Lectures on Algebraic Topology

I Preliminaries on Categories, Abelian Groups, and Homotopy.- §1 Categories and Functors.- §2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- §3 Homotopy.- II Homology of Complexes.- §1 Complexes.- §2 Connecting Homomorphism, Exact Homology Sequence.- §3 Chain-Homotopy.- §4 Free Complexes.- III Singular Homology.- §1 Standard Simplices and Their Linear Maps.- §2 The Singular Complex.- §3 Singular Homology.- §4 Special Cases.- §5 Invariance under Homotopy.- §6 Barycentric Subdivision.- §7 Small Simplices. Excision.- §8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- §1 Standard Maps between Cells and Spheres.- §2 Homology of Cells and Spheres.- §3 Local Homology.- §4 The Degree of a Map.- §5 Local Degrees.- §6 Homology Properties of Neighborhood Retracts in ?n.- §7 Jordan Theorem, Invariance of Domain.- §8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- §1 Cellular Spaces.- §2 CW-Spaces.- §3 Examples.- §4 Homology Properties of CW-Spaces.- §5 The Euler-Poincaré Characteristic.- §6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- §7 Simplicial Spaces.- §8 Simplicial Homology.- VI Functors of Complexes.- §1 Modules.- §2 Additive Functors.- §3 Derived Functors.- §4 Universal Coefficient Formula.- §5 Tensor and Torsion Products.- §6 Horn and Ext.- §7 Singular Homology and Cohomology with General Coefficient Groups.- §8 Tensorproduct and Bilinearity.- §9 Tensorproduct of Complexes. Künneth Formula.- §10 Horn of Complexes. Homotopy Classification of Chain Maps.- §11 Acyclic Models.- §12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- §1 The Scalar Product.- §2 The Exterior Homology Product.- § 3 The Interior Homology Product (Pontijagin Product).- § 4 Intersection Numbers in ?n.- §5 The Fixed Point Index.- §6 The Lefschetz-Hopf Fixed Point Theorem.- §7 The Exterior Cohomology Product.- § 8 The Interior Cohomology Product (?-Product).- § 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- §10 Hopf Algebras.- §11 The Cohomology Slant Product.- §12 The Cap-Product (?-Product).- § 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- §1 Elementary Properties of Manifolds.- §2 The Orientation Bundle of a Manifold.- §3 Homology of Dimensions ? n in n-Manifolds.- §4 Fundamental Class and Degree.- §5 Limits.- §6 ?ech Cohomology of Locally Compact Subsets of ?n.- §7 Poincaré-Lefschetz Duality.- §8 Examples, Applications.- §9 Duality in ?-Manifolds.- §10 Transfer.- §11 Thom Class, Thom Isomorphism.- §12 The Gysin Sequence. Examples.- §13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- §1 Limits of Functors.- §2 Polyhedrons under a Space, and Partitions of Unity.- §3 Extending Functors from Polyhedrons to More General Spaces.