Infinite loop spaces from operads with homological stability

TL;DR

This paper shows that operads satisfying homological stability conditions produce infinite loop spaces, linking moduli space operads to stable homotopy theory and K-theory via recent stability results.

Contribution

It establishes that operads with homological stability are infinite loop space operads, extending the connection between moduli spaces and stable homotopy theory.

Findings

Operads with homological stability produce infinite loop spaces.

Recent stability results enable new examples of such operads.

The map to K-theory from diffeomorphism actions is an infinite loop map.

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 the map to $K$-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.