Basic algebraic topology and its applications

This book provides an accessible introduction to algebraic topology, a field at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book offers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.

Prerequisite Concepts and Notations -- Homotopy Theory: Elementary Basic Concepts -- The Fundamental Groups -- Covering Spaces -- Fibre Bundles, Vector Bundles and K-theory -- Geometry of Simplicial Complexes and Fundamental Groups of Polyhedra -- Higher Homotopy Groups -- CW-complexes and Homotopy -- Products in Homotopy Groups -- Homology and Cohomology Theories -- Eilenberg-MacLane Spaces -- Eilenberg-Steenrod Axioms for Homology and Cohomology Theories -- Consequences of the Eilenberg-Steenrod Axioms -- Applications -- Spectral Homology and Cohomology Theories -- Obstruction Theory -- More Relations Between Homology and Homotopy -- A Brief History of Algebraic Topology.