Topology. (Russian) Zbl 0668.55001

Novikov, S. P.

Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 12, 5-252 (1986).

This 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.

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

Keywords:

simplicial complexes; Gauß-Bonnet formulas; linkig coefficient; fibrations; cellular complexes; homology; cohomology; obstruction theory; algebraic topology of smooth manifolds; characteristic classes; bordism; cobordism