Topology.

*(Russian)*Zbl 0668.55001This work is concerned with some special problems of topology having numerous applications in mechanis and mathematical physics.

In the first chapter the author presents some elementary topological notions and properties such as: Euler’s theorem, the Gauß and Gauß-Bonnet formulas, the linkig coefficient, etc. The second chapter is devoted to a concise introduction in the study of topological spaces, fibrations and homotopy theory. Then the main problems of the work are presented. The author first introduces, in the third chapter, simplicial and cellular complexes, homology and cohomology, and the connection with homotopy theory and obstruction theory. In the fourth chapter the algebraic topology of smooth manifolds is studied. Among other things, this chapter deals with: homology and homotopy of smooth manifolds, a study of characteristic classes, bordism and cobordism, a study of the smooth manifolds of small dimension \((n=2\), 3), the classification theory of non-connected manifolds of dimension \(\geq 5\), and some elements of geometric topology.

The reviewer considers this work a very good survey in algebraic topology which contains a detailed study about some very important problems in this field of contemporary mathematics.

In the first chapter the author presents some elementary topological notions and properties such as: Euler’s theorem, the Gauß and Gauß-Bonnet formulas, the linkig coefficient, etc. The second chapter is devoted to a concise introduction in the study of topological spaces, fibrations and homotopy theory. Then the main problems of the work are presented. The author first introduces, in the third chapter, simplicial and cellular complexes, homology and cohomology, and the connection with homotopy theory and obstruction theory. In the fourth chapter the algebraic topology of smooth manifolds is studied. Among other things, this chapter deals with: homology and homotopy of smooth manifolds, a study of characteristic classes, bordism and cobordism, a study of the smooth manifolds of small dimension \((n=2\), 3), the classification theory of non-connected manifolds of dimension \(\geq 5\), and some elements of geometric topology.

The reviewer considers this work a very good survey in algebraic topology which contains a detailed study about some very important problems in this field of contemporary mathematics.

Reviewer: Ioan Pop (Iaşi)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R19 | Algebraic topology on manifolds and differential topology |

57R20 | Characteristic classes and numbers in differential topology |

57R75 | \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism |

55N10 | Singular homology and cohomology theory |

55R05 | Fiber spaces in algebraic topology |

55S35 | Obstruction theory in algebraic topology |