Elements of Combinatorial and Differential TopologyModern topology uses very diverse methods. This book is devoted largely to methods of combinatorial topology, which reduce the study of topological spaces to investigations of their partitions into elementary sets, and to methods of differential topology, which deal with smooth manifolds and smooth maps. Many topological problems can be solved by using either of these two kinds of methods, combinatorial or differential. In such cases, both approaches are discussed. One of the maingoals of this book is to advance as far as possible in the study of the properties of topological spaces (especially manifolds) without employing complicated techniques. This distinguishes it from the majority of other books on topology. The book contains many problems; almost all of them are suppliedwith hints or complete solutions. |
Contents
| 1 | |
| 5 | |
Chapter 2 Topology in Euclidean Space | 55 |
Chapter 3 Topological Spaces | 87 |
Chapter 4 TwoDimensional Surfaces Coverings Bundles and Homotopy Groups | 139 |
Chapter 5 Manifolds | 181 |
Chapter 6 Fundamental Groups | 257 |
Hints and Solutions | 291 |
| 317 | |
| 325 | |
Back Cover | 335 |
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Common terms and phrases
assume barycentric subdivision base point belongs Borsuk-Ulam theorem boundary cell choose circle Clearly closed manifold coincides completely labeled connected components Consider the map constant map construct contained continuous map corresponding critical points CW-complex cycle defined deleted denote determined diffeomorphism dimension disjoint disk domains easy to verify edges element embedding exists fibration follows function f fundamental group graph G Hausdorff Hausdorff space hence homeomorphic homotopy equivalent hyperplane identity map intersection isomorphic Jacobian joined lemma loop manifold map f map g matrix Möbius band Morse function neighborhood obtain one-to-one open cover open set open subset orientation orthogonal pairwise path path-connected planar graph plane point a e polyhedron polynomial preimage Problem projection Proof prove pseudomanifold regular value simplex simplicial complex singular points smooth map SO(n sphere subgraph subgroup subspace Suppose tangent vector Theorem topological space triangulation two-dimensional surface vector field vertex vertices