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**Algebraic topology.**
*(English)*
Zbl 1156.55001

EMS Textbooks in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-048-7/hbk). xi, 567 p. (2008).

This book, part of the European Mathematical Society series of Text books in Mathematics is aimed at both students and professional mathematicians seeking an introduction into a particular field. In this case Algebraic Topology. The author chooses to treat homotopy theory before homology theory, homology is introduced in parallel with homological algebra, manifolds are introduced after homology theory but before cohomology. From there the author proceeds to duality, and via the Euler class to characteristic classes. The final chapters cover the relation between homology and homotopy and bordism theory. Clearly this is a very comprehensive introduction to algebraic topology. There are extensive exercises throughout the book. The reader who absorbs all this material will be well placed for further study in the field, although there will still be a large distance to the frontiers of the field if one is interested in undertaking research.

The general approach is algebraic, rather than geometric. A typical example of this is the use of the groupoid approach in the first computation of the fundamental group of the circle. The reader who needs visual aids to “see” what is going on will need to draw her own pictures. The few pictures that do appear are in the early pages of the book. There were occasions when I had to convince myself that a very algebraic proof did correspond to my geometrical intuition. At the risk of making an already long book even longer it could have been helpful if the author had provided more insight into the way that proofs work. This is not a book for the faint hearted. It is only a bit of an exaggeration to say that the first chapter on topological spaces is a one trimester course. On the other hand to get to the point where one can prove interesting results in algebraic topology, such as the Hurewicz theorem does require a considerable amount of machinery. The result is a valuable book, crammed with material that the algebraic topologist needs.

The general approach is algebraic, rather than geometric. A typical example of this is the use of the groupoid approach in the first computation of the fundamental group of the circle. The reader who needs visual aids to “see” what is going on will need to draw her own pictures. The few pictures that do appear are in the early pages of the book. There were occasions when I had to convince myself that a very algebraic proof did correspond to my geometrical intuition. At the risk of making an already long book even longer it could have been helpful if the author had provided more insight into the way that proofs work. This is not a book for the faint hearted. It is only a bit of an exaggeration to say that the first chapter on topological spaces is a one trimester course. On the other hand to get to the point where one can prove interesting results in algebraic topology, such as the Hurewicz theorem does require a considerable amount of machinery. The result is a valuable book, crammed with material that the algebraic topologist needs.

Reviewer: Jonathan Hodgson (Philadelphia)

### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |