In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined topological operads with homological stability (OHS) and proved that the group completion of an algebra over an OHS is weakly equivalent to an infinite loop space. The most interesting examples of OHSs come from moduli spaces of manifolds. These operads can be used to put a new infinite loop space structure on certain stable moduli spaces of high-dimensional manifolds, which were already known to be infinite loop spaces by the work of Galatius and Randal-Williams.
In this talk, I shall outline a construction which to an algebra \(A\) over an OHS associates a new infinite loop space. Under mild conditions on the operad, this space is equivalent as an infinite loop space to the group completion of \(A\). This generalises a result of Wahl on the equivalence of the two infinite loop space structures constructed by Tillmann on the classifying space of the stable mapping class group, and could possibly lead to a proof that the infinite loop space structures on the stable moduli spaces of high-dimensional manifolds arising from the OHSs are in fact equivalent to those given by Galatius and Randal-Williams.