Anatoly Timofeevich Fomenko is Chair of Differential Geometry and Applications in the Department of Mathematics and Mechanics at Lomonosov Moscow State University. He is a full member of the Russian Academy of Sciences, and a member of the Moscow Mathematical Society. He is the author of several books, including Visual Geometry and Topology, Modeling for Visualization (with T.L. Kunii), and Modern Geometry: Methods and Applications (with B.A. Dubrovin and S.P. Novikov).

Dmitry Borisovich Fuchs is Professor Emeritus of Mathematics at the University of California, Davis. He earned his [...]. from Moscow State University, and his [...]. at Tblisi State University. His research interests include topology and the theory of foliations, homological algebra, and representation theory. His main body of work deals with representations and cohomology of infinite-dimensional Lie algebras. This work has consequences in string theory and conformal quantum field theory as codified in the mathematical theory of vertex operator algebras. He is the author of over 25 articles, and has served as thesis advisor to several well-known mathematicians, including Boris Feigin, Fedor Malikov, and Vladimir Rokhlin.

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.