##
**A basic course in algebraic topology.**
*(English)*
Zbl 0725.55001

Graduate Texts in Mathematics, 127. New York etc.: Springer-Verlag. xiv, 428 p. DM 108.00 (1991).

The text of this book contains material from the author’s former books [Algebraic topology: An introduction (1981; Zbl 0457.55001); Singular homology theory (1980; Zbl 0442.55001)]. As the author writes, it is intended as a textbook for a course in algebraic topology at the beginning graduate level.

Roughly, the book has two parts. The first part contains the classification theorem for compact surfaces, the fundamental group of the circle, Seifert-van Kampen’s theorem, and the theory of covering spaces.

In the second part, homology theory is considered. Based on cubical rather than on simplicial chains, singular homology is defined and its properties are verified. The cellular chain complex of a CW-complex is introduced. Also the necessary homological algebra for the universal coefficient theorem and the Künneth-formula is developed.

Of course, cohomology, various products and the duality theorems for manifolds are also considered. As an application, the cohomology ring of the projective spaces is calculated.

An appendix contains the proof of de Rham’s theorem.

The book is well-written and can be recommended to the student, since the author avoids confusing his reader by too much generality. There are a lot of examples and exercises in the book, and also some historical remarks are not forgotten.

Roughly, the book has two parts. The first part contains the classification theorem for compact surfaces, the fundamental group of the circle, Seifert-van Kampen’s theorem, and the theory of covering spaces.

In the second part, homology theory is considered. Based on cubical rather than on simplicial chains, singular homology is defined and its properties are verified. The cellular chain complex of a CW-complex is introduced. Also the necessary homological algebra for the universal coefficient theorem and the Künneth-formula is developed.

Of course, cohomology, various products and the duality theorems for manifolds are also considered. As an application, the cohomology ring of the projective spaces is calculated.

An appendix contains the proof of de Rham’s theorem.

The book is well-written and can be recommended to the student, since the author avoids confusing his reader by too much generality. There are a lot of examples and exercises in the book, and also some historical remarks are not forgotten.

Reviewer: H.M.Unsöld (Berlin)

### MSC:

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

57M05 | Fundamental group, presentations, free differential calculus |

57M10 | Covering spaces and low-dimensional topology |

55N10 | Singular homology and cohomology theory |