. Inessentia l mappings 9 9 5. Projection s 10 1 6. Separatin g R" 10 1 7. Fixe d points 10 2

Chapter VIII . Linkin g 1. Chains , cycles, bounding cycles 10 5 2. Linkin g polygons 10 8 3. Linkin g curves 10 9 4. Alexande r addition theorem 10 9 5. Ball s do not link I l l 6. Nonlinkin g curves 11 2 7. Homolog y groups 11 3

Chapter IX . Separatio n 1. Genera l position approximations 11 5 2. Separatio n by spheres 11 7 3. Jordan-Brouwe r theorem . . . . 11 8 4. Loca l separation 11 9

Chapter X . Pullin g Bac k Feeler s 1. Pull-bac k theorem s 12 3 2. Reimbeddin g a crumpled cube 12 6 3. Repeate d pull-backs 12 6 4. Sewin g cubes together 12 8 5. Pullin g feelers off a null sequence of disks 12 9

TABLE O F CONTENT S Vl i

Chapter XI. Intersection s of Surfaces with 1-Simplexes 1. Pickin g a triangulation 13 1 2. Simplifyin g K1 f l A 1 13 2 3. Shrinkin g at the waist 13 3 4. Removin g noncircling components 13 5 5. Removin g circling continua 13 7 6. Locatin g h(K2) 13 7

Chapter XII. Intersection s of Surfaces with Skeleta 1. Expandin g closed sets 14 1 2. Pushin g on a membrane 14 3 3. Eliminatin g nonpiercing points 14 4 4. Reducin g components 14 6 5. Simplifyin g T 2 n S 15 0

Chapter XIII. Sid e Approximation Theorem 1. Ver y special case 15 2 2. Modificatio n of very special case 15 4 3. Specia l case 15 5 4. Genera l case 15 8

Chapter XIV. Th e PL Schoenflies Theorem for R3

1. Extendin g the PL Schoenflies theorem 16 1 2. Pushin g disks about on tetrahedra 16 1 3. Extendin g a 9 curve result 16 3 4. Specia l case of the PL Schoenflies theorem for R3 16 4 5. Genera l case of the PL Schoenflies theorem for R3 16 7 6. Shellin g 16 9

A. The-house-with-two-room s 17 0 B. Cub e with knotted spanning 1-simplex 17 2

Chapter XV. Coverin g Spaces 1. Example s and definitions 17 5 2. Usin g splittings 17 6 3. Liftin g paths 17 7 4. Usin g paths 17 9 5. Two-sheete d coverings 18 0

Chapter XVI. Dehn' s Lemma 1. Singula r disks 18 3 2. Nic e singularities 18 3 3. Splittin g and resewing 18 4 4. Eliminatin g branch points 18 7 5. Tryin g to remove more singularities 19 2 6. Thickenin g disks 19 3 7. Histor y of Dehn's lemma 19 8 8. A strong version of Dehn's lemma 19 9

viii TADLI ; O F CONTENT S

Chapter XVII . Loo p Theore m 1. Version s of th e loop theorem 20 3 2. Eliminatin g branch points 20 6 3. Thickenin g canonical disks 20 6 4. Proo f of loo p theorem 20 7 5. Histor y 21 0

Chapter XVIII . Relate d Result s 1. Approximatin g 2-complexes 21 1 2. Approximatin g homeomorphisms on 3-manifoIds 21 2 3. Triangulatin g 3-manifolds 21 3 4. Locall y tame sets are tame 21 4 5. Tamenes s from the side 21 5 6. Reembeddin g crumpled cubes 21 7 7. Tam e sets in wild surfaces 21 8 8. Characterization s 21 9 9. Decomposition s 22 0

10. Othe r references 22 1 Appendix - Som e Standard Result s i n Topolog y

1. Metri c spaces 22 3 2. Plana r results 22 4 3. Result s about 2-spheres 22 5 4. Cylinder s in R3 22 6 5. Decomposition s 22 6

References 22 9 Index 23 5

PREFACE

This book contains an elaboration of some aspects of my Colloquium Lectures given befor e th e America n Mathematica l Societ y i n 1970 . I t represent s a con -tinuation of my efforts t o bridge the gap between the study of topological objects in 3-space and the corresponding polyhedral objects.

