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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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Inverse of a path class 153 Inverse, image 342 Inverse, left 343 Inverse, map 342 Inverse, right 343 346 Isolated singular point 192 Isometry 8 Isomorphic coverings 258 Isomorphism in a category 171 Isomorphism of coverings 258 Isomorphism of groups 354 Isomorphism, problem 203 Isomorphism, simplicial 96 Isomorphism, theorem, covering 260 Isomorphism, theorem, first 356 Isotropic spacetime 14 k-skeleton see “Skeleton” Ker see “Kernel” Kernel 354 Kernel is a subgroup 354 Kernel is normal 355 Klein bottle 126 Klein bottle, covering 253 Klein bottle, presentation 133 largest element 341 Latitude 3 Laws of motion, Newton’s 12 Least upper bound 343 Lebesgue number 76 Lebesgue number, lemma 76 Left action 59 Left coset 354 Left coset space 61 Left inverse 343 Left translation 59 63 Length 7 347 Lens space 269 Lens space, coverings of 286 Lie group 10 Lie group, abelian 11 Lift 179 180 237 Lifting criterion 240 Lifting problem 239 Lifting property, homotopy 238 Lifting property, homotopy of the circle 181 Lifting property, path 238 Lifting property, path of the circle 181 Lifting property, unique 237 Lifting property, unique of the circle 181 Limit of a sequence in a discrete space 20 Limit of a sequence in a Hausdorff space 32 Limit of a sequence in a metric space 349 Limit of a sequence in a topological space 20 Limit of a sequence in a trivial space 20 Limit point 26 348 Limit point and closed sets 26 Limit point, compact 76 Limit point, compact vs. compact 77 78 Limit point, compact vs. sequentially compact 77 78 Line with two origins 62 Line, long 88 Line, segment 347 linear combination 204 Linear combination, formal 97 Linear ordering 341 Linear transformation 20 Linearly independent 204 Local criterion for continuity 21 Local finiteness see “Locally finite” Local homeomorphism 24 Local homeomorphism, openness 24 Local section 184 236 Local section of a covering map 236 Locally compact 81 Locally compact, Hausdorff space 81 82 89 Locally connected 72 Locally Euclidean 4 30 Locally Euclidean, implies first countable 38 Locally finite 93 96 Locally path connected 72 Locally simply connected 262 Logarithmic function 20 Long exact homology sequence 311 Long line 88 longitude 3 Loop 151 Loop, based at a point 151 Loop, constant 151 Lorentz metric 14 Lower bound 341 Main theorem on compactness 73 Main theorem on connectedness 67 Manifold 1 4 33 Manifold is locally compact Hausdorff 83 Manifold is locally path connected 72 Manifold with boundary 34 334 Manifold with boundary, 2-dimensional 191 Manifold, boundary 35 Manifold, classification 6 Manifold, complex 33 Manifold, countable fundamental group 189 Manifold, homology of 334 Manifold, product of 51 Manifold, Riemannian 8 Manifold, smooth 33 Manifold, topological 33 Manifold, zero-dimensional 37 MAP 341 Mapping 341 Mapping cylinder 167 Markov, A. A. 7 Mathematical object 338 Maximal 341 Maximal tree 214 Mayer — Vietoris sequence, cohomology 335 Mayer — Vietoris sequence, simplicial 325 Mayer — Vietoris sequence, singular 309 Mayer — Vietoris theorem, cohomology 335 Mayer — Vietoris theorem, simplicial 325 Mayer — Vietoris theorem, singular 292 309 Mechanics, classical 12 Member of a set 338 membership 338 mesh 317 Metric 348 Metric, discrete 348 Metric, Euclidean 348 Metric, hyperbolic 271 Metric, Lorentz 14 Metric, space 348 Metric, space, first countable 38 Metric, space, Hausdorff 31 Metric, space, second countable 38 Metric, space, subspace of 40 Metric, topology 19 Minimal 341 Mobius, band 105 176 Mobius, group 272 Mobius, transformation 272 Modulo n 356 Moise, Edwin 105 Monodromy theorem 239 Morphism 170 Morphism, induced 172 Motion, Newton’s laws of 12 Multiple-valued function 8 Multiplication of cosets 355 Multiplication