  Two DGAs are called topologically equivalent if 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 counter-example, Dugger and Shipley showed that there
are DGAs that are non-trivially topologically equivalent, i.e. 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 non-trivially $E_\infty$ topologically equivalent $E_\infty$ DGAs.
Also, we show using these obstruction theories that for co-connective
$E_\infty$ DGAs, $E_\infty$ topological equivalences and quasi-isomorphisms
agree. For $E_\infty$ $\mathbb{F}_p$-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 $
$\mathbb{F}_p$-equivalence.
