Preface.

Algebraic topology is one of the most important creations in mathematics which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism (though usually classify up to homotopy equivalence). The most important of these invariants are homotopy groups, homology groups, and cohomology groups (rings). The main purpose of this book is to give an accessible presentation to the readers of the basic materials of algebraic topology through a study of homotopy, homology, and cohomology theories. Moreover, it covers a lot of topics for advanced students who are interested in some applications of the materials they have been taught. Several basic concepts of algebraic topology, and many of their successful applications in other areas of mathematics and also beyond mathematics with surprising results have been given. The essence of this method is a transformation of the geometric problem to an algebraic one which offers a better chance for solution by using standard algebraic methods. The monumental work of Poincaré in “Analysis situs”, Paris, 1895, organized the subject for the first time. This work explained the difference between curves deformable to one another and curves bounding a larger space. The first one led to the concepts of homotopy and fundamental group; the second one led to the concept of homology. Poincaré is the first mathematician who systemically attacked the problems of assigning algebraic invariants to topological spaces. His vision of the key role of topology in all mathematical theories began to materialize from 1920. This subject is an interplay between topology and algebra and studies algebraic invariants provided by homotopy, homology, and cohomology theories. The twentieth century witnessed its greatest development. The literature on algebraic topology is very vast. Based on the author’s teaching experience of 50 years, academic interaction with Prof. B. Eckmann and Prof. P.J. Hilton at E.T.H., Zurich, Switzerland, in 2003, and lectures at different institutions in India, USA, France, Switzerland, Greece, UK, Italy, Sweden, Japan, and many other countries, this book is designed to serve as a basic text of modern algebraic topology at the undergraduate level. A basic course in algebraic topology