Topological equivalences of E-infinity differential graded algebras

Haldun Özgür Bayındır

doi:10.2140/agt.2018.18.1115

Haldun Özgür Bayındır

Open Access

https://doi.org/10.2140/agt.2018.18.1115

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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.

Highlights

Dugger and Shipley defined a new equivalence relation between associative differential graded algebras that they call topological equivalences [DS07]
Fp–DGAs with trivial first homology, we show that an topological equivalence induces an isomorphism in homology that preserves the
Two R–DGAs X and Y are said to be topologically equivalent if the corresponding HR– algebras HX and HY are weakly equivalent as S–algebras where S denotes the sphere spectrum

Summary

Introduction

Dugger and Shipley defined a new equivalence relation between associative differential graded algebras (which we call DGAs) that they call topological equivalences [DS07]. Another theorem in [DS07] states that there are no examples of non-trivial topological equivalences in Q–DGAs, ie topologically equivalent Q–DGAs are quasi-isomorphic. Before stating our non-existence results, we note that topologically equivalent DGAs have isomorphic homology rings. This is because the Quillen equivalence between R– DGAs and HR–algebras gives an isomorphism between the homology ring of an R–DGA and the homotopy ring of the corresponding ring spectra. This says that there is an E∞ Z–DGA corresponding to a co-connective commutative S–algebra By this and Theorem 1.3, we deduce that weak equivalence classes of coconnective commutative S–algebras are uniquely determined by the quasi-isomorphism classes of the corresponding E∞ Z–DGAs. In Example 5.1, we construct E∞ Fp–DGAs that are non-trivially E∞ topologically equivalent. The category of spectra we use is symmetric spectra in topological spaces with the positive model structure as in Mandell, May, Schwede and Shipley [MMSS01]

Previous examples of topological equivalences

Obstruction theories for ring spectra

Dyer–Lashof operations and the dual Steenrod algebra