#### Volume 18, issue 2 (2018)

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Topological equivalences of E-infinity differential graded algebras

### Haldun Özgür Bayındır

Algebraic & Geometric Topology 18 (2018) 1115–1146
##### 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 ${E}_{\infty }\phantom{\rule{0.3em}{0ex}}{\mathbb{F}}_{p}\phantom{\rule{0.3em}{0ex}}$–DGAs, ${E}_{\infty }$ topological equivalences and quasi-isomorphisms agree. For ${E}_{\infty }\phantom{\rule{0.3em}{0ex}}{\mathbb{F}}_{p}\phantom{\rule{0.3em}{0ex}}$–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 ${H}_{\infty }\phantom{\rule{0.3em}{0ex}}{\mathbb{F}}_{p}\phantom{\rule{0.3em}{0ex}}$–equivalence.

##### Keywords
commutative ring spectra, homological algebra, E-infinity DGAs
##### Mathematical Subject Classification 2010
Primary: 18G55, 55P43, 55S12, 55S35, 55U99