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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∞ topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially E∞ topologically equivalent E∞ DGAs. Also, we show using these obstruction theories that for coconnective E∞Fp–DGAs, E∞ topological equivalences and quasi-isomorphisms agree. For E∞Fp–DGAs with trivial first homology, we show that an E∞ topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an H∞Fp–equivalence.
Keywords: commutative ring spectra, homological algebra, E-infinity DGAs
Bayındır, Haldun Özgür 1
@article{10_2140_agt_2018_18_1115,
author = {Bay{\i}nd{\i}r, Haldun \"Ozg\"ur},
title = {Topological equivalences of {E-infinity} differential graded algebras},
journal = {Algebraic and Geometric Topology},
pages = {1115--1146},
publisher = {mathdoc},
volume = {18},
number = {2},
year = {2018},
doi = {10.2140/agt.2018.18.1115},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1115/}
}
TY - JOUR AU - Bayındır, Haldun Özgür TI - Topological equivalences of E-infinity differential graded algebras JO - Algebraic and Geometric Topology PY - 2018 SP - 1115 EP - 1146 VL - 18 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1115/ DO - 10.2140/agt.2018.18.1115 ID - 10_2140_agt_2018_18_1115 ER -
%0 Journal Article %A Bayındır, Haldun Özgür %T Topological equivalences of E-infinity differential graded algebras %J Algebraic and Geometric Topology %D 2018 %P 1115-1146 %V 18 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1115/ %R 10.2140/agt.2018.18.1115 %F 10_2140_agt_2018_18_1115
Bayındır, Haldun Özgür. Topological equivalences of E-infinity differential graded algebras. Algebraic and Geometric Topology, Tome 18 (2018) no. 2, pp. 1115-1146. doi: 10.2140/agt.2018.18.1115
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