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Lee J.M. — Introduction to Topological Manifolds
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Название: Introduction to Topological ManifoldsАвтор: Lee J.M. Аннотация: This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields. In his beautifully conceived introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout. John M. Lee is currently Professor of Mathematics at the University of Washington.
Язык: Рубрика: Математика /Геометрия и топология /Общая топология /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2000Количество страниц: 385Добавлена в каталог: 19.04.2005Операции: Положить на полку |
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First category 86 First countable 32 First countable, locally Euclidean spaces 38 First countable, metric spaces 38 First isomorphism theorem 356 Five lemma 313 Fixed point theorem, Brouwer 192 Folding 134 Forgetful functor 172 Formal linear combination 97 Free Abelian group 204 Free abelian group, characteristic property 204 Free abelian group, uniqueness 208 Free abelian subgroup 205 Free group 200 Free group action 60 Free group, characteristic property 200 Free group, uniqueness 201 Free product 195 Free product, characteristic property 198 Free product, uniqueness 199 Free vector space 97 Freedman, Michael 7 Freedom, degrees of 2 Full subcategory 171 Function 341 Function, multiple-valued 8 Functor, cohomology 330 Functor, contravariant 172 Functor, covariant 172 Functor, exact 335 Functor, forgetful 172 Functor, fundamental group 172 Functor, homology 296 Fundamental group 6 152 Fundamental group and homology 305 Fundamental group and surface presentation 217 Fundamental group is a group 154 Fundamental group of a graph 215 Fundamental group of a manifold is countable 189 Fundamental group of a polyhedron 230 Fundamental group of a product 189 Fundamental group of a surface 220 Fundamental group of a surface, abelianized 228 Fundamental group of a topological group 191 Fundamental group of a wedge of spaces 213 Fundamental group of spheres 188 217 Fundamental group of the circle 180 Fundamental group of the projective plane 247 Fundamental group of the torus 189 Fundamental group, action on fiber 245 Fundamental group, change of base point 154 Fundamental group, functor 172 Fundamental group, homotopy invariance 161 Fundamental group, topological invariance 159 Fundamental theorem of algebra 191 Gauss — Bonnet theorem 8 General linear group 10 59 60 General linear group, complex 10 General position 92 General relativity 14 Generating circle 45 Generator of a cyclic group 356 Generator of a group 199 Generator of a presentation 201 Genus 144 Geodesic, hyperbolic 271 Geodesic, polygon 273 Geodesic, polygon, regular 274 Geometric realization 97 99 Geometric realization of a polygonal presentation 131 Geometric realization, functor 172 Geometrization conjecture 7 Geometry, algebraic 11 Geometry, Euclidean 7 Geometry, plane 7 Geometry, Riemannian 8 Geometry, solid 7 Gluing lemma 46 Gluing lemma, counterexample 62 Graph 100 Graph of a continuous function 2 43 Graph of a continuous functionis a manifold 44 Graph of a relation 8 Graph, finite 101 Graph, fundamental group 215 Graph, simple 101 Gravitation 14 Gravity, center of 110 Greatest lower bound 343 Group 352 GROUP (category of groups) 171 Group as a category 173 Group, abelian 11 203 Group, action 59 266 Group, action, continuous 60 Group, action, free 60 Group, action, proper 266 Group, action, quotient by 61 Group, action, transitive 60 Group, automorphism, of covering 248 Group, complex general linear 10 Group, complex special linear 11 Group, covering 248 Group, direct product 353 Group, discrete 58 Group, divisible 335 Group, free 200 Group, free abelian 204 Group, fundamental 6 152 Group, general linear 10 59 60 Group, homotopy 169—170 Group, injective 335 Group, Lie 10 Group, orthogonal 11 59 60 Group, permutation 353 Group, presentation 201 202 Group, special linear 11 Group, special orthogonal 11 Group, special unitary 11 Group, symmetric 353 Group, theory, combinatorial 203 Group, topological 58 Group, topological, quotient 63 Group, unitary 11 Groups, category of 171 Hairy ball theorem 322 Half space, upper 34 Ham sandwich theorem 254 handedness 106 Handle 9 129 Hauptvermutung 112 Hausdorff, if diagonal is closed 62 Hausdorff, product space 50 Hausdorff, space 31 Hausdorff, sub space 42 Heegaard, Poul 137 Heine — Borel theorem 351 hole 5 6 147 Holomorphic 8 Hom(X, Y) (set of group homomorphisms) 173 Homeomorphic 4 22 Homeomorphic is an equivalence relation 22 Homeomorphism 4 22 Homeomorphism, local 24 Homeomorphism, local, openness 24 