This is essentially a book on singular homology and cohomology with special emphasis on products and manifolds. It does not treat homotopy theory except for some basic notions, some examples, and some applica­ tions of (co-)homology to homotopy. Nor does it deal with general(-ised) homology, but many formulations and arguments on singular homology are so chosen that they also apply to general homology. Because of these absences I have also omitted spectral sequences, their main applications in topology being to homotopy and general (co-)homology theory. Cech­ cohomology is treated in a simple ad hoc fashion for locally compact subsets of manifolds; a short systematic treatment for arbitrary spaces, emphasizing the universal property of the Cech-procedure, is contained in an appendix. The book grew out of a one-year's course on algebraic topology, and it can serve as a text for such a course. For a shorter basic course, say of half a year, one might use chapters II, III, IV (§§ 1-4), V (§§ 1-5, 7, 8), VI (§§ 3, 7, 9, 11, 12). As prerequisites the student should know the elementary parts of general topology, abelian group theory, and the language of categories - although our chapter I provides a little help with the latter two. For pedagogical reasons, I have treated integral homology only up to chapter VI; if a reader or teacher prefers to have general coefficients from the beginning he needs to make only minor adaptions.

Inhalt

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 Cohornology 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. Künneth Formulas for Spaces.- VII Products.- § 1 The Scalar Product.- § 2 The Exterior Homology Product.- § 3 The Interior Homology Product (Pontrjagin 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 Pontrjagin 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.