Homotopical topology This classic text of the renowned Moscow mathematical school equips the aspiring mathematician with a solid grounding in the core of topology from a homotopical perspective. Its comprehensiveness and depth of treatment are unmatched among topology textbooks: in addition to covering the basics the fundamental notions and constructions of homotopy theory covering spaces and the fundamental group CW complexes homology and cohomology homological algebra the book treats essential advanced topics such as obstruction theory characteristic classes Steenrod squares K theory and cobordism theory and with distinctive thoroughness and lucidity spectral sequences. The organization of the material around the major achievements of the golden era of topology the Adams conjecture Bott periodicity the Hirzebruch Riemann Roch theorem the Atiyah Singer index theorem to name a few paints a clear picture of the canon of the subject. Grassmannians loop spaces and classical groups play a central role in mathematics and therefore in the presentation of this book as well. A judicious focus on the key ideas at an appropriate magnification of detail enables the reader to navigate the breadth of material confidently without the disorientation of algebraic minutiae. Many exercises are integrated throughout the text to build up the reader’s mastery of concepts and techniques. Numerous technical illustrations elucidate geometric constructions and the mechanics of spectral sequences and other sophisticated methods. Over fifty hauntingly captivating images by A. T. Fomenko artistically render the wondrous beauty and mystery of the subject Introduction Homotopy Homology Spectral Sequences of Fibrations Cohomology Operations The Adams Spectral Sequence K Theory and Other Extraordinary Cohomology Theories