Topological equivalences of E-infinity differential graded algebras

Bayındır, Haldun Özgür

doi:10.2140/agt.2018.18.1115

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Topological equivalences of E-infinity differential graded algebras

Haldun Özgür Bayındır

Algebraic & Geometric Topology 18 (2018) 1115–1146

DOI: 10.2140/agt.2018.18.1115

Abstract

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 $E_{\infty}$ topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially $E_{\infty}$ topologically equivalent $E_{\infty}$ DGAs. Also, we show using these obstruction theories that for coconnective $E_{\infty} F_{p}$ –DGAs, $E_{\infty}$ topological equivalences and quasi-isomorphisms agree. For $E_{\infty} F_{p}$ –DGAs with trivial first homology, we show that an $E_{\infty}$ topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an $H_{\infty} F_{p}$ –equivalence.

Keywords

commutative ring spectra, homological algebra, E-infinity DGAs

Mathematical Subject Classification 2010

Primary: 18G55, 55P43, 55S12, 55S35, 55U99

References

Publication

Received: 22 May 2017

Revised: 18 October 2017

Accepted: 27 October 2017

Published: 12 March 2018

Authors

Haldun Özgür Bayındır
Department of Mathematics University of Illinois Chicago, IL United States

Volume 18, issue 2 (2018)