Speaker: Tomáš Zeman (Stockholm University)
Title: On operads with homological stability
Abstract: 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.
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. I shall talk about applications of this construction to stable moduli spaces of high-dimensional manifolds in the sense of Galatius and Randal-Williams.