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Contents

• V Full version
• 0 Introduction
• 1 Definitions
• 1.1 Some recollections and conventions
• 1.2 Cell complexes
• 2 Homotopy and the fundamental group
• 2.1 Motivation
• 2.2 Homotopy
• 2.3 Paths
• 2.4 The fundamental group
• 3 Covering spaces
• 3.1 Covering space
• 3.2 The fundamental group of the circle and its applications
• 3.3 Universal covers
• 3.4 The Galois correspondence
• 4 Some group theory
• 4.1 Free groups and presentations
• 4.2 Another view of free groups
• 4.3 Free products with amalgamation
• 5 Seifert-van Kampen theorem
• 5.1 Seifert-van Kampen theorem
• 5.2 The effect on pi1 of attaching cells
• 5.3 A refinement of the Seifert-van Kampen theorem
• 5.4 The fundamental group of all surfaces
• 6 Simplicial complexes
• 6.1 Simplicial complexes
• 6.2 Simplicial approximation
• 7 Simplicial homology
• 7.1 Simplicial homology
• 7.2 Some homological algebra
• 7.3 Homology calculations
• 7.4 Mayer-Vietoris sequence
• 7.5 Continuous maps and homotopy invariance
• 7.6 Homology of spheres and applications
• 7.7 Homology of surfaces
• 7.8 Rational homology, Euler and Lefschetz numbers