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This rapid and concise presentation of the essential ideas and results of algebraic topology follows the axiomatic foundations pioneered by Eilenberg and Steenrod. The approach of the book is pragmatic: while most proofs are given, those that are particularly long or technical are omitted, and results are stated in a form that emphasizes practical use over maximal generality. Moreover, to better reveal the logical structure of the subject, the separate roles of algebra and topology are illuminated.

Assuming a background in point-set topology, Fundamentals of Algebraic Topology covers the canon of a first-year graduate course in algebraic topology: the fundamental group and covering spaces, homology and cohomology, CW complexes and manifolds, and a short introduction to homotopy theory. Readers wishing to deepen their knowledge of algebraic topology beyond the fundamentals are guided by a short but carefully annotated bibliography.

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Preface.-1. The Basics.- 2. The Fundamental Group.- 3. Generalized Homology Theory.- 4. Ordinary Homology Theory.- 5. Singular Homology Theory.- 6. Manifolds.- 7. Homotopy Theory.- 8. Homotopy Theory.- A. Elementary Homological Algebra.- B. Bilinear Forms.- C. Categories and Functors.- Bibliography.- Index.

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"Weinberg front-loads the Eilenberg-Steenrod axioms, thus rendering singular homology, where most authors start, merely an important example that emerges well on in the text. Certain readers either primarily interested in making calculations or in ´extraordinary´ theories such as K-theories or (co)bordism will appreciate this emphasis. Summing Up: Recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 52 (10), June, 2015)

"The fundamental group, covering spaces, a heavy dose of homology theory, applications to manifolds, and the higher homotopy groups is what the book is all about. ... The exposition is exquisite, making reading the book very enjoyable. The book certainly has its place among the existing literature, as it offers something different from its peers. ... All in all, what the book does it does very well, and it achieves a lot. ... Certainly a recommended read." (Ittay Weiss, MAA Reviews, March, 2015)
"This new booklet by the renowned textbook author Steven H. Weintraub is to serve as a quick guide to the fundamental concepts and results of classical algebraic topology. ... the present book is certainly a highly useful and valuable companion for a first-year graduate course in algebraic topology, as well for ambitious students as for instructors." (Werner Kleinert, zbMATH, Vol. 1305, 2015)

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Steven H. Weintraub is Professor of Mathematics at Lehigh University. He is the author of Galois Theory and Algebra: An Approach via Module Theory (with W. A. Adkins).

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