Title: Topological Equivalences of $E_\infty$ DGAs
Speaker: Özgür Bayindir (University of Illinois at Chicago)
Abstract: In this talk, I present an idea for studying $E_\infty$ differential graded algebras ($E_\infty$ DGAs) using stable homotopy theory. Namely, I discuss new equivalences between $E_\infty$ DGAS that are defined using commutative ring spectra. We say $E_\infty$ DGAs are $E_\infty$ topologically equivalent when the corresponding commutative ring spectra are equivalent. Quasi-isomorphic $E_\infty$ DGAs are $E_\infty$ topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are $E_\infty$ DGAs that are $E_\infty$ topologically equivalent but not quasi-isomorphic. This says that between $E_\infty$ DGAs, we have more equivalences than just the quasi-isomorphisms. I also discuss interaction of $E_\infty$ topological equivalences with the Dyer-Lashof operations and cases where $E_\infty$ topological equivalences and quasi-isomorphisms agree.
Seminar URL: https://people.math.osu.edu/valenzuelavasquez.2/hts/