Elements of Homology Theory
This book is a translation of the Russian original (published in 2005). It is a continuation of the author's earlier book “Elements of Combinatorial and Differential Topology”. It is convenient to have the companion volume at hand because the book refers to it for basic definitions and concepts. In the first half of the book, the author builds the theory of simplicial homology and cohomology and describes some of its applications (most of the space being devoted to characteristic classes in the simplicial setting). It is followed by a description of singular homology. The penultimate chapter defines Čech cohomology and de Rham cohomology and it proves the de Rham theorem and its simplicial analogue. The last chapter “Miscellany” contains material on invariants of links, embeddings and immersions of manifolds, and a section on cohomology of Lie Groups and H-Spaces.
The book contains 136 problems (mostly in the chapters on simplicial homology and cohomology and their applications). For the majority of problems, there are solutions or hints in a 37-page appendix. Together with its companion volume, the book can be recommended as a basis for an introductory course in algebraic topology.