You are here

Algebraic Topology

Allen Hatcher
Cambridge University Press
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
John Martino
, on
The field of algebraic topology is concerned with the use of algebraic techniques on topological spaces. For this reason, students are not usually exposed to the subject until late in their academic careers, and they may find the algebraic and topological demands of the course exceed their introductory courses. 
The text Algebraic Topology by Allen Hatcher is designed to meet these students’ needs. The text is divided into four chapters: the fundamental group, homology, cohomology and the higher homotopy groups. The basic text includes discussions of covering spaces, Poincaré duality and the Hurewicz theorem. Each chapter ends with a section called Additional Topics, which allows the instructor to build on the core materials. The additional topics make up a large part of the text, which means the core book is not as large as the tome initially appears to be when you first pick it up. The additional topics are independent of the core material and may be read separately by the interested student. 
A distinguishing feature of Hatcher’s text is its emphasis on the underlying geometry of the algebraic functors that make up the core of algebraic topology. The book contains numerous illustrations to aid in the geometric understanding of the constructions and arguments. 
The book is designed to be read by the students, not merely used as a reference. The text takes a narrative approach to the material with definitions buried in the bodies of the paragraphs. The author has taken great care in ensuring the accessibility of the arguments to students who have only been exposed to undergraduate algebra and topology. 
Another feature of the text is its index. The reader is directed to not only the initial reference to an idea but also to secondary references allowing the reader to find the main theorems involving that idea. Furthermore, the text has many instructive exercises for the students.
The emphasis on geometry and readability makes the text ideal for an introductory course in algebraic topology, immediately following introductory algebra and topology. 


John Martino is Professor of Mathematics at Western Michigan University.