Abstract
The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown–Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us to conclude that the 2-primary Brown–Peterson spectrum does not admit the structure of an $E_n$-algebra for any $n \geq 12$, answering a question of May in the negative.
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[may-quinn-ray-ringspectra]
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[may-sigurdsson-parametrized]
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[priddy-cellularBP]
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author = {Robinson, Alan},
title = {Obstruction theory and the strict associativity of {M}orava {$K$}-theories},
booktitle = {Advances in Homotopy Theory},
venue = {{C}ortona, 1988},
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[senger-obstr] A. Senger, On the realization of truncated Brown–Peterson spectra as $E_\infty$ ring spectra.
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[strickland-products]
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@ARTICLE{strickland-products,
author = {Strickland, N. P.},
title = {Products on {${\rm MU}$}-modules},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {351},
year = {1999},
number = {7},
pages = {2569--2606},
issn = {0002-9947},
mrclass = {55N22 (55N20)},
mrnumber = {1641115},
mrreviewer = {W. Stephen Wilson},
doi = {10.1090/S0002-9947-99-02436-8},
url = {https://doi.org/10.1090/S0002-9947-99-02436-8},
zblnumber = {0924.55005},
} -
[tomdieck-cobordismoperations]
T. tom Dieck, "Steenrod-Operationen in Kobordismen-Theorien," Math. Z., vol. 107, pp. 380-401, 1968.
@ARTICLE{tomdieck-cobordismoperations,
author = {tom Dieck, Tammo},
title = {Steenrod-{O}perationen in {K}obordismen-{T}heorien},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {107},
year = {1968},
pages = {380--401},
issn = {0025-5874},
mrclass = {55.34 (57.00)},
mrnumber = {0244989},
mrreviewer = {C. M. Naylor},
doi = {10.1007/BF01110069},
url = {https://doi.org/10.1007/BF01110069},
zblnumber = {0167.51801},
} -
[tilson-kunneth] S. Tilson, Power operations in the Künneth spectral sequence and commutative $\mathrm{H}\mathbb{F}_p$-algebras, 2016.
@MISC{tilson-kunneth,
author = {Tilson, Sean},
title = {Power operations in the {K}{ü}nneth spectral sequence and commutative {$\mathrm{H}\mathbb{F}_p$}-algebras},
year = {2016},
arxiv = {1602.06736},
} -
[turner-dyerlashof]
P. R. Turner, "Dyer-Lashof operations in the Hopf ring for complex cobordism," Math. Proc. Cambridge Philos. Soc., vol. 114, iss. 3, pp. 453-460, 1993.
@ARTICLE{turner-dyerlashof,
author = {Turner, Paul R.},
title = {Dyer-{L}ashof operations in the {H}opf ring for complex cobordism},
journal = {Math. Proc. Cambridge Philos. Soc.},
fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
volume = {114},
year = {1993},
number = {3},
pages = {453--460},
issn = {0305-0041},
mrclass = {55S12 (55N22)},
mrnumber = {1235993},
mrreviewer = {Mark Hovey},
doi = {10.1017/S0305004100071747},
url = {https://doi.org/10.1017/S0305004100071747},
zblnumber = {0797.55013},
} -
@ARTICLE{vogt-hocolim,
author = {Vogt, Rainer M.},
title = {Homotopy limits and colimits},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {134},
year = {1973},
pages = {11--52},
issn = {0025-5874},
mrclass = {55D15},
mrnumber = {0331376},
mrreviewer = {Brayton Gray},
doi = {10.1007/BF01219090},
url = {https://doi.org/10.1007/BF01219090},
zblnumber = {0276.55006},
}