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**Topology: A geometric account of general topology, homotopy types and the fundamental groupoid. Rev., updated and expanded ed.**
*(English)*
Zbl 0655.55001

Ellis Horwood Series in Mathematics and its Applications. Chichester (UK): Ellis Horwood Ltd.; New York etc.: Halsted Press. xvii, 460 p. £38.50 (1988).

The title, along with the table of contents give an accurate impression of this work. An abbreviated version of the table of contents is given below:

Chapter 1: Some topology on the real line (neighborhoods, continuity, open sets, generalizations) 14 pages.

Chapter 2: Topological spaces (axioms, products, continuity, homeomorphisms, metric spaces, normed vector spaces, Hausdorff spaces) 39 pages.

Chapter 3: Connected spaces, compact spaces (sums, connected, locally connected, path-connected, compactness, Lebesgue covering lemma, compactifications) 27 pages.

Chapter 4: Identification spaces and cell complexes (identification topology and maps, cells, spheres, adjunction spces, cell complexes) 33 pages.

Chapter 5: Projective and other spaces (quaternions, projective spaces, isometries, simplicial complexes, bases, initial topologies, joins, smash product, c-o topology) 39 pages.

Chapter 6: The fundamental groupoid (categories, construction of the fundamental groupoid, properties of groupoids, functions and morphisms, homotopies, coproducts and push outs, fundamental groupoid of a union) 37 pages.

Chapter 7: Cofibrations (track groupoid, fibrations of groupoids, homotopy equivalences of unions, homotopy type of adjunction spaces, cellular approximation theorem) 48 pages.

Chapter 8: Computation of the fundamental groupoid (universal morphisms, free groupoids, quotient groupoids, Van Kampen theorem, Jordan curve theorem) 34 pages.

Chapter 9: Covering spaces (covering maps, homotopies, and groupoids, lifting sums and morphisms, lifted topologies, equivalence of categories, induced coverings and pullbacks, applications to group theory, orbit spaces and groupoids, quotients, semi-direct products) 54 pages.

Appendix: Functions and cardinality (functions, sets, products, axiom of choice, universal construction) 13 pages.

Glossary of terms from set theory. References. Glossary of symbols. Index.

Preserving the best and most important from the past, and pressing forward with new material or a fresh viewpoint are two goals the author seems to be striving for in this graduate text. The body of the book divides naturally into three pieces. The first five chapters constitute part of a text on topology with classic content. The next three chapters contain an exposition of those aspects of category theory needed to construct, apply, and extend the notion of the fundamental groupoid. These three chapters provide a source of material difficult to find elsewhere. They contain many changes from the first edition, published as “Elements of modern topology” by McGraw-Hill in 1968 (see Zbl 0159.522). The final chapter is a nice fusion of old and newer ideas, and indicates application of this union to various areas, including for example the Kurosch and Grushko theorems from group theory.

A particular strong point of the book is the inclusion of interesting and informative notes following each chapter. These will be of particular interest to topologists but are readable by all. The ratio of examples to theorems and corollaries falls off in Chapters 6 and 7, rising again in 8 and 9. Perhaps as a consequence the reviewer found chapters 6 and 7 a bit heavy going, but others may not find them so.

Major omissions from standard course material include the following: homology and cohomology, proofs of the Tychonoff and Tieze extension theorems, paracompactness, dimension, inverse and direct limits. The emphasis is geometric, so that absence of these topics is perhaps balanced by the inclusion of such things as projective spaces, adjunction spaces, quaternionic space, joins, smash products, and a rather extensive exposition of homotopy type including a proof of the cellular approximation theorem.

Chapter 1: Some topology on the real line (neighborhoods, continuity, open sets, generalizations) 14 pages.

Chapter 2: Topological spaces (axioms, products, continuity, homeomorphisms, metric spaces, normed vector spaces, Hausdorff spaces) 39 pages.

Chapter 3: Connected spaces, compact spaces (sums, connected, locally connected, path-connected, compactness, Lebesgue covering lemma, compactifications) 27 pages.

Chapter 4: Identification spaces and cell complexes (identification topology and maps, cells, spheres, adjunction spces, cell complexes) 33 pages.

Chapter 5: Projective and other spaces (quaternions, projective spaces, isometries, simplicial complexes, bases, initial topologies, joins, smash product, c-o topology) 39 pages.

Chapter 6: The fundamental groupoid (categories, construction of the fundamental groupoid, properties of groupoids, functions and morphisms, homotopies, coproducts and push outs, fundamental groupoid of a union) 37 pages.

Chapter 7: Cofibrations (track groupoid, fibrations of groupoids, homotopy equivalences of unions, homotopy type of adjunction spaces, cellular approximation theorem) 48 pages.

Chapter 8: Computation of the fundamental groupoid (universal morphisms, free groupoids, quotient groupoids, Van Kampen theorem, Jordan curve theorem) 34 pages.

Chapter 9: Covering spaces (covering maps, homotopies, and groupoids, lifting sums and morphisms, lifted topologies, equivalence of categories, induced coverings and pullbacks, applications to group theory, orbit spaces and groupoids, quotients, semi-direct products) 54 pages.

Appendix: Functions and cardinality (functions, sets, products, axiom of choice, universal construction) 13 pages.

Glossary of terms from set theory. References. Glossary of symbols. Index.

Preserving the best and most important from the past, and pressing forward with new material or a fresh viewpoint are two goals the author seems to be striving for in this graduate text. The body of the book divides naturally into three pieces. The first five chapters constitute part of a text on topology with classic content. The next three chapters contain an exposition of those aspects of category theory needed to construct, apply, and extend the notion of the fundamental groupoid. These three chapters provide a source of material difficult to find elsewhere. They contain many changes from the first edition, published as “Elements of modern topology” by McGraw-Hill in 1968 (see Zbl 0159.522). The final chapter is a nice fusion of old and newer ideas, and indicates application of this union to various areas, including for example the Kurosch and Grushko theorems from group theory.

A particular strong point of the book is the inclusion of interesting and informative notes following each chapter. These will be of particular interest to topologists but are readable by all. The ratio of examples to theorems and corollaries falls off in Chapters 6 and 7, rising again in 8 and 9. Perhaps as a consequence the reviewer found chapters 6 and 7 a bit heavy going, but others may not find them so.

Major omissions from standard course material include the following: homology and cohomology, proofs of the Tychonoff and Tieze extension theorems, paracompactness, dimension, inverse and direct limits. The emphasis is geometric, so that absence of these topics is perhaps balanced by the inclusion of such things as projective spaces, adjunction spaces, quaternionic space, joins, smash products, and a rather extensive exposition of homotopy type including a proof of the cellular approximation theorem.

Reviewer: L.Neuwirth

### MSC:

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

54-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology |

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

18F99 | Categories in geometry and topology |

57M10 | Covering spaces and low-dimensional topology |

54B30 | Categorical methods in general topology |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |