Topological Equivalences of E∞ DGAs

Haldun Özgür Bayındır, The University of Haifa.
Date: 6th of February, 2019, Thursday. Time: 11:00 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In algebraic topology we often encounter chain complexes with extra multiplicative structure. For example, the cochain complex of a topological space has what is called the E-algebra structure which comes from the cup product. In this talk I present an idea for studying such chain complexes, E differential graded algebras (E DGAs), using stable homotopy theory. Namely, I discuss new equivalences between E DGAS that are defined using commutative ring spectra.We say E DGAs are E topologically equivalent when the corresponding commutative ring spectra are equivalent. Quasi-isomorphic E DGAs are E topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are E DGAs that are E topologically equivalent but not quasi-isomorphic. This says that between E DGAs, we have more equivalences than just the quasi-isomorphisms. I also discuss interaction of E topological equivalences with the Dyer-Lashof operations and cases where E topological equivalences and quasi-isomorphisms agree.