Lectures on Algebraic Topology
1st Edition
Published on 6. December 2012
XI, 380 pages
978-3-662-00756-3 (ISBN)
Description
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.
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Content
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.
