Higher genus surface operad detects infinite loop spaces
Spaces with various structures have been organized by homotopy theorists using the notion of an operad. It is the case that algebraic notions are not always homotopy theoretic notions, and so families of higher homotopies are needed in order for the algebraic structure to manifest itself up to homotopy. The algebraic structures that make an important appearance in conformal field theory are organized by an operad \({\mathcal M}\) whose \(n\)-th space consists of Riemann surfaces of any genus with boundary having one incoming and \(n\) outgoing components. The operad structure is based on the operation of glueing incoming components to outgoing ones. The relations between operads imply ways in which the existence on one structure on a space obtains another structure. For example, if we restrict to genus zero Riemann surfaces in \({\mathcal M}\), then we obtain a suboperad \({\mathcal RB}\) (ribbon braid group operad). \textit{E. Y. Miller} [J. Differ. Geom. 24, 1-14 (1986; Zbl 0618.57005)] showed that a space with an action of the operad \({\mathcal RB}\) is a double loop space. The main result of this paper is that a space \(X\) (or a group completion of \(X\) when \(\pi_0(X)\) is not a group) with an action of the operad \({\mathcal M}\) is an infinite loop space. With this structure, Dyer-Lashof operations exist on the homology of such a space with Adem relations and a Cartan formula. The proof proceeds by identifying the operad \({\mathcal M}\) in such a way that its product is strictly associative and comparing \({\mathcal M}\) with the operad \(\Gamma\) for infinite loop spaces introduced by \textit{M. G. Barratt} and \textit{P. J. Eccles} [Topology 13, 25-45 (1974; Zbl 0292.55010)].
scientific article; zbMATH DE number 1502101
scientific article