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Coll032-Endmatter.Pdf http://dx.doi.org/10.1090/coll/032 AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 32 Topology of Manifolds Raymond Louis Wilder American Mathematical Society Providence, Rhode Island International Standard Seria l Numbe r 0065-925 8 International Standar d Boo k Number 0-8218-1032- 4 Library o f Congress Catalog Card Numbe r 49-672 2 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given. Republication, systematic copying, or multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mail t o reprint-permissionQmath.ams.org . Copyright © 1949 , 196 3 by the American Mathematica l Societ y Revised edition, 196 3 Revised edition , fourt h printing , with corrections, 197 9 The America n Mathematica l Societ y retain s al l right s except thos e grante d to the United States Government . Printed i n the Unite d State s of America . @ Th e pape r use d i n this book i s acid-free an d fall s withi n th e guideline s established to ensure permanence and durability . 10 9 8 7 6 0 1 0 0 9 9 9 8 TABLE O F CONTENT S PREFACE . vi i INTRODUCTION T O THE 196 3 EDITIO N x i NOTES T O THE 196 3 EDITIO N x i NOTES T O TH E 197 9 PRINTIN G xii i I. ELEMENTAR Y CONCEPTS ; CHARACTERIZATION S O F E l AN D S l 1 1. Set s 1 2. Space s 2 3. Metri c space s 4 4. Close d and open subsets o f a space 5 5. Mappings ; homeomorphisms 6 6. Historica l remark s 1 0 7. Connecte d space s 1 6 8. Components ; quasi-components 1 8 9. Connecte d spaces satisfying the 2nd Hausdorff axio m and the weak separation axiom 1 9 10. Space s irreducibly connected abou t a subset 2 1 11. Th e simple ar c and the 1-spher e 2 7 12. Som e fundamental lemma s 3 3 IL LOCALL Y CONNECTED SPACES ; FUNDAMENTAL PROPERTIES OF THE EUCUDEAN TI-SPHER E 4 0 1. Loca l connectedness 4 0 2. Irreducibl e lc-connexes; recognition o f E l an d S 1 amon g lc spaces ... 4 2 3. Som e general properties o f lc spaces . 4 5 4. "Phragmen-Brouwe r properties " and their equivalence in lc spaces .... 4 7 5. Som e topology o f th e n-spher e 5 1 III. PEAN O SPACES ; CHARACTERIZATION S O F S* AN D TH E 2-MANIFOLD S 6 9 1. Pean o continua ; mapping theorems 6 9 2. Topologica l characterizatio n o f Peano continu a 7 4 3. Pean o spaces 7 6 4. Recognitio n o f the 2-spher e 8 7 5. Recognitio n o f th e close d 2-cell 9 2 6. Recognitio n o f the 2-manifolds . • 9 4 IV. NON-METRI C L C SPACES, WIT H APPLICATION S T O SUBSET S O F TH E 2-SPHER E .. 9 9 1. Component s o f locall y compac t Hausdorf f space s 9 9 2. A characterization o f locally compact, connecte d space s that fai l to be l c 10 2 3. Som e characterizations o f locally compac t l c spaces 10 4 4. Relation s between lc , S and ulc properties 10 7 5. Accessibilit y 11 0 6. Mor e properties o f the 2-sphere 11 2 7. Recognitio n o f Pean o continua i n S* by accessibilit y propertie s 11 6 8. Remark s 11 8 iii iv TABL E O F CONTENT S V. BASI C ALGEBRAI C TOPOLOG Y 12 0 1. Complexe s 12 0 2. Algebrai c apparatu s 12 0 3. Chai n groups 12 2 4. Homolog y group s 12 3 5. Importan t specia l case s and geometric interpretation s 12 4 6. Som e fundamental lemma s 12 6 