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A Nontrivial Product in the Stable Homotopy of Spheres

Received: 7 August 2017     Published: 7 August 2017
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Abstract

Let  be an arbitrary odd prime number greater than eleven andbe the mod  Steenrod algebra. In this paper, it has proved that the product  is nontrivial and converges to  nontrivially of order  in , where , by making use of the Adams spectral sequence.

Published in Applied and Computational Mathematics (Volume 6, Issue 4)
DOI 10.11648/j.acm.20170604.17
Page(s) 196-201
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Steenrod Algebra, Cohomology, May Spectral Sequence, Stable Homotopy of Spheres

References
[1] Cohen R. L., Odd primary families in stable homotopy theory, Mem. Amer. Math. S oc., 1981, 242: 1-92.
[2] J. F.Adams,Stable Homotopy and Generalised Homology, Chicago: University of Chicago Press, 1974.
[3] A. Liulevicius, The factorizations of cyclic reduced powersbysecondary cohomology operations, Mem. Amer. Math. Soc. 42 1962:1-112.
[4] Miller H. R., Ravenel D. C. and Wilson W. S., Periodic phenome-na in the Adams-Novikov spectral sequence, Ann. of Math., 1977, 106:469-516.
[5] T. Aikawa, 3-dimensional cohomology of the modSteenrod algebra, Math. Scand. 47 (1980), 91--115.
[6] X. Liu and H. Zhao, On a product in the classical Adams spectral sequence, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2489-2496.
[7] X. Wang and Q. Zheng, The convergence of , Sci. China Ser. A 41 (1998), no. 6, 622-628.
[8] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Orlando: Academic Press, 1986.
[9] X. Liu, Some infinite elements in the Adams spetral sequence for the sphere spectrum, J. Math. Kyoto Univ.48 (2008), 617-629.
[10] Hao Zhao, Xiangjun Wang, Linan Zhong. The convergence of the product  in the Adams spectral sequence. Forum Mathematicum, 2015, 27 (3):1613-1637.
[11] Zhong Linan, X. Liu. Non-Triviality of the Product in the Adams Spectral Sequence. Acta Mathematica Scientia, 2014, 34 (2):274-282.
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    Wang Chong. (2017). A Nontrivial Product in the Stable Homotopy of Spheres. Applied and Computational Mathematics, 6(4), 196-201. https://doi.org/10.11648/j.acm.20170604.17

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    ACS Style

    Wang Chong. A Nontrivial Product in the Stable Homotopy of Spheres. Appl. Comput. Math. 2017, 6(4), 196-201. doi: 10.11648/j.acm.20170604.17

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    AMA Style

    Wang Chong. A Nontrivial Product in the Stable Homotopy of Spheres. Appl Comput Math. 2017;6(4):196-201. doi: 10.11648/j.acm.20170604.17

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  • @article{10.11648/j.acm.20170604.17,
      author = {Wang Chong},
      title = {A Nontrivial Product in the Stable Homotopy of Spheres},
      journal = {Applied and Computational Mathematics},
      volume = {6},
      number = {4},
      pages = {196-201},
      doi = {10.11648/j.acm.20170604.17},
      url = {https://doi.org/10.11648/j.acm.20170604.17},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170604.17},
      abstract = {Let  be an arbitrary odd prime number greater than eleven andbe the mod  Steenrod algebra. In this paper, it has proved that the product  is nontrivial and converges to  nontrivially of order  in , where , by making use of the Adams spectral sequence.},
     year = {2017}
    }
    

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    AB  - Let  be an arbitrary odd prime number greater than eleven andbe the mod  Steenrod algebra. In this paper, it has proved that the product  is nontrivial and converges to  nontrivially of order  in , where , by making use of the Adams spectral sequence.
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Author Information
  • College of Mathematical and Statistics, Cangzhou Normal University, Cangzhou, China

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