Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 2 (2018), 1115-1146.
Topological equivalences of E-infinity differential graded algebras
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 topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially topologically equivalent DGAs. Also, we show using these obstruction theories that for coconnective –DGAs, topological equivalences and quasi-isomorphisms agree. For –DGAs with trivial first homology, we show that an topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an –equivalence.
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 1115-1146.
Received: 22 May 2017
Revised: 18 October 2017
Accepted: 27 October 2017
First available in Project Euclid: 22 March 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 18G55: Homotopical algebra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55S12: Dyer-Lashof operations 55S35: Obstruction theory 55U99: None of the above, but in this section
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