of path classes 153 Multiplication of paths 152 Multiplication of words 194 Multiplication, group 352 n-dimensional manifold 33 n-dimensional topological manifold 33 n-holed torus 129 n-holed torus, universal covering 275 n-manifold 33 n-sphere 44 n-sphere, singular homology 309 n-torus 51 n-torus as a coset space of 61 n-torus, fundamental group 189 n-tuple, ordered 345 Naive set theory 337 Natural numbers 344 Natural orientation 106 Naturality of connecting homomorphisms 312 Nearness 18 Neighborhood 18 Neighborhood, basis 32 Neighborhood, basis, countable 32 Neighborhood, basis, nested 77 Neighborhood, Euclidean 30 Neighborhood, regular hyperbolic 278 Nested cubes 351 Nested neighborhood basis 77 Nested sets 76 Newton’s laws of motion 12 Nondegenerate base point 212 Nonorientable surface 144 Nonorientable surface, covering of 253 Norm 89 347 Normal closure 201 Normal covering 245 Normal subgroup 354 Normal subgroup, image 356 North pole 3 45 Nowhere dense 85 nth homotopy group 170 nth power map 191 235 Null homotopic 151 O(n) (orthogonal group) 11 59 60 Object in a category 170 Object, mathematical 338 Odd map 253 Odd permutation 353 One-point compactihcation 89 One-point space, singular homology 299 One-point union 55 One-to-one correspondence 342 One-to-one function 342 Onto 342 Open ball 348 Open ball is an open set 349 Open cover 32 73 Open cover in a metric space 350 Open cover of a subset 73 Open map 24 Open map, product of 62 Open map, vs. homeomorphism 27 Open set as a topological space 19 Open set in a metric space 348 Open set in a topological space 18 Open set is a manifold 34 Open set is Hausdorff 31 Open set is second countable 33 Open set, criterion for continuity 350 Open set, intersection, in a metric space 349 Open set, intersection, in a topological space 18 Open set, union, in a metric space 349 Open set, union, in a topological space 18 Open simplex 93 Open star 114 Orbit 60 245 Orbit, criterion 249 Orbit, space 61 266 Orbit, space by free proper group action 268 Order of a group 353 Order topology 37 Ordered field 343 Ordered n-tuple 345 Ordered pair 340 Ordered set, partially 341 Ordered set, totally 37 341 Ordered well 88 341 Ordering, linear 341 Ordering, partial 341 Ordering, simple 341 Ordering, total 341 Orientability is combinatorially invariant 115 Orientable pseudomanifold 334 Orientable simplicial complex 107 Orientable surface 144 229 Orientation of a simplex 105 Orientation of a simplicial complex 107 Orientation, induced 107 Orientation, natural 106 Oriented presentation 144 Oriented simplex 106 Origins, line with two 62 Orthogonal group 11 59 60 Orthogonal group, special 11 Pair, ordered 340 Pancakes 234 parabola 2 Paraboloid 3 Parameters 1 Partial ordering 341 Partially ordered set 341 Particle, elementary 14 Partition 52 340 Passing to the quotient 56 57 Passing to the quotient, homomorphism 356 Pasting 134 Path 69 150 Path, class 151 Path, class identity 153 Path, class inverse 153 Path, class multiplication 153 Path, class multiplication, associativity 153 Path, class product 153 Path, class product, associativity 153 Path, component 72 Path, connected 69 Path, connected, locally 72 Path, connectivity relation 71 Path, homotopic 151 Path, homotopy 151 Path, homotopy and composition 158 Path, homotopy is an equivalence relation 151 Path, lifting property 238 Path, lifting property of the circle 181 Path, multiplication 152 Path, multiplication, grouping 154 Path, multiplication, homotopy invariance 152 Path, product 152 Path, product, grouping 154 Path, product, homotopy invariance 152 Path, reverse 153 Periodic edge path 119 Periodic trajectory 14 Permutation 342 Permutation, even 353 Permutation, group 353 Permutation, odd 353 Plane 3 Plane, curve 2 Plane, geometry 7 Plane, projective 119 Poincare, conjecture 6 Poincare, Henri 4 6
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