# Topological Equivalences of E-infinity Differential Graded Algebras

## TL;DR

This paper explores the concept of topological equivalences among $E_$-algebra DGAs, constructing examples of non-trivially equivalent cases and analyzing conditions where topological equivalence aligns with quasi-isomorphism.

## Contribution

It introduces $E_$ topological equivalences, constructs first examples of non-trivially $E_$ topologically equivalent DGAs, and compares these with quasi-isomorphisms under various conditions.

## Key findings

- Constructed examples of non-trivially $E_$ topologically equivalent DGAs.
- Showed that for co-connective $E_$ DGAs, topological equivalence and quasi-isomorphism coincide.
- Proved that for $E_$ $_p$-DGAs with trivial first homology, topological equivalence preserves homology and Dyer-Lashof operations.

## 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.