Homeomorphism, vs. closed map 27 Homeomorphism, vs. open map 27 Homogeneity of norm 89 Homogeneous, space 63 Homogeneous, spacetime 14 Homological algebra 297 Homological degree 320 Homologous 295 Homology and the fundamental group 305 Homology of a compact polyhedron 334 Homology of a contractible space 303 Homology of a disconnected space 298 Homology of a one-point space 299 Homology of a pseudomanifold 334 Homology of a simplex 324 Homology of a triangulable manifold 334 Homology of a wedge 334 Homology of spheres 309 Homology, class 295 Homology, functor 296 Homology, groups of a chain complex 297 Homology, groups, simplicial 324 Homology, groups, singular 295 Homology, homomorphism, induced by a chain map 297 Homology, homomorphism, induced by a continuous map 296 Homology, homotopy invariance 300 Homology, sequence, long exact 311 Homology, simplicial 324 Homology, simplicial, vs. singular 326 Homology, singular 295 Homology, singular, vs. simplicial 326 Homology, topological invariance 296 Homology, zero-dimensional 298 Homomorphism from a quotient group 355 Homomorphism of cyclic group 357 Homomorphism of group 354 Homomorphism of topological groups 270 Homomorphism, covering 258 Homomorphism, covering is a covering map 258 Homomorphism, criterion, covering 260 Homomorphism, fundamental group, induced by a continuous map 159 Homomorphism, homology, induced by a chain map 297 Homomorphism, homology, induced by a continuous map 296 Homotopic 148 Homotopic, degree 320 Homotopic, maps and fundamental group homomorphisms 164 Homotopic, maps and homology homomorphisms 300 Homotopic, path 151 Homotopic, relative to a subspace 151 Homotopy is an equivalence relation 148 Homotopy of maps 148 Homotopy, category 173 Homotopy, category, pointed 173 Homotopy, chain 303 317 Homotopy, equivalence 161 Homotopy, equivalence and deformation retraction 166 Homotopy, equivalence is an equivalence relation 161 Homotopy, equivalent 161 Homotopy, groups 169—170 Homotopy, invariance of path product 152 Homotopy, invariance of singular homology 300 Homotopy, invariance of the fundamental group 161 Homotopy, lifting property 238 Homotopy, lifting property of the circle 181 Homotopy, path 151 Homotopy, path and composition 158 Homotopy, path is an equivalence relation 151 Homotopy, preserved by composition 149 Homotopy, relative 151 Homotopy, straight-line 150 Homotopy, theory 170 Homotopy, type 161 Hull, convex 92 Hurewicz, homomorphism 308 Hurewicz, theorem 308 Hurewicz, Witold 308 Hyperbolic disk 271 Hyperbolic geodesic 271 Hyperbolic metric 271 Hyperbolic metric, triangle inequality 289 Hyperbolic neighborhood, regular 278 Hyperplane, affine 92 i (imaginary unit) 344 I (unit interval) 54 Ideal point 12 Identification space 52 Identity in a category 171 Identity in a group 352 Identity in a group, uniqueness 353 Identity, map 342 Identity, map, continuity 21 Identity, path class 153 Image is a subgroup 354 Image of a function 342 Image of a homomorphism 354 Image of a normal subgroup 356 Image, inverse 342 Image, set 342 Imaginary unit 344 Imf (image of f) 354 Inclusion map 342 Inclusion map, continuity 41 Increasing function 345 Independent, linearly 204 Index of a subgroup 354 Index of a vector field 192 Index set 345 Indexed collection 345 Indexed collection, intersection 345 Indexed collection, union 345 Induced homomorphism of fundamental groups 159 Induced homomorphism of fundamental groups by homotopic maps 164 Induced homomorphism, in homology 296 297 Induced morphism 172 Induced orientation 107 Induced subgroup 239 Infimum 343 Infinite cyclic group 200 356 Infinite dimensional simplicial complex 96 Infinite product 49 Infinite set 344 Initial point of a path 150 Initial vertex 131 Injection in a category 175 Injection into direct sum 177 Injection into disjoint union 345 Injection into free group 200 Injection into free product 197 Injective 342 Injective, group 335 Injectivity theorem 239 Inside out sphere 5 int see “Interior” Integers 344 Integers modulo n 356 Interior 25 26 Interior of a manifold with boundary 34 38 Interior of a simplex 93 Intermediate Value Theorem 65 68 Intersection of an indexed collection 345 Intersection of closed sets in a metric space 349 Intersection of closed sets in a topological space 24 Intersection of open sets in a metric space 349 Intersection of open sets in a topological space 18 Intersection of sets 339 Intertwined edge pairs 140 interval 344 Interval is connected 68 Interval, unit 54 Invariance of dimension 318 319 Invariant, combinatorial 113 142 Invariant, topological 6 Inverse in a group 352 Inverse in a group, uniqueness 353
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