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.
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