## Algebraic & Geometric Topology

### Topological equivalences of E-infinity differential graded algebras

Haldun Özgür Bayındır

#### 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 $E ∞$ 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 $E ∞ F p$–DGAs, $E ∞$ topological equivalences and quasi-isomorphisms agree. For $E ∞ F p$–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 $H ∞ F p$–equivalence.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 1115-1146.

Dates
Revised: 18 October 2017
Accepted: 27 October 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684032

Digital Object Identifier
doi:10.2140/agt.2018.18.1115

Mathematical Reviews number (MathSciNet)
MR3773750

Zentralblatt MATH identifier
06859616

#### Citation

Bayındır, Haldun Özgür. Topological equivalences of E-infinity differential graded algebras. Algebr. Geom. Topol. 18 (2018), no. 2, 1115--1146. doi:10.2140/agt.2018.18.1115. https://projecteuclid.org/euclid.agt/1521684032

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