##
**General topology and homotopy theory.**
*(English)*
Zbl 0562.54001

New York etc.: Springer-Verlag. VII, 248 p. DM 108.00; $ 42.40 (1984).

From the author’s preface: ”In this book I have tried to take a fresh look at some of (...) basic material and to organize it in a coherent fashion.” The book consists of the Introduction and eight chapters: 1. The Basic Framework, 2. The Axioms of Topology, 3. Spaces Under and Spaces Over, 4. Topological Transformation Groups, 5. The Notion of Homotopy, 6. Cofibrations and Fibrations, 7. Numerable Coverings, 8. Extensors and Neighbourhood Extensors.

In Chapter 1 the author gives a category-theoretical background. She defines and discusses the notions of category, functor, Cartesian and co- Cartesian square, pull-back and push-out, product and coproduct, adjoint functor, binary and cobinary system, and action. Examples usually concern the category of sets. The language of category theory introduced in Chapter 1 is consequently used throughout the subsequent chapters. Chapter 2 covers a part of general topology. The author says (p.31): ”... by selecting only a relevant material and omitting everything else I believe it is possible to give a clearer picture of this part of the subject, at least from the point of view I have adopted.” This ”relevant material” is much broader than is suggested by the title. It contains topological spaces, continuous functions, open maps and closed maps, initial and final topologies determined by functions, connected spaces, topological sum and product, endofunctors, compact spaces and maps, separation axioms (only Hausdorff spaces and regular spaces), and compact-open topology. Chapter 3 is characterized by the author as follows (comp. Introduction): ”The third chapter is concerned with the various ’comma’ categories associated with the basic category of topology, notably the category of spaces over a base, which is of central importance (...). This is relevant to fibre bundle theory....” This chapter splits into two parts. The first part (concerning spaces under a given space) deals with pointed spaces, wedge sum and smash product, and compactification (understood as the one-point compactification). The second part (concerning spaces over a given space) deals with fibre product, fibre join, general topology over a base, fibrewise compact-open topology, fibrewise compactification, and fibre-smash product. As the author writes in the Introduction, Chapter 4 is motivated by ”the increase of interest in equivariant topology” in the past decade. It is devoted to topological groups, in particular topological transformation groups, compact transformation groups, discrete transformation groups, covering spaces. The remaining four chapters follow on directly from chapters 1-4 and are devoted to various aspects of homotopy theory. Chapter 5 concerns homotopy properties of mapping spaces, homotopy under a given space, fundamental groupoid, group-like spaces, homotopy over a given space, fibrewise pointed homotopy, fibrewise group-like spaces, equivariant homotopy, mapping cylinder, mapping path-space. Chapter 6 is devoted to cofibrations, Strøm structures, homotopy cofibres, fibrations, homotopy fibres. In Chapter 7 the author introduces two more separation axioms: she defines normal and paracompact spaces. Next she discusses partition of unity, nilpotency, numerable bundles, fibre homotopy equivalences. Finally, Chapter 8 is devoted to extensors and neighbourhood extensors and their fibrewise and equivariant counterparts.

The book is rather self-contained, using only basic notions of point-set topology and group theory. It can be interesting and useful as well for specialists as for non-specialists, and can be recommended for graduate students.

Let me however make some critical remarks:

1. Though the only reference to Chapter 1 is MacLane, Categories for the Working Mathematician, there are essential differences in terminology between the book under review and MacLane’s book. The author uses the term ”equivalence” instead of ”isomorphism” (Def. (1.3) p. 4 and Def. (1.14) p. 12), while Maclane uses ”equivalence” in an essentially more general sense (comp. Maclane p. 18).

2. Pull-back is first defined for a cotriad as a certain morphism (p. 14) and then for a triad as a certain cotriad (p. 15). There is no link between these two definitions; in fact the second one is proper. Similarly for push-outs. Further, in applications to topology (Chapter 2), push-outs are used to define some functors; the expression ”and similarly for maps” is unclear as well on page 47 line 9 from the bottom as on page 53 line 1 from the bottom. On page 14 line 13 from the bottom the formula \(\xi f=\eta g\) should be replaced by \(f\xi =g\eta\).

In Chapter 1 the author gives a category-theoretical background. She defines and discusses the notions of category, functor, Cartesian and co- Cartesian square, pull-back and push-out, product and coproduct, adjoint functor, binary and cobinary system, and action. Examples usually concern the category of sets. The language of category theory introduced in Chapter 1 is consequently used throughout the subsequent chapters. Chapter 2 covers a part of general topology. The author says (p.31): ”... by selecting only a relevant material and omitting everything else I believe it is possible to give a clearer picture of this part of the subject, at least from the point of view I have adopted.” This ”relevant material” is much broader than is suggested by the title. It contains topological spaces, continuous functions, open maps and closed maps, initial and final topologies determined by functions, connected spaces, topological sum and product, endofunctors, compact spaces and maps, separation axioms (only Hausdorff spaces and regular spaces), and compact-open topology. Chapter 3 is characterized by the author as follows (comp. Introduction): ”The third chapter is concerned with the various ’comma’ categories associated with the basic category of topology, notably the category of spaces over a base, which is of central importance (...). This is relevant to fibre bundle theory....” This chapter splits into two parts. The first part (concerning spaces under a given space) deals with pointed spaces, wedge sum and smash product, and compactification (understood as the one-point compactification). The second part (concerning spaces over a given space) deals with fibre product, fibre join, general topology over a base, fibrewise compact-open topology, fibrewise compactification, and fibre-smash product. As the author writes in the Introduction, Chapter 4 is motivated by ”the increase of interest in equivariant topology” in the past decade. It is devoted to topological groups, in particular topological transformation groups, compact transformation groups, discrete transformation groups, covering spaces. The remaining four chapters follow on directly from chapters 1-4 and are devoted to various aspects of homotopy theory. Chapter 5 concerns homotopy properties of mapping spaces, homotopy under a given space, fundamental groupoid, group-like spaces, homotopy over a given space, fibrewise pointed homotopy, fibrewise group-like spaces, equivariant homotopy, mapping cylinder, mapping path-space. Chapter 6 is devoted to cofibrations, Strøm structures, homotopy cofibres, fibrations, homotopy fibres. In Chapter 7 the author introduces two more separation axioms: she defines normal and paracompact spaces. Next she discusses partition of unity, nilpotency, numerable bundles, fibre homotopy equivalences. Finally, Chapter 8 is devoted to extensors and neighbourhood extensors and their fibrewise and equivariant counterparts.

The book is rather self-contained, using only basic notions of point-set topology and group theory. It can be interesting and useful as well for specialists as for non-specialists, and can be recommended for graduate students.

Let me however make some critical remarks:

1. Though the only reference to Chapter 1 is MacLane, Categories for the Working Mathematician, there are essential differences in terminology between the book under review and MacLane’s book. The author uses the term ”equivalence” instead of ”isomorphism” (Def. (1.3) p. 4 and Def. (1.14) p. 12), while Maclane uses ”equivalence” in an essentially more general sense (comp. Maclane p. 18).

2. Pull-back is first defined for a cotriad as a certain morphism (p. 14) and then for a triad as a certain cotriad (p. 15). There is no link between these two definitions; in fact the second one is proper. Similarly for push-outs. Further, in applications to topology (Chapter 2), push-outs are used to define some functors; the expression ”and similarly for maps” is unclear as well on page 47 line 9 from the bottom as on page 53 line 1 from the bottom. On page 14 line 13 from the bottom the formula \(\xi f=\eta g\) should be replaced by \(f\xi =g\eta\).

Reviewer: M.Moszyńska

### MSC:

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

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

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

54H15 | Transformation groups and semigroups (topological aspects) |

55Pxx | Homotopy theory |

54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |