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		Contents

Preface xi 1 Background in Physics 1 1.1 Path Integral and Quantum Field Theories 1 1.2 Gauge Theories 10 1.3 Magnetic Monopoles 14 1.4 Instantons 17 1.5 Orders in Condensed Matter Systems 19 1.6 General Relativity 27 1.7 Berry's Phase 29 1.8 String Theory 30 Problems 1 33 2 Mathematical Preliminaries 35 2.1 Maps 35 2.2 Vector Spaces 43 2.3 Topological Spaces 48 2.4 Homeomorphisms and Topological Invariants 53 Problems 2 59 3 Homology Groups 61 3.1 Abelian Groups 61 3.2 Simplexes and Simplicial Complexes 66 3.3 Homology Groups of Simplicial Complexes 69 3.4 General Properties of Homology Groups 84 4 Homotopy Groups 89 4.1 Fundamental Groups 89 4.2 General Properties of Fundamental Groups 94 4.3 Examples of Fundamental Groups 99 4.4 Fundamental Groups of Polyhedra 102 4.5 Higher Homotopy Groups 113 4.6 General Properties of Higher Homotopy Groups 116