Lie algebras over operads and their applications in homotopy theory

V. A. Smirnov

Izvestiya. Mathematics , Tome 62 (1998) no. 3, pp. 549-580

Résumé

This paper is devoted to the development of algebraic devices that are necessary for describing the homotopy groups of topological spaces. As shown in the note [1], the notion of a Lie algebra over the operad $E_\infty$ must be used for this purpose. Here we consider this notion in more detail, prove its most important properties and clarify the question about the algebraic structure on the homotopy groups of topological spaces. In particular, we show that the homotopy groups of a topological space possess the structure of a Lie $E_\infty$-algebra which determines the homotopy type of the space in the simply-connected case. Since the notion of an operad is analogous to that of algebra, we begin with recalling the notions of algebra and coalgebra and reviewing the main constructions over them. Then we transfer these constructions to the operad case and use them to investigate the structure of Lie algebras and coalgebras over an operad.