Simplicial and operad methods in algebraic topology (Q2707659)

\textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275-312 (1963; Zbl 0114.39402)] introduced the \(A_\infty\)-structure to describe loop spaces. The concept of an operad proposed by \textit{J. P. May} [The geometry of iterated loop spaces, Lect. Notes Math. 271 (1972; Zbl 0244.55009)] was used to show that on any \(n\)-fold loop space there is an action of the operad \(E_n\), and each arcwise connected \(E_n\)-space has the weak homotopy type of an \(n\)-fold loop space. Then \textit{T. V. Kadeishvili} [Russ. Math. Surv. 35, No. 3, 231-238 (1980); translation from Usp. Mat. Nauk 35, No. 3(213), 183-188 (1980; Zbl 0521.55015)] and the author [Russ. Math. Surv. 35, No. 3, 294-298 (1980); translation from Usp. Mat. Nauk 35, No. 3(213), 227-230 (1980; Zbl 0471.55010)] generalized the notion of operad to the case of the category of cochain and chain complexes. Thus many problems of homotopy theory can be reduced to a study of the \(E_\infty\)-structure on those complexes. NEWLINENEWLINENEWLINEFor solving some very serious problems of algebraic topology the language of operads arises naturally on topological spaces and their homology and homotopy as well. There are ten chapters in the book under review; everything depends on the first four chapters and the remaining material often reflects the original nature of the project. NEWLINENEWLINENEWLINEChapters I and II contain fundamental organizing constructions and theorems of operads in the category of topological spaces and simplicial sets. Chapters III and IV deal with necessary facts concerning chain complexes and \(A\)-\(\infty\)-structures which are further repositories of things used later. Chapter V offers a discussion of algebras and coalgebras over operads. Homology and cohomologyof \(E_\infty\)-operads are studied and elements of the Dyer-Lashof, the mod-\(p\) Steenrod algebra and its dual are obtained. NEWLINENEWLINENEWLINEHomology groups \(H_\ast(\Omega^n\Sigma^nX)\) of the \(n\)-fold loop space of the \(n\)-fold suspension of a space \(X\) were computed by \textit{J. P. May} [loc. cit.]. Homology groups \(H_\ast(\Omega^nX)\) of an iterated loop space are studied in Chapter VI. In particular, homology of iterated loop spaces of real and projective spaces are computed. Chapter VII outlines the necessary foundations of abstract homotopy theory introduced by D.G. Quillen. Curtis and Bousfield-Kan spectral sequences are studied and homotopy groups of symmetric products are computed. The main purpose of Chapter VIII are algebraic structures on generalized homology and cohomology of topological spaces, in particular in cobordism theories. The Dyer-Lashof algebra and the Steenrod algebra for a generalized homology and cohomology corresponding to an \(E_\infty\)-multiplicative spectrum is defined. Actions of these algebras for cobordisms of topological spaces are computed and the Steenrod operations are related to the Landweber-Novikov operations [\textit{P. S. Landweber}, Trans. Am. Math. Soc. 129, 94-110 (1967; Zbl 0169.54602); \textit{S. P. Novikov}, Sov. Math., Dokl. 8, 27-31 (1967); translation from Dokl. Akad. Nauk SSSR 172, 33-36 (1967; Zbl 0155.50902)]. Operad methods are applied in Chapter IX to describe cohomology of groups, Banach algebras and the \(E_2\)-term of the Adams spectral sequence for stable homotopy groups of topological spaces. Chapter X takes first into account functional and functorial homology operations. Stasheff's \(A_\infty\)-structures are used to reformulate the result by \textit{J. M. Cohen} [Ann. Math. (2) 87, 305-320 (1968; Zbl 0162.55102)] on the \(E_\infty\)-term of the Adams spectral sequence for the stable homotopy groups of spheres. Then indecomposable elements of the term \(E_\infty(S^\ast)\) of the Bousfield-Kan spectral sequence for the unstable homotopy groups of spheres are found. NEWLINENEWLINENEWLINEThe book should prove enlightening to a broad range of readers including prospective students and researchers who want to apply operad and simplicial techniques for whatever reason.