Elements of Homology Theory

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 Künneth 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)