# Geometry and Topology Seminar, 18 OCT, Ozgur Bayindir. Title “Topological Equivalences of E-infinity DGAs”

Department of Mathematics                                                           University of  Haifa

Geometry & Topology Seminar

Speaker:  Ozgur Bayindir (University of Haifa)

Topic:  Topological Equivalences of E_\infty DGAs

Place:    room 614 in the Science & Education Building

Time   12:10

Date:  Thursday, October 18, 2018

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 $E_\infty$-algebra structure which comes from the cup product.

In this talk I present an idea for studying such chain complexes, $E_\infty$ 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.

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