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. Front Matter....Pages i-xxix Prerequisite Concepts and Notations....Pages 1-44 Homotopy Theory: Elementary Basic Concepts....Pages 45-106 The Fundamental Groups....Pages 107-145 Covering Spaces....Pages 147-196 Fiber Bundles, Vector Bundles and K-Theory....Pages 197-247 Geometry of Simplicial Complexes and Fundamental Groups of Polyhedra....Pages 249-271 Higher Homotopy Groups....Pages 273-304 CW-Complexes and Homotopy....Pages 305-327 Products in Homotopy Theory....Pages 329-346 Homology and Cohomology Theories....Pages 347-406 Eilenberg–MacLane Spaces....Pages 407-417 Eilenberg–Steenrod Axioms for Homology and Cohomology Theories....Pages 419-431 Consequences of the Eilenberg–Steenrod Axioms....Pages 433-443 Applications....Pages 445-473 Spectral Homology and Cohomology Theories....Pages 475-509 Obstruction Theory....Pages 511-531 More Relations Between Homology and Homotopy ....Pages 533-545 A Brief History of Algebraic Topology....Pages 547-568 Back Matter....Pages 569-615