By V. A. Smirnov
Lately, for fixing difficulties of algebraic topology and, specifically, tough difficulties of homotopy idea, algebraic constructions extra advanced than simply a topological monoid, an algebra, a coalgebra, etc., were used progressively more frequently. A handy language for describing quite a few buildings bobbing up clearly on topological areas and on their cohomology and homotopy teams is the language of operads and algebras over an operad. This language used to be proposed by way of J. P. might within the Seventies to explain the constructions on a variety of loop areas. This ebook offers a close research of the concept that of an operad within the different types of topological areas and of chain complexes. The notions of an algebra and a coalgebra over an operad are brought, and their homes are investigated. The algebraic constitution of the singular chain complicated of a topological area is defined, and it's proven how the matter of homotopy type of topological areas might be solved utilizing this constitution. For algebras and coalgebras over operads, general buildings are outlined, relatively the bar and cobar buildings. Operad equipment are utilized to computing the homology of iterated loop areas, investigating the algebraic constitution of generalized cohomology theories, describing cohomology of teams and algebras, computing differential within the Adams spectral series for the homotopy teams of the spheres, and a few different difficulties.