Advertisement

Relative homotopy abelian H-spaces

  • S. Theriault
  • J. Wu
Open Access
Article
  • 50 Downloads

Abstract

We introduce the notion of a relatively homotopy associative and homotopy commutative H-space, construct one for any path-connected space X, and describe several useful properties, including exponent properties.

Mathematics Subject Classification

Primary 55P45 Secondary 55P35 55Q52 

References

  1. 1.
    Cohen, F.R., Moore, J.C., Neisendorfer, J.A.: Torsion in homotopy groups. Ann. Math. 109, 121–168 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cohen, F.R., Moore, J.C., Neisendorfer, J.A.: Exponents in homotopy theory, algebraic topology and algebraic K-theory. In: Browder, W. (ed.) Annals of Mathematics Studies, pp. 3–34. Princeton University Press, Princeton (1987)Google Scholar
  3. 3.
    Cohen, F.R., Neisendorfer, J.A.: A construction of \(p\)-local \(H\)-spaces, pp. 351–359. Lecture Notes in Math. Vol. 1051, Springer, Berlin (1984)Google Scholar
  4. 4.
    Dold, A., Lashof, R.: Principal quasifibrations and fiber homotopy equivalence of bundles. Ill. J. Math. 3, 285–305 (1959)zbMATHGoogle Scholar
  5. 5.
    Ganea, T.: A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39, 295–322 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gray, B.: Homotopy commutativity and the \(EHP\) sequence. Contemp. Math. 96, 181–188 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gray, B.: On decompositions in homotopy theory. Trans. Am. Math. Soc. 358, 3305–3328 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gray, B.: Universal abelian \(H\)-spaces. Topol. Appl. 159, 209–224 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gray, B.: Abelian properties of Anick spaces, Mem. Amer. Math. Soc. 246, No. 1162 (2017)Google Scholar
  10. 10.
    Grbić, J.: Universal spaces of two-cell complexes and their homotopy exponents. Quart. J. Math. Oxford 57, 355–366 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Grbić, J., Theriault, S., Wu, J.: Suspension splittings and James–Hopf invariants. Proc. Roy. Soc. Edinburgh Sect. A 144, 87–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    James, I.M.: Reduced product spaces. Ann. Math. 62, 170–197 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    McGibbon, C.A.: Homotopy commutativity in localized groups. Am. J. Math 106, 665–687 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Neisendorfer, J.A.: Properties of certain \(H\)-spaces. Quart. J. Math. 34, 201–209 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Selick, P., Wu, J.: On natural decompositions of loop suspensions and natural coalgebra decompositions of tensor algebras, Mem. Amer. Math. Soc. 148, No. 701 (2000)Google Scholar
  16. 16.
    Stasheff, J.: On homotopy abelian \(H\)-spaces. Math. Proc. Cambridge Philos. Soc. 57, 734–745 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sugawara, M.: On a condition that a space is an \(H\)-space. Math. J. Okayama Univ. 6, 109–129 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Theriault, S.D.: Properties of Anick’s spaces. Trans. Am. Math. Soc. 353, 1009–1037 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Theriault, S.D.: The \(H\)-structure of low rank torsion free \(H\)-spaces. Quart. J. Math. Oxford 56, 403–415 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Theriault, S.D.: The odd primary \(H\)-structure of Lie groups of low rank and its application to exponents. Trans. Am. Math. Soc. 359, 4511–4535 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Williams, F.D.: A theorem on homotopy-commutativity. Michigan Math. J. 18, 51–53 (1971)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is 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, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical SciencesUniversity of SouthamptonSouthamptonUK
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations