Secondary power operations and the Brown–Peterson spectrum at the prime $2$

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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      url = {https://doi.org/10.1017/CBO9780511529955.009},
      zblnumber = {1086.55006},
      }
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      author = {Gepner, David and Haugseng, Rune},
      title = {Enriched {$\infty$}-categories via non-symmetric {$\infty$}-operads},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {279},
      year = {2015},
      pages = {575--716},
      issn = {0001-8708},
      mrclass = {18D20 (18D10 18D50)},
      mrnumber = {3345192},
      mrreviewer = {Christopher L. Rogers},
      doi = {10.1016/j.aim.2015.02.007},
      url = {https://doi.org/10.1016/j.aim.2015.02.007},
      zblnumber = {1342.18009},
      }
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    @INCOLLECTION{goerss-dieudonne,
      author = {Goerss, Paul G.},
      title = {Hopf rings, {D}ieudonné modules, and {$E_*\Omega^2S^3$}},
      booktitle = {Homotopy Invariant Algebraic Structures},
      venue = {{B}altimore, {MD},
      1998},
      series = {Contemp. Math.},
      volume = {239},
      pages = {115--174},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1999},
      mrclass = {57T05 (16W30 55N20 55P35 55S05)},
      mrnumber = {1718079},
      mrreviewer = {Neil P. Strickland},
      doi = {10.1090/conm/239/03600},
      url = {https://doi.org/10.1090/conm/239/03600},
      zblnumber = {0954.55006},
      }
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      author = {Harper, John R.},
      title = {Secondary Cohomology Operations},
      series = {Grad. Stud. in Math.},
      volume = {49},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2002},
      pages = {xii+268},
      isbn = {0-8218-3198-4},
      mrclass = {55S20 (55-01 55R20 55S10)},
      mrnumber = {1913285},
      mrreviewer = {Lionel Schwartz},
      doi = {10.1090/gsm/049},
      url = {https://doi.org/10.1090/gsm/049},
      zblnumber = {1003.55001},
      }
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      author = {Hu, P. and Kriz, I. and May, J. P.},
      title = {Cores of spaces, spectra, and {$E_\infty$} ring spectra},
      note = {Equivariant stable homotopy theory and related areas (Stanford, CA, 2000)},
      journal = {Homology Homotopy Appl.},
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      volume = {3},
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      pages = {341--354},
      issn = {1532-0081},
      mrclass = {55P15 (55P43)},
      mrnumber = {1856030},
      mrreviewer = {J. M. Boardman},
      zblnumber = {0987.55009},
      doi = {10.4310/HHA.2001.v3.n2.a3},
      }
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      author = {Hovey, Mark and Shipley, Brooke and Smith, Jeff},
      title = {Symmetric spectra},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {13},
      year = {2000},
      number = {1},
      pages = {149--208},
      issn = {0894-0347},
      mrclass = {55P42 (18D10 18D50 18G30 18G55 55U10 55U35)},
      mrnumber = {1695653},
      mrreviewer = {J. P. C. Greenlees},
      doi = {10.1090/S0894-0347-99-00320-3},
      url = {https://doi.org/10.1090/S0894-0347-99-00320-3},
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      }
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      issn = {0166-8641},
      mrclass = {55P43 (55P42)},
      mrnumber = {2670503},
      mrreviewer = {Yutaka Hemmi},
      doi = {10.1016/j.topol.2010.06.007},
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      fjournal = {Transactions of the American Mathematical Society},
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      issn = {0002-9947},
      mrclass = {55F45 (55G99)},
      mrnumber = {0331386},
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      doi = {10.2307/1996429},
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      author = {Kriz, I.},
      title = {Towers of {$E_\infty$}-ring spectra with an application to {$BP$}},
      note = {unpublished},
      year = {1995},
      zblnumber = {},
      }
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      author = {Lazarev, A.},
      title = {Homotopy theory of {$A_\infty$} ring spectra and applications to {$M{\rm U}$}-modules},
      journal = {$K$-Theory},
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      volume = {24},
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      mrclass = {55P43 (55N20 55P42 55S35 55T25)},
      mrnumber = {1876800},
      mrreviewer = {Andrew J. Baker},
      doi = {10.1023/A:1013394125552},
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      }
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      author = {Lewis, Jr., L. G. and May, J. P. and Steinberger, M. and McClure, J. E.},
      title = {Equivariant Stable Homotopy Theory},
      series = {Lecture Notes in Math.},
      volume = {1213},
      note = {with contributions by J. E. McClure},
      publisher = {Springer-Verlag, Berlin},
      year = {1986},
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      isbn = {3-540-16820-6},
      mrclass = {55-02 (55Nxx 55Pxx 57S99)},
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      mrreviewer = {T. tom Dieck},
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      author = {Lawson, Tyler and Naumann, Niko},
      title = {Strictly commutative realizations of diagrams over the {S}teenrod algebra and topological modular forms at the prime 2},
      journal = {Int. Math. Res. Not. IMRN},
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      mrnumber = {3214285},
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      }
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      mrclass = {18-02 (18B25 18E35 18G30 18G55 55U40)},
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      author = {May, J. P.},
      title = {Problems in infinite loop space theory},
      booktitle = {Conference on Homotopy Theory},
      venue = {{E}vanston, {I}ll., 1974},
      series = {Notas Mat. Simpos.},
      volume = {1},
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      publisher = {Soc. Mat. Mexicana, México},
      year = {1975},
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      }
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      author = {May, J. Peter},
      title = {{$E\sb{\infty }$} Ring Spaces and {$E\sb{\infty }$} Ring Spectra},
      series = {Lecture Notes in Math.},
      volume = {577},
      note = {with contributions by Frank Quinn, Nigel Ray, and J\o rgen Tornehave},
      publisher = {Springer-Verlag, New York},
      year = {1977},
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      mrclass = {55D35},
      mrnumber = {0494077},
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      }
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    @ARTICLE{may-idempotent,
      author = {May, J. P.},
      title = {Idempotents and {L}andweber exactness in brave new algebra},
      note = {Equivariant stable homotopy theory and related areas (Stanford, CA, 2000)},
      journal = {Homology Homotopy Appl.},
      fjournal = {Homology, Homotopy and Applications},
      volume = {3},
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      pages = {355--359},
      issn = {1532-0081},
      mrclass = {55P43 (18G55)},
      mrnumber = {1856031},
      mrreviewer = {A. D. Elmendorf},
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      }
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      title = {A spectral sequence for the homology of an infinite delooping},
      journal = {Pacific J. Math.},
      fjournal = {Pacific Journal of Mathematics},
      volume = {79},
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      number = {1},
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      author = {Mandell, M. A. and May, J. P. and Schwede, S. and Shipley, B.},
      title = {Model categories of diagram spectra},
      journal = {Proc. London Math. Soc. (3)},
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      mrclass = {55P42 (18A25 18E30 55U35)},
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      mrreviewer = {Mark Hovey},
      doi = {10.1112/S0024611501012692},
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      }
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      author = {May, J. P. and Sigurdsson, J.},
      title = {Parametrized Homotopy Theory},
      series = {Math. Surveys Monogr.},
      volume = {132},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2006},
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      isbn = {978-0-8218-3922-5; 0-8218-3922-5},
      mrclass = {55P42 (19L99 55N20 55N22 55P91)},
      mrnumber = {2271789},
      mrreviewer = {A. A. Ranicki},
      doi = {10.1090/surv/132},
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      }
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      author = {McClure, James E. and Smith, Jeffrey H.},
      title = {Cosimplicial objects and little {$n$}-cubes. {I}},
      journal = {Amer. J. Math.},
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      mrclass = {55P48 (18D50 55U10)},
      mrnumber = {2089084},
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      journal = {Geom. Topol.},
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      doi = {10.2307/1995637},
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      author = {Priddy, Stewart},
      title = {Dyer-{L}ashof operations for the classifying spaces of certain matrix groups},
      journal = {Quart. J. Math. Oxford Ser. (2)},
      fjournal = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      volume = {26},
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      issn = {0033-5606},
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      doi = {10.1093/qmath/26.1.179},
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      title = {A cellular construction of {BP} and other irreducible spectra},
      journal = {Math. Z.},
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      doi = {10.1007/BF01215522},
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      author = {Peterson, F. P. and Stein, N.},
      title = {Secondary cohomology operations: two formulas},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
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      title = {On the formal group laws of unoriented and complex cobordism theory},
      journal = {Bull. Amer. Math. Soc.},
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      url = {https://doi.org/10.1090/S0002-9904-1969-12401-8},
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      }

Authors

Tyler Lawson

University of Minnesota, Minneapolis, MN