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Topological equivalences of E-infinity differential graded algebras

Haldun Özgür Bayındır

Algebraic & Geometric Topology 18 (2018) 1115–1146

Two DGAs are said to be topologically equivalent when the corresponding Eilenberg–Mac Lane ring spectra are weakly equivalent as ring spectra. Quasi-isomorphic DGAs are topologically equivalent, but the converse is not necessarily true. As a counterexample, Dugger and Shipley showed that there are DGAs that are nontrivially topologically equivalent, ie topologically equivalent but not quasi-isomorphic.

In this work, we define E topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially E topologically equivalent E DGAs. Also, we show using these obstruction theories that for coconnective EFp–DGAs, E topological equivalences and quasi-isomorphisms agree. For EFp–DGAs with trivial first homology, we show that an E topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an HFp–equivalence.

commutative ring spectra, homological algebra, E-infinity DGAs
Mathematical Subject Classification 2010
Primary: 18G55, 55P43, 55S12, 55S35, 55U99
Received: 22 May 2017
Revised: 18 October 2017
Accepted: 27 October 2017
Published: 12 March 2018
Haldun Özgür Bayındır
Department of Mathematics
University of Illinois
Chicago, IL
United States