Topological equivalences of E-infinity differential graded algebras

Abstract

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 $E_{\infty }$ topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially $E_{\infty }$ topologically equivalent $E_{\infty }$ DGAs. Also, we show using these obstruction theories that for coconnective $E_{\infty }{\mathbb{F}}_p$–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.