Abstract

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined operads with homological stability (OHS) and showed that algebras over an OHS group-complete to infinite loop spaces. This can in particular be used to put a new infinite loop space structure on stable moduli spaces of high-dimensional manifolds in the sense of Galatius and Randal-Williams, which are known to be infinite loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of the OHS condition and construct yet another infinite loop space structure on these stable moduli spaces. This structure turns out to be equivalent to that constructed by Basterra et al. It is believed hat the infinite loop space structure due to Galatius—Randal-Williams is also equivalent to these two structures.