One of th e main goals is to give an understandable proof o f th e side approxi-mation theorem . Thi s theore m wa s introduce d i n th e earl y 60' s an d ha s bee n useful i n studying topologica l surfaces . Ther e have been alternativ e proof s o f i t suggested i n researc h paper s bu t perhap s non e so extensive a s tha t include d i n Chapters X-XIII. I have considered severa l versions of thi s proof bu t doubt tha t the best treatment is yet at hand.

Topics relate d t o th e sid e approximatio n theore m includ e wil d surface s (Chapters IV, VI), the Schoenflies theorem (Chapters III, V, XIV), Dehn's lemma (Chapters XV, XVI), and the loop theorem (Chapter XVII). Applications of these topics are listed in Chapter XVIII.

So a s t o mak e th e boo k somewha t self-contained , som e preliminar y result s from P L topology, homotopy theory , and homology theory that ar e of particula r use in th e study o f 3-manifol d ar e treated i n Chapter s I , II , VII , VIII, IX. The material is traditional but it is treated in the down-to-earth context in which it is applied. This provides a straightforward approac h fo r thos e who like to develop mathematics from the specific to the general.

I was very fortunate to have Professor R . L. Moore as a teacher. His success in leading student s t o formulat e theorem s an d proof s i s legendary . Durin g thes e student days I profited fro m knowin g such mathematical leaders as E. E. Moise, F. B . Jones, R . D . Anderson , C . E . Burgess , Mar y Elle n Rudin , Bill y J o Ball , Mary Elizabet h Hamstrom . Later , a s colleagues , E d Floyd , E d Fadell , Dean e Montgomery, Joe Martin, Bil l Eaton, Steve Armentrout, Jim Cannon, and Mik e Starbird had a big impact on my research activities. Work with graduate students has bee n ver y stimulating , an d i n particular , tha t wit h Mor t Brown , Rus s McMillan, Do n Sanderson , Bo b Daverman , Davi d Gillman , Davi d Henderson , and Joh n Hempe l i s relevan t t o th e content s o f thi s book . Graduat e student s Robert L . Dawes , Russel l Rose , an d Richar d Skor a wer e helpfu l i n reviewin g versions of the book.

IX

X PRL«PAcr:

My mothe r ha d mathematic s a s a hobby , an d I learned fro m he r even befor e starting schoo l tha t arithmeti c wa s fu n i f don e rapidl y an d accurately . Sh e late r encouraged th e notio n tha t geometri c proof s wer e t o b e discovere d an d prove d rather than learned.

In addition t o al l thi s mathematical help , i receive d financia l researc h suppor t from th e Universit y o f Wisconsin , Th e Universit y o f Texas , an d th e Nationa l Science Foundation. I also recognize the encouragement of my wife, Mary , which made i t possibl e fo r m e t o spen d s o muc h tim e wit h mathematic s an d students .

R. H . BIN G

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203-216. [S9] , Null sequence cellular decompositions of E3, Fund . Math. 112(1981) , 81-87 . [SW2] M . Starbir d an d E . Woodruff , Decompositions of E 3 with countably many non-degenerate

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Paris, 1963 .

