Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-08T13:58:45.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization, Algebraic Loops and H-Spaces II

Published online by Cambridge University Press:  20 November 2018

Albert O. Shar
Albert O. Shar*
Affiliation:
University of New Hampshire, Durham, New Hampshire
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree.

The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 812 - 817
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Math., 304 (Springer, 1972).Google Scholar
2. Bruck, R. H., A survey of binary systems (Springer-Verlag, 1958).Google Scholar
3. Gruenberg, K. W., Residual properties of infinite soