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.