| Preface |
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vii | |
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Chapter 1 Basic Concepts Of Manifolds |
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1 | (42) |
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1.1 Two definitions of a manifold and examples |
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1 | (9) |
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1.2 Smooth maps between manifolds |
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10 | (4) |
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1.3 Induced smooth structures |
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14 | (2) |
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1.4 Immersions and Submersions |
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16 | (6) |
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22 | (4) |
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1.6 Further examples of manifolds |
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26 | (4) |
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30 | (6) |
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1.8 Manifolds with boundary and corner |
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36 | (7) |
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Chapter 2 Approximation Theorems And Whitney's Embedding |
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43 | (26) |
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2.1 Smooth partition unity |
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43 | (8) |
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2.2 Smooth approximations to continuous maps |
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51 | (3) |
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54 | (5) |
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2.4 Approximations by immersions |
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59 | (3) |
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2.5 Whitney's embedding theorem |
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62 | (1) |
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2.6 Homotopy of smooth maps |
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63 | (2) |
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2.7 Stability of smooth maps |
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65 | (4) |
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Chapter 3 Linear Structures on Manifolds |
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69 | (36) |
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3.1 Tangent spaces and derivative maps |
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69 | (7) |
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3.2 Vector Fields and Flows |
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76 | (6) |
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82 | (6) |
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88 | (2) |
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3.5 Derivations of algebra of differential forms |
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90 | (5) |
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3.6 Darboux-Weinstein theorems |
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95 | (10) |
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Chapter 4 Riemannian Manifolds |
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105 | (28) |
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105 | (7) |
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4.2 Geodesics on a Manifold |
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112 | (4) |
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4.3 Riemannian connection and geodesics |
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116 | (6) |
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122 | (4) |
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126 | (3) |
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4.6 Totally geodesic submanifolds |
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129 | (4) |
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Chapter 5 Vector Bundles On Manifolds |
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133 | (36) |
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133 | (9) |
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5.2 Construction of vector bundles |
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142 | (3) |
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5.3 Homotopy property of vector bundles |
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145 | (3) |
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5.4 Subbundle and quotient bundle |
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148 | (3) |
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151 | (6) |
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5.6 Reduction of structure group of a vector bundle |
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157 | (2) |
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5.7 Homology characterisation of orientation |
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159 | (4) |
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5.8 Integration of differential forms on manifolds |
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163 | (6) |
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169 | (30) |
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6.1 ε-neighbourhood of submanifold of Euclidean space |
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169 | (4) |
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173 | (6) |
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6.3 Compact one-manifolds and Brouwer's theorem |
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179 | (3) |
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6.4 Boundary and preimage orientations |
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182 | (3) |
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6.5 Intersection numbers, and Degrees of maps |
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185 | (6) |
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6.6 Hopf's degree theorem |
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191 | (8) |
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Chapter 7 Tubular Neighbourhoods |
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199 | (26) |
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7.1 Tubular neighbourhood theorems |
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199 | (3) |
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7.2 Collar neighbourhoods |
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202 | (4) |
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7.3 Isotopy extension theorem |
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206 | (6) |
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7.4 Uniqueness of tubular neighbourhoods |
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212 | (4) |
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7.5 Manifolds with corner and straightening them |
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216 | (3) |
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7.6 Construction of manifolds by gluing process |
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219 | (6) |
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Chapter 8 Spaces of Smooth Maps |
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225 | (42) |
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225 | (8) |
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8.2 Weak and strong topologies |
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233 | (7) |
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8.3 Continuity of maps between spaces of smooth maps |
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240 | (4) |
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8.4 Spaces of immersions and embeddings |
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244 | (4) |
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8.5 Baire property of the space of smooth maps |
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248 | (2) |
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8.6 Smooth structures on jet spaces |
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250 | (4) |
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8.7 Thom's Transversality Theorem |
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254 | (6) |
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8.8 Multi-jet transversality |
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260 | (2) |
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8.9 More results on Whitney's immersion and embedding |
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262 | (5) |
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267 | (32) |
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267 | (5) |
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9.2 Critical levels and attaching handles |
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272 | (14) |
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286 | (6) |
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9.4 Perfect Morse functions |
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292 | (2) |
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9.5 Triangulations of manifolds |
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294 | (5) |
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Chapter 10 Theory Of Handle Presentations |
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299 | (42) |
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10.1 Existence of handle presentation |
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300 | (5) |
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305 | (7) |
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10.3 Normalisation of presentation |
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312 | (2) |
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10.4 Cancellation of handles |
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314 | (5) |
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10.5 Classification of closed surfaces |
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319 | (2) |
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10.6 Removal of intersection points |
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321 | (9) |
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330 | (4) |
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10.8 Simplification of handle presentations |
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334 | (3) |
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10.9 h-cobordism theorem and generalised Poincare conjecture |
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337 | (4) |
| Bibliography |
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341 | (4) |
| Index |
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345 | |
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