代数拓扑讲义

Chapter Ⅰ Preliminaries on Categories，Abelian Groups， and Homotopy

§1 Categories and Functors

§2 Abelian Groups (Exactness， Direct Sums，Free Abelian Groups)

§3 Homotopy

Chapter Ⅱ Homology of Complexes

§1 Complexes

§2 Connecting Homomorphism，Exact Homology Sequence

§3 Chain-Homotopy

§4 Free Complexes

Chapter 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

Chapter 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 IRn

§7 Jordan Theorem， Invariance of Domain

§8 Euclidean Neighborhood Retracts (ENRs)

Chapter 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

Chapter Ⅵ Functors of Complexes

§1 Modules

§2 Additive Functors

§3 Derived Functors

§4 Universal Coefficient Formula

§5 Tensor and Torsion Products

§6 Hom 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

Chapter Ⅶ Products

§1 The Scalar Product

§2 The Exterior Homology Product

§3 The Interior Homology Product(Pontrjagin Product

§4 Intersection Numbers in IRn

§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

§ll The Cohomology Slant Product

§12 The Cap-Product(Product)

§13 The Homology Slant Product，and the Pontrjagin Slant Product Manffolds

Chapter Ⅷ Manifolds

§l Elementary Properties of Manifolds

§2 The Orientation Bundle of a Manifold

§3 Homology of Dimension≧n in n．Manifolds

§4 Fundamental Class and Degree

§5 Limits

§6 Cech Cohomology of Locally Compact Subsets of

§7 Poincar6-Lefschetz Duality

§8 Examples，Applications

§9 Duality in a-Manifolds

§10 Transfer

§11 Thom Class，Thorn Isomorphism

§12 The Gysin Sequence．Examples

§13 Intersection of Homology Classes Kan．and Cech-Extensions of Functors

Appendix

§l Limits of Functors

§2 Polyhcdtons under a Space，and Partitions of Unity

§3 Extending Functors from Polyhedrons to more General Spaces Bibliography SubjectIndex

Bibliography

Subject Index

This is essentially a book on singular homology and cohomology withspecial emphasis on products and manifolds. It does not treat homotopytheory except for some basic notions， some examples， and some applica-tions of homology to homotopy. Nor does it deal with general（ised）homology， but many formulations and arguments on singular homologyare so chosen that they also apply to general homology. Because of theseabsences I have also omitted spectral sequences， their main applicationsin topology being to homotopy and general homology theory. ech-cohomology is treated in a simple ad hoc fashion for locally compactsubsets of manifolds; a short systematic treatment for arbitrary spaces，emphasizing the universal property of the （ech-procedure， is containedin an appendix.The book grew out of a one-years course on algebraic topology， and itcan serve as a text for such a course. For a shorter basic course， say ofhalf a year， one might use chapters Ⅱ Ⅲ Ⅳ（§1-4）， Ⅴ（§I-5， 7， 8），Ⅵ（§ 3， 7， 9， 11， 12）. As prerequisites the student should know theelementary parts of general topology， abelian group theory， and thelanguage of categories-although our chapter Ⅰprovides a little helpwith the latter two. For pedagogical reasons， I have treated integralhomology only up to chapter Ⅵ if a reader or teacher prefers tohave general coefficients from the beginning he needs to make only minoradaptions.As to the outlay of the book， there are eight chapters， Ⅰ-Ⅷand nappendix， A; each of these is subdivided into several