7. Cec h cycles and homolog y group s 12 9 8. Coverin g lemma s 13 3 9. Vecto r spaces 13 5 10. Existenc e theorem s 13 8 11. Som e applications t o connectedness an d local connectednes s 14 1 12. Fundamenta l system s o f cycle s fo r a compact metri c spac e 14 5 13. Alternativ e definition s 14 6 14. Dua l homomorphism s 14 8 15. Cocycles ; cohomology group s 14 9 16. Chai n product s fo r a complex 15 3 17. Extensio n to topological space s 15 6 18. Scala r products and dua l pairing s 15 9 19. Application s to homology propertie s o f spaces 16 7 20. Homologie s i n non-compact space s 16 8 21. Approximat e homologie s . 17 2 VI. LOCA L CONNECTEDNES S AN D LOCA L CO-CONNECTEDNES S 17 6 1. Loca l connectedness in dimensio n n 17 6 2. Chain-realization s 17 7 3. Complex-lik e characte r o f compac t lc n spaces 18 0 4. Non-compac t case s 18 3 5. Fundamenta l system s o f cycle s • 18 5 6. Loca l co-connectedness; local connectivity number s and local dualities . 18 9 7. Propert y (P , Q) n , . 19 3 8. Othe r types o f highe r dimensional loca l connectednes s 19 8 VII. APPLICATIO N OF HOMOLOGY AND COHOMOLOG Y THEORY T O THE THEORY OF CONTINUA 20 0 1. Fundamenta l lemma s 20 0 2. Existenc e lemma s regardin g carrier s of cycle s and homologies 20 4 3. Separation s o f continu a by close d subsets 21 1 4. Non-r-cu t an d r-avoidable point s 21 8 5. r-extendabilit y 22 4 6. Non-cut-point s an d avoidable point s 22 7 7. Propert y £„ 23 4 8. Set-avoidabilit y 23 8 9. A n additio n theore m 24 1 VIII. GENERALIZE D MANIFOLDS ; DUALITIE S O F THE POINCAR E AN D ALEXANDE R TYP E . 24 4 1. Genera l propertie s 24 4 2. Orientabilit y 24 6 3. Th e orientabl e n-gc m 25 0 4. Th e Poincar S dualit y fo r an orientabl e n-gc m 25 2 5. Th e ope n n-gm 25 4 6. Th e Alexande r type o f dualit y fo r a closed subset o f a n n-gcm. Firs t proo f 26 1 7. Th e Alexander type of duality for a closed subset of an n-gcm. Secon d proof 26 3 8. Linkin g theorem s 26 6 9. A dualit y fo r non-closed set s 26 9 TABLE O F CONTENT S v IX. FURTHE R PROPERTIE S O F n-GMs ; REGULA R MANIFOLD S AN D GENERALIZE D TI-CELL S 27 1 1. Cas e n = 1 27 1 2. Cas e n = 2 27 2 3. Avoidabilit y propertie s 27 2 4. Characterizatio n b y mean s o f loca l linking 27 4 5. Th e cas e n = 2 without th e orientability conditio n 27 5 6. Th e general non-orientable cas e 28 0 7. Compariso n o f the case n > 2 with the classical case; regular manifold s an d generalized n-cell s 28 2 X. SUBMANIFOLD S OF A MANIFOLD; DECOMPOSITIO N INT O CELL S 29 0 1. Positiona l invariants 29 1 2. Unifor m loca l co-connectedness; duality o f r-ul c and ( n — r)-coulc .... 29 2 3. Th e Jordan-Brouwer separatio n theore m i n an n-gcm , an d it s converse . 29 4 4. Generalizatio n 29 6 5. Additiona l positiona l propertie s 29 8 6. Th e boundary o f a ulc n~2 domain i n a manifol d 30 4 7. Additiona l converse s o f the Jordan-Brouwer separatio n theore m 30 7 8. Th e genera l ulc n~2 open subse t o f an n-gc m 30 9 9. Decompositio n o f the spherelik e n-gc m int o two generalized close d n-cells . 31 1 XI. Lc * SUBSET S OF A N n-G M 31 6 1.
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