Infinite loop spaces from operads with homological stability

Abstract

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces.The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces. Furthermore, the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces.