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Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra

Received: 8 October 2019     Accepted: 4 December 2019     Published: 23 April 2020
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Abstract

Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted Sq and pi respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 2)
DOI 10.11648/j.pamj.20200902.12
Page(s) 37-45
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), 2020. Published by Science Publishing Group

Keywords

π-Strongly Homotopy Commutative Hopf Algebra, Cohomology Operations, Gerstenhaber Algebra

References
[1] J. Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. U. S. A. 38, (1952), 720-726.
[2] A. K. Bousfield, V. K. A. M. Gugenheim, On PL Rham theory and rational homotopy type, Mem. Amer. Soc. 179 (1976) 1-94.
[3] H. Cartan, Sur l’it´eration des op´erations de Steenrod, Comment. Math. Helv. 29, (1955), 40-58.
[4] S. Eilenberg and J. C. Moore, Homotopy and fibrations I, Comment. Math. Helv. 40 (1966) 199-236.
[5] Y. Felix, S. Halperin, J. C. Thomas, Differential graded algebras in topology, I. M. James(Ed), Hand book of algebraic Topology, North-Holland, Amsterdam, 1995, (chapter 16). Advances in Math.
[6] Y. Felix, S. Halperin, J. C. Thomas, Gorenstein spaces, Advances in Math. 71 (1988) 92-112.
[7] M. Gerstenhaber, The cohomology structure of an associative ring, Annals of Math. 78 (1963) 267-288.
[8] El Hassan Idrissi, shc-Equivalences, prepublication, University of angers, (Novembre 1999).
[9] J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403-423.
[10] J. P. May, A General Algebraic Approach to Steenrod Operations, Springer Le ture Notes in Math., 168 1970, 153-231.
[11] H. J. Munkholm, Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure. Appl. Alg. 5 (1974) 1-50.
[12] B. Ndombol and J.-C. Thomas, Steenrod operations in shc-algebras, J. Pure Appl. Algebra 192 (2004), 239-264.
[13] B. Ndombol and J.-C. Thomas, On the cohomology algebra of free loop space, Topology 41 (2002) 85-106.
[14] J. D. Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra 89 (1993) 231-335.
[15] N. E. Steenrod, Products of cocycles and extensions of mappings, Ann. of Math.(2) 48, (1947), 290-320.
[16] N. E. Steenrod, Reduced powers of cohomology classes, Ann. of Math. (2) 56, (1952), 47-67.
[17] C. Tcheka, Steenrod operation on the negative cyclic homology, Homology, Homotopy and Applications. 11, (2009) 1-39.
[18] C. Tcheka, Steenrod operation, Hochschild cohomology and negative cyclic cohomology, JP Journal of Geometry and Topology. 13, (2013) 1-39.
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    Calvin Tcheka. (2020). Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra. Pure and Applied Mathematics Journal, 9(2), 37-45. https://doi.org/10.11648/j.pamj.20200902.12

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    ACS Style

    Calvin Tcheka. Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra. Pure Appl. Math. J. 2020, 9(2), 37-45. doi: 10.11648/j.pamj.20200902.12

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    AMA Style

    Calvin Tcheka. Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra. Pure Appl Math J. 2020;9(2):37-45. doi: 10.11648/j.pamj.20200902.12

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  • @article{10.11648/j.pamj.20200902.12,
      author = {Calvin Tcheka},
      title = {Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra},
      journal = {Pure and Applied Mathematics Journal},
      volume = {9},
      number = {2},
      pages = {37-45},
      doi = {10.11648/j.pamj.20200902.12},
      url = {https://doi.org/10.11648/j.pamj.20200902.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200902.12},
      abstract = {Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted Sq and pi respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.},
     year = {2020}
    }
    

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    AB  - Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted Sq and pi respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.
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Author Information
  • Department of Mathematics and Computer Sciences, University of Dschang, Dschang, Cameroon

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