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 $E_{\infty }$ 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.

References in corpus (2)

• An algebraic model for commutative HZ-algebras
• Postnikov extensions of ring spectra

Cited by in corpus (1)

• Extension DGAs and topological Hochschild homology