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INDEX

absolute retrac t = AR , 29 8 absolute neighborhoo d retrac t = ANR , 29 8 accessible, 2 8 Alexander, J . W. , 33 , 38, 47,16 1 Alexander additio n theorem , 10 9 Alexander horne d sphere , 34 , 38, 41 Alexandroff, P . S. , 22 2 Alford, W . B. , 33 , 44 almost approximat e from , 126 , 15 1 annulus conjecture , 6 8 Antoine, L. , 44 , 47 Antoine's necklace , 4 4 Antoine's wil d sphere , 4 7 approximation theorems , 116,151,152 , 15 4 arc theorem, 22 3 Armentrout, Steve , 22 1 Artin, E. , 33 , 47 associativity, 7 1

Baire categor y theorem , 22 3 ball, 2 , 33 Ball, B . J. , 33 , 44 barycentric subdivision , 6 base point , 7 1 Bessel-Hagen, 22 1 bicollared, 5 8 Bing, R. H. , 26 , 33, 37, 43, 53, 95, 102,116,121 ,

132,151,212,214,215, 219,220 , 22 1 Birman, Joan , 22 2 Borsuk's theorem , 10 1 bounding cycle , 10 5 bounds, 10 5 branch point , 75,18 4 brick partitioning , 2 6 Browder, William , 22 2 Brown, Morton , 58 , 65 Burgess, C . E. , 217 , 22 2

Cairn's Stewar t S. , 1 9 Cannon, J . W. , 217 , 22 1

Cannon, L . O. , 33 , 44, 217 canonical collare d Schoenflie s theorem , 62 ,

73 canonical norma l disk , 20 6 canonical n-sphere , 33,19 7 Caratheodory, C , 1 9 cartesian produc t neighborhood , 5 8 Casler, B . G. , 33 , 44 cell, 2 cellular, 49 , 22 4 cellular decomposition , 22 1 cellular partitioning , 21 9 cellular shelling , 17 0 cellular subdivision , 80,17 0 center push , 14 3 chain, 10 5 Chillingsworth, D . R . J. , 17 4 circles, 45,133 , 137 , 226 Coelho, R . P. , 4 5 collapse, 17 3 collar, 5 8 compact support , 1 3 compatible, 1 7 complete, 22 4 complex, 1 coning, 6 conjugacy class , 20 4 Connelly, Robert , 6 5 contains most , 15 1 converges, 14 0 covering, 70 , 225, 22 6 Craggs, R. , 13 2 crosses, 11 6 Crowell, R . H. , 9 5 Crumpled cube , 3 4 curvilinear triangulation , 2 cycle, 10 5

Daverman, R . J. , 33 , 121

235

236 INDEX

decompositions, 220 , 226 decreasing sequence , 22 5 degenerate, 139,14 2 Dehn, Max , 184 , 19 8 Dehn disk , 18 4 Dehn's lemma , 19 8 d(/,Id), 1 4 dog-bone-space, 22 1 double line , 18 4 Doyle, P . H. , 217 , 21 8

Eaton, W . T. , 33 , 40, 44,161, 217, 221 c-approximation, 115,12 5 edge, 2 0 c-homeomorphism, 1 6 e-map, 12 5 end o f doubl e line , 20 5 end poin t relation , 74 , 78 equivalence clas s o f paths , 7 0 equivalence clas s o f cycles , 11 3 e-set, 11 8 essential, 10 0 eye bolt , 3 9 expansion, 141 , 225 extensions o f triangulations , 4

faces, 2 feeler, 4 7 finite graph , 2 4 fixed point s property , 10 2 flat, 2 7 Fort, M . K. , Jr. , 3 3 Fox, R . H. , 33 , 47, 95 Fox-Arten spheres , 4 7 Freedman, Michael , 22 0 free face , 17 3 free surfac e problem , 21 6 fundamental group , 7 2

general positio n = GP, 13 , 115,11 8 generalized Schoenflie s theorem , 7 2 geometrical collapse , 165,17 3 geometric complex , 3 4 Gillman, D . S. , 55 , 217 Graueb, W. , 16 1 Griffith, H . C. , 21 7

Haken, Wolfgang , 22 0 Hall, Dic k Wick , 22 1 Hamilton, A . J . S. , 21 4 handle, 20 3 Harrold, D . G. , Jr. , 211,21 7 Heegaard splitting , 22 0 Hempel, John , 33 , 210, 216 , 22 1 Henderson, D . W. , 19 8 Hocking, J . G. , 11 8 Homma, T. , 19 8

homology group , 11 3 Hn(U,Z2), 159,18 0 homology cube , 9 4 nomotopic to , 41 , 69, 99 homotopy, 7 0 homotopy 3-balls , 17 0 homotopy extensio n theorem , 10 0 hooked rugs , 5 3 Hosay, Norman , 21 7 house- with- two-rooms, 17 0 Hsiang, W.-c , 22 2 Hudson, J . F . P. , 22 2 Hurewicz, W. , 97 , 22 2

inaccessible, 21 7 inessential, 99,10 1 initial point , 6 9 invariance o f domain , 10 2 invariant, 13 3 inverse, 6 9 involution, 4 3 irreducible separator , 13 3

Janiszewski, S . (Z.) , 2 6 Johansson, I. , 19 8 join, 6,16 4 Jordan-Brouwer theorem , 27,11 7 Jordan curv e theorem , 10 2

Kerekjarto, B . von , 22 1 Kirby, R . C , 6 3 Kister, James , 1 6 Klein bottle , 18 6 Kline spher e theorem , 218 , 226 Kneser, H. , 198 , 210 knotted spannin g arc , 17 2

lamp cor d trick , 5 2 Lawson, Blaine , 22 2 lift, 17 7 linear, 9 , 1 1 Lininger, L . L. , 122 , 217 link, 13 , 108, 109,110 , 11 1 Lister, F . M. , 14 1 locally collared , 6 1 locally compatible , 1 7 locally connecte d = L.C., 46 , 224 locally finite, 13 4 locally flat, 6 4 locally n-connecte d =71.1.0., 11 2 locally PL, 203 , 211 locally polyhedral , 21 1 local separation , 11 9 locally simpl y connected , 46 , 21 6 locally tame , 21 4 longitudinal, 4 5 loop, 71 , 203

INDEX

loop theorem , 20 3 Loveland, L . D. , 21 7

manifold, 3 7 map, 9 Martin, J . M. , 33 , 44, 219, 221 Massey, W . S. , 13 1 Mauldin, Dan , 10 3 Mazur, Barry , 5 8 McMillan, D . Ft. , Jr. , 21 8 meridional, 4 5 mesh, 4 4 meshing, 13 1 Milnor, John , 22 1 mod-2 boundary , 10 5 mod-2 sum , 105 , 10 8 Moise, E . E. , 19,161 , 210, 212, 214, 217, Montgomery, Deane , 44 , 220 Moore, R . L. , 27 , 22 1 Moore decompositio n theorem , 22 7 Moore space , 22 3 Morse, Marston , 5 8

n-connected, 11 2 neighborhood, 3 5 Newman, H . M . A. , 22 1 n-locally connecte d (i n homology )

= n-lc, 11 2 n-locally connecte d (i n homology )

= n-LC, 4 6 n-manifold, 3 7 noarcseparates R 2, 2 6 non-circling continue , 13 5 non-degenerate, 131,13 4 non-hour-glass property , 151 , 225 non-linking curves , 11 2 normal, 73 , 75 normal curve , 184 , 204 normal Dehn disk , 20 4 no-triod property , 14 3 n-sheeted covering , 17 5 n-simplex, 2 n-sphere, 3 3 null sequence , 12 9 1-connected, 4 1 1-handle, 20 3 1-locally connected = 1-LC , 4 6

opposite side s o f disk , 14 4

Papakyriakopoulos, C . D. , 184 , 210 partitioning, 26 , 38 path, 6 9 path inverse , 2 0 periodic homeomorphism , 43 , 220 piecewise linea r = PL , 11 , 22, 33 PL disk , 22 , 34, 18 3

PL map , 11,13 2 PL neighborhood , 9 9 PL Schoenflie s theorem , 20,161,16 4 pierces, 14 4 pill box , 20 3 Poincare. J. , 93 , 220 Poincare conjecture , 38 , 93,170, 22 0 polygon, 2 0 polyhedral, 34,16 2 polyhedron, 3 4 presentation o f groups , 8 3 preserves linearity , 9 preserves ratios , 9 products o f paths , 6 9 projection, 73,101,17 5 proper spannin g segment , 2 2 pulling bac k feelers , 123,126,12 9 push, 13,17 7 pushing cente r o f a membrane , 14 3

raise-from-dead, 5 9 rectilinear, 3 reembedding crumple d cube , 12 6 refinement, 2 regular neighborhood , 161,19 4 regular position , 7 3 relations, 7 8 retraction, 9 7 rigid motion , 4 3 Rolfsen, Dale , 33 , 41, 221 Rosen, Harvey , 3 3 Rourke, C , 22 2 Rudin, Mar y Ellen , 17 4 Rushing, T . Benny , 22 2

S° = 0-sphere , 3 4 51 = 1-sphere , 3 4 5 2 = 2-sphere , 3 4 5 3 = 3-sphere , 3 4 Samelson, Hans , 44 , 220 Sanderson, B. , 22 2 Sanderson, D . E. , 21 2 Schoenflies theorem , 20 , 29 second barycentri c subdivisio n

= secon d derived , 7 Seifert, H. , 22 2 selection theorems , 8 0 semipolygon, 2 7 separating spheres , 11 7 sequentially unicoheremt , 3 8 sewing, 12 8 Shalen, Peter , 214 , 222 Shapiro, A. , 198 , 210 shelling, 80 , 165 , 16 9 shield, 4 shrink t o a point , 41 , 203 side o f cylinder , 126

238 INDEX

side approximatio n theorem , 126 , 15 2 Sierpinski curve , 21 7 simplex, 2 simplicial collapse , 17 3 simply conneted , 4 1 singularities, 6 1 singularities o f disk , 18 3 singularities o f map , 18 3 skeleton, 3 skew curve , 2 3 Smale, S. , 22 0 Smith, P . a. , 4 3 solid Alexande r horne d sphere , 4 2 solid torus , 3 6 spanning arc , 2 2 spanning disk , 12 5 Spencer, G . S. , 22 2 spinning abou t axis , 144 , 14 6 splitting an d resewing , 176,18 4 sphere, 33 , 34, 38 Stallings, J . R. , 22 0 Starbird, Michael , 22 1 stable, 67 , 20 6 star, 13 , 22 stellar, 6 subdivisions, 2 , 4 Sullivan, Dennis , 22 2

tame, 33 , 214, 215, 218 taming set , 21 5 terminal point , 6 9 tetrahedral, 16 3 Threlfell, W. , 22 2 Thurston, Willia m P. , 22 2 Tietze extensio n theorem , 98 , 223 tips o f horns , 3 9 totally disconnected , 131,14 2 to-the-boundary-theorem, 22 4 triangulation, 2 triod, 14 2

triple point , 18 4 2-manifold, 1 3 2-sheeted covering , 18 0 2-simplex, 2 2-sphere, 3 3 3-manifold, 20 4 3-sphere, 34 , 38

undercrossing point , 73 , 79 undercrossing relation , 79 , 80 unicoherent, 13 3 universal coverin g space , 17 9 ulc = unifor m locall y connecte d

(in homology) , 11 2 ULC = uniforml y locall y connecte d

(in homology) , 112,11 8 upper semicontinuou s decomposition , 22 1 Urysohn's lemma , 22 3

van Kampen , E . R. , 9 5 variable cartesia n product , 6 5 vertex, 2 0

Waldhausen, F. , 22 0 Wallace, A . H. , 22 0 Wallman, H. , 69 , 22 2 Whitehead, J . H . C , 38,184,198 , 210 , 220 Whyburn, G . T. , 22 2 Wilder, R . L. , 109,113 , 22 2 wild, 3 3 Woodruff, Edythe , 22 1

Young, G . S. , 11 8

Zeeman, E . G. , 220 , 222 0-chain, 10 5 0-cycle, 10 5 Zippen, Leo , 44 , 220