Topological Equivalences of E-infinity Differential Graded Algebras
Abstract
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∞ topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of non-trivially E∞ topologically equivalent E∞ DGAs. Also, we show using these obstruction theories that for co-connective E∞ DGAs, E∞ topological equivalences and quasi-isomorphisms agree. For E∞ Fp-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∞ Fp-equivalence.