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**Topology for physicists. Transl. from the Russian by Silvio Levy.**
*(English)*
Zbl 0858.55001

Grundlehren der Mathematischen Wissenschaften. 308. Berlin: Springer. xi, 296 p. DM 178.00; öS 1.388.40; sFr 178.00/hbk (1994).

This book is based on the author’s Russian original ‘Quantum field theory and topology’ (1989; Zbl 0682.58001) and its English translation (1993; Zbl 0789.58004), but it provides generally the topological background of topics of interests of physicists, not only of topics in quantum field theory; it is therefore completely reorganized.

Overview of the contents: Chapter 0 covers the background necessary for the understanding of the book. Chapter 1 defines and illustrates some fundamental topological notions – in particular, homotopies and homotopy equivalences. Chapters 2 and 3 are devoted to the degree of a map, the fundamental group, and covering spaces. Chapter 4 gives the basic definitions of the theory of smooth manifolds. Next come differential forms and homology theory; Chapter 5 deals with the case of open subsets of \(\mathbb{R}^n\), while Chapter 6 discusses the general case. Throughout Chapter 5 and the beginning of Chapter 6, little rigor is used in introducing homology theory; the details come in Sections 6.5 and 6.6. Chapter 7 studies homotopy groups for simply connected spaces, and Chapter 8 the same groups for arbitrary spaces. Chapter 9 gives the main definitions of the theory of fiber spaces, and Chapter 10 indicates ways of computing homotopy groups using fiber spaces, by listing several relations between the homotopy groups of the base, the fiber and the total space. The homotopy groups of certain important examples are also given in Chapter 10. Chapters 12 and 13 contain a brief summary of the theory of Lie groups and Lie algebras from our point of view. Chapter 14 studies the homotopy and homology groups of Lie algebras and homogeneous spaces (on the whole one limits oneself to homology and homotopy in dimensions up to three, since these are the dimensions that occur most often in physics). Chapter 15 is devoted to the geometry and topology of gauge fields.

Finally, there is a set of problems of varying complexity, from simple exercises to important and subtle results not contained in the main text.

Overview of the contents: Chapter 0 covers the background necessary for the understanding of the book. Chapter 1 defines and illustrates some fundamental topological notions – in particular, homotopies and homotopy equivalences. Chapters 2 and 3 are devoted to the degree of a map, the fundamental group, and covering spaces. Chapter 4 gives the basic definitions of the theory of smooth manifolds. Next come differential forms and homology theory; Chapter 5 deals with the case of open subsets of \(\mathbb{R}^n\), while Chapter 6 discusses the general case. Throughout Chapter 5 and the beginning of Chapter 6, little rigor is used in introducing homology theory; the details come in Sections 6.5 and 6.6. Chapter 7 studies homotopy groups for simply connected spaces, and Chapter 8 the same groups for arbitrary spaces. Chapter 9 gives the main definitions of the theory of fiber spaces, and Chapter 10 indicates ways of computing homotopy groups using fiber spaces, by listing several relations between the homotopy groups of the base, the fiber and the total space. The homotopy groups of certain important examples are also given in Chapter 10. Chapters 12 and 13 contain a brief summary of the theory of Lie groups and Lie algebras from our point of view. Chapter 14 studies the homotopy and homology groups of Lie algebras and homogeneous spaces (on the whole one limits oneself to homology and homotopy in dimensions up to three, since these are the dimensions that occur most often in physics). Chapter 15 is devoted to the geometry and topology of gauge fields.

Finally, there is a set of problems of varying complexity, from simple exercises to important and subtle results not contained in the main text.

### MSC:

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

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

00A79 | Physics |

55M25 | Degree, winding number |

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

55N10 | Singular homology and cohomology theory |