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.
"Topological equivalences of E-infinity differential graded algebras." Algebr. Geom. Topol. 18 (2) 1115 - 1146, 2018. https://doi.org/10.2140/agt.2018.18.1115