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1 Elements of homology theory 1

1.1 Categories and functors 1

1.2 Some geometric properties of RN 4

1.3 Chain complexes 7

1.4 Homology groups of a simplicial complex 10

1.5 Simplicial maps 13

1.6 Induced homomorphisms of homology groups 17

1.7 Degrees of maps between manifolds 18

1.8 Applications of the degree of a map 23

1.9 Relative homology 32

1.10 The exact homology sequence 33

1.11 Axiomatic point of view on homology 37

1.12 Digression to the theory of Abelian groups 39

1.13 Calculation of homology groups 41

1.14 Cellular homology 43

1.15 Lefschetz fixed point theorem 47

1.16 Homology with coefficients 50

1.17 Elements of cohomology theory 53

1.18 The Poincare duality 58

2 Elements of homotopy theory 61

2.1 Definition of the fundamental group 61

2.2 Independence of the choice of the base point 63

2.3 Presentations of groups 65

2.4 Calculation of fundamental groups 68

2.5 Wirtinger's presentation 72

2.6 The higher homotopy groups 74

2.7 Bundles and exact sequences 76

2.8 Coverings 79

Answers, hints, solutions 83

Bibliography 95

Index 97