Basic Concepts of Algebraic Topology

1 Geometric Complexes and Polyhedra.- 1.1 Introduction.- 1.2 Examples.- 1.3 Geometric Complexes and Polyhedra.- 1.4 Orientation of Geometric Complexes.- 2 Simplicial Homology Groups.- 2.1 Chains, Cycles, Boundaries, and Homology Groups.- 2.2 Examples of Homology Groups.- 2.3 The Structure of Homology Groups.- 2.4 The Euler-Poincaré Theorem.- 2.5 Pseudomanifolds and the Homology Groups of Sn.- 3 Simplicial Approximation.- 3.1 Introduction.- 3.2 Simplicial Approximation.- 3.3 Induced Homomorphisms on the Homology Groups.- 3.4 The Brouwer Fixed Point Theorem and Related Results.- 4 The Fundamental Group.- 4.1 Introduction.- 4.2 Homotopic Paths and the Fundamental Group.- 4.3 The Covering Homotopy Property for S1.- 4.4 Examples of Fundamental Groups.- 4.5 The Relation Between H1(K) and ?1(|K|).- 5 Covering Spaces.- 5.1 The Definition and Some Examples.- 5.2 Basic Properties of Covering Spaces.- 5.3 Classification of Covering Spaces.- 5.4 Universal Covering Spaces.- 5.5 Applications.-6 The Higher Homotopy Groups.- 6.1 Introduction.- 6.2 Equivalent Definitions of ?n(X, x0).- 6.3 Basic Properties and Examples.- 6.4 Homotopy Equivalence.- 6.5 Homotopy Groups of Spheres.- 6.6 The Relation Between Hn(K) and ?n(|K|).- 7 Further Developments in Homology.- 7.1 Chain Derivation.- 7.2 The Lefschetz Fixed Point Theorem.- 7.3 Relative Homology Groups.- 7.4 Singular Homology Theory.- 7.5 Axioms for Homology Theory.- Appendix 1. Set Theory.- Appendix 2. Point-set Topology.- Appendix 3. Algebra.