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Book DetailsGraduate Studies in MathematicsVolume: 35; 2001; 367 ppMSC: Primary 55; 57;
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs linebyline to understanding the overall structure of proofs of difficult theorems.
To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.
Prerequisites for using this book include basic settheoretic topology, the definition of CWcomplexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.
The topics discussed by the authors include typical material for first and secondyear graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic \(K\)theory and the scobordism theorem.
A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars.
The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.ReadershipGraduate students and research mathematicians interested in geometric topology and homotopy theory.

Table of Contents

Chapters

Chapter 1. Chain complexes, homology, and cohomology

Chapter 2. Homological algebra

Chapter 3. Products

Chapter 4. Fiber bundles

Chapter 5. Homology with local coefficients

Chapter 6. Fibrations, cofibrations and homotopy groups

Chapter 7. Obstruction theory and EilenbergMacLane spaces

Chapter 8. Bordism, spectra, and generalized homology

Chapter 9. Spectral sequences

Chapter 10. Further applications of spectral sequences

Chapter 11. Simplehomotopy theory


Additional Material

Reviews

Many exercises and comments in the book, which complement the material, as well as suggestions for further study, presented in the form of projects … The book is a nice advanced textbook on algebraic topology and can be recommended to anybody interested in modern and advanced algebraic topology.
European Mathematical Society Newsletter 
The book might well have been titled ‘What Every Young Topologist Should Know’ … presents, in a selfcontained and clear manner, all classical constituents of algebraic topology … recommend this book as a valuable tool for everybody teaching graduate courses as well as a selfcontained introduction … for independent reading.
Mathematica Bohemica


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The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs linebyline to understanding the overall structure of proofs of difficult theorems.
To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.
Prerequisites for using this book include basic settheoretic topology, the definition of CWcomplexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.
The topics discussed by the authors include typical material for first and secondyear graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic \(K\)theory and the scobordism theorem.
A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars.
The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.
Graduate students and research mathematicians interested in geometric topology and homotopy theory.

Chapters

Chapter 1. Chain complexes, homology, and cohomology

Chapter 2. Homological algebra

Chapter 3. Products

Chapter 4. Fiber bundles

Chapter 5. Homology with local coefficients

Chapter 6. Fibrations, cofibrations and homotopy groups

Chapter 7. Obstruction theory and EilenbergMacLane spaces

Chapter 8. Bordism, spectra, and generalized homology

Chapter 9. Spectral sequences

Chapter 10. Further applications of spectral sequences

Chapter 11. Simplehomotopy theory

Many exercises and comments in the book, which complement the material, as well as suggestions for further study, presented in the form of projects … The book is a nice advanced textbook on algebraic topology and can be recommended to anybody interested in modern and advanced algebraic topology.
European Mathematical Society Newsletter 
The book might well have been titled ‘What Every Young Topologist Should Know’ … presents, in a selfcontained and clear manner, all classical constituents of algebraic topology … recommend this book as a valuable tool for everybody teaching graduate courses as well as a selfcontained introduction … for independent reading.
Mathematica Bohemica