Lectures On Algebraic Topology (Paperback)

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-Poincare 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. Kunneth 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 Poincare-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.