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Building on rudimentary knowledge of real analysis, point-set topology, and basic algebra, Basic Algebraic Topology provides plenty of material for a two-semester course in algebraic topology.
The book first introduces the necessary fundamental concepts, such as relative homotopy, fibrations and cofibrations, category theory, cell complexes, and simplicial complexes. It then focuses on the fundamental group, covering spaces and elementary aspects of homology theory. It presents the central objects of study in topology visualization: manifolds. After developing the homology theory with coefficients, homology of the products, and cohomology algebra, the book returns to the study of manifolds, discussing Poincaré duality and the De Rham theorem. A brief introduction to cohomology of sheaves and Čech cohomology follows. The core of the text covers higher homotopy groups, Hurewicz’s isomorphism theorem, obstruction theory, Eilenberg-Mac Lane spaces, and Moore-Postnikov decomposition. The author then relates the homology of the total space of a fibration to that of the base and the fiber, with applications to characteristic classes and vector bundles. The book concludes with the basic theory of spectral sequences and several applications, including Serre’s seminal work on higher homotopy groups.
Thoroughly classroom-tested, this self-contained text takes students all the way to becoming algebraic topologists. Historical remarks throughout the text make the subject more meaningful to students. Also suitable for researchers, the book provides references for further reading, presents full proofs of all results, and includes numerous exercises of varying levels.
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Dr. Anant R. Shastri is a professor in the Department of Mathematics at the Indian Institute of Technology Bombay, where he has been teaching for over 20 years. His research focuses on the topology of matrix varieties.Review:
"... a good graduate text: the book is well written and there are many well-chosen examples and a decent number of exercises. It meets its ambitious goals and should succeed in leading a lot of solid graduate students, as well as working mathematicians from other specialties seeking to learn this subject, deeper and deeper into its workings and subtleties."
―Michael Berg, MAA Reviews, February 2014
"Similar to his other well-written textbook on differential topology, Professor Shastri’s book gives a detailed introduction to the vast subject of algebraic topology together with an abundance of carefully chosen exercises at the end of each chapter. The content of Professor Shastri’s book furnishes the necessary background to access many major achievements ... [and] to explore current research works as well as possible applications to other branches of mathematics of modern algebraic topology."
―From the Foreword by Professor Peter Wong, Bates College, Lewiston, Maine, USA
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