Elements of Homology Theory (Graduate Studies in Mathematics, 81)

The Book Is A Continuation Of The Previous Book By The Author (elements Of Combinatorial And Differential Topology, Graduate Studies In Mathematics, Volume 74, American Mathematical Society, 2006). It Starts With The Definition Of Simplicial Homology And Cohomology, With Many Examples And Applications. Then The Kolmogorov-alexander Multiplication In Cohomology Is Introduced. A Significant Part Of The Book Is Devoted To Applications Of Simplicial Homology And Cohomology To Obstruction Theory, In Particular, To Characteristic Classes Of Vector Bundles. The Later Chapters Are Concerned With Singular Homology And Cohomology, And Cech And De Rham Cohomology. The Book Ends With Various Applications Of Homology To The Topology Of Manifolds, Some Of Which Might Be Of Interest To Experts In The Area. The Book Contains Many Problems; Almost All Of Them Are Provided With Hints Or Complete Solutions.--jacket. Simplicial Homology -- Cohomology Rings -- Applications Of Simplicial Homology -- Singular Homology -- Čech Cohomology And De Rham Cohomology -- Miscellany. V.v. Prasolov. Includes Bibliographical References (p. 403-409) And Index. Prasolov developed from his seminars on topology for second-year graduate students at the Independent U. of Moscow, so his treatment of homology and cohomology takes a very structured form rather than emerging as a series of mathematical vignettes. Prasolov starts with the definition of simplicial homology and cohomology and backs this up with examples and applications, describes calculations, the Euler characteristic and the Lefschetz theorem. He then introduces cohomology rings in terms of the Kolmogotov-Alexander multiplication in cohomology, the homology and cohomology of manifolds, and the Knneth theorem, then turns to applications of simplicial homology, including homology's relationship with homotopy, characteristic classes, group actions and Stenrod squares, singular homology, Cech cohomology and de Rham cohomology. Other topics include the Alexander polynomial, the Arf invariant, embeddings and immersions, complex manifolds, Lie groups and H-spaces. Prasolov includes solutions for selected exercises. Annotation 2007 Book News, Inc., Portland, OR (booknews.com) The book is a continuation of the previous book by the author (Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, Volume 74, American Mathematical Society, 2006). It starts with the definition of simplicial homology and cohomology, with many examples and applications. Then the Kolmogorov–Alexander multiplication in cohomology is introduced. A significant part of the book is devoted to applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles. The later chapters are concerned with singular homology and cohomology, and Čech and de Rham cohomology. The book ends with various applications of homology to the topology of manifolds, some of which might be of interest to experts in the area. The book contains many problems; almost all of them are provided with hints or complete solutions. Chapter 1. Simplicial homology Chapter 2. Cohomology rings Chapter 3. Applications of simplicial homology Chapter 4. Singular homology Chapter 5. Čech cohomology and de Rham cohomology Chapter 6. Miscellany Hints and solutions