Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications. The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Sergey V. Matveev
Publisher: European Mathematical Society
Year: 2006
Language: English
Pages: 107
Preface......Page 5
Contents......Page 7
1.1 Categories and functors......Page 9
1.2 Some geometric properties of R^N......Page 12
1.3 Chain complexes......Page 15
1.4 Homology groups of a simplicial complex......Page 18
1.5 Simplicial maps......Page 21
1.6 Induced homomorphisms of homology groups......Page 25
1.7 Degrees of maps between manifolds......Page 26
1.8 Applications of the degree of a map......Page 31
1.9 Relative homology......Page 40
1.10 The exact homology sequence......Page 41
1.11 Axiomatic point of view on homology......Page 45
1.12 Digression to the theory of Abelian groups......Page 47
1.13 Calculation of homology groups......Page 49
1.14 Cellular homology......Page 51
1.15 Lefschetz fixed point theorem......Page 55
1.16 Homology with coefficients......Page 58
1.17 Elements of cohomology theory......Page 61
1.18 The Poincaré duality......Page 66
2.1 Definition of the fundamental group......Page 69
2.2 Independence of the choice of the base point......Page 71
2.3 Presentations of groups......Page 73
2.4 Calculation of fundamental groups......Page 76
2.5 Wirtinger’s presentation......Page 80
2.6 The higher homotopy groups......Page 82
2.7 Bundles and exact sequences......Page 84
2.8 Coverings......Page 87
Answers, hints, solutions......Page 91
Bibliography......Page 103
Index......Page 105
