Homotopical Topology

Graduate Texts in Mathematics #273

by Anatoly Fomenko, Dmitry Fuchs

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, loops 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.

Genres: Mathematics

638 pages, Hardcover

First published January 2, 2016

Reviews

[Andrew · Rating 5 out of 5]

My favorite math textbook...

Very clear and thorough approach to algebraic topology, with a sharp aesthetic sensibility (not just in the art pages scattered about, but with the presentation and choice of the material). Somehow, this book covers what I had once thought to be difficult and opaque subjects in a lucid and natural manner. Others may have different experiences, but reading the book was like having math explained to me: it always told me what I needed to hear.

In particular, the sections on obstruction theory, spectral sequences, and Steenrod squares are particularly well-explained.

Edit a few months later:
I think I simply find the homotopical way of thinking to be very appealing. Others probably have their own experiences.