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.
Article PDF
Similar content being viewed by others
References
Cohen, F.R., Moore, J.C., Neisendorfer, J.A.: Torsion in homotopy groups. Ann. Math. 109, 121–168 (1979)
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)
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)
Dold, A., Lashof, R.: Principal quasifibrations and fiber homotopy equivalence of bundles. Ill. J. Math. 3, 285–305 (1959)
Ganea, T.: A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39, 295–322 (1965)
Gray, B.: Homotopy commutativity and the \(EHP\) sequence. Contemp. Math. 96, 181–188 (1989)
Gray, B.: On decompositions in homotopy theory. Trans. Am. Math. Soc. 358, 3305–3328 (2006)
Gray, B.: Universal abelian \(H\)-spaces. Topol. Appl. 159, 209–224 (2012)
Gray, B.: Abelian properties of Anick spaces, Mem. Amer. Math. Soc. 246, No. 1162 (2017)
Grbić, J.: Universal spaces of two-cell complexes and their homotopy exponents. Quart. J. Math. Oxford 57, 355–366 (2006)
Grbić, J., Theriault, S., Wu, J.: Suspension splittings and James–Hopf invariants. Proc. Roy. Soc. Edinburgh Sect. A 144, 87–108 (2014)
James, I.M.: Reduced product spaces. Ann. Math. 62, 170–197 (1955)
McGibbon, C.A.: Homotopy commutativity in localized groups. Am. J. Math 106, 665–687 (1984)
Neisendorfer, J.A.: Properties of certain \(H\)-spaces. Quart. J. Math. 34, 201–209 (1983)
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)
Stasheff, J.: On homotopy abelian \(H\)-spaces. Math. Proc. Cambridge Philos. Soc. 57, 734–745 (1961)
Sugawara, M.: On a condition that a space is an \(H\)-space. Math. J. Okayama Univ. 6, 109–129 (1957)
Theriault, S.D.: Properties of Anick’s spaces. Trans. Am. Math. Soc. 353, 1009–1037 (2001)
Theriault, S.D.: The \(H\)-structure of low rank torsion free \(H\)-spaces. Quart. J. Math. Oxford 56, 403–415 (2005)
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)
Williams, F.D.: A theorem on homotopy-commutativity. Michigan Math. J. 18, 51–53 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Wu: Partially supported by the Singapore Ministry of Education research Grant (AcRF Tier 1 WBS No. R-146-000-222-112) and a Grant (No. 11329101) from the NSFC of China.
Rights and permissions
Open Access This 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.
About this article
Cite this article
Theriault, S., Wu, J. Relative homotopy abelian H-spaces. manuscripta math. 159, 301–319 (2019). https://doi.org/10.1007/s00229-018-1060-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-018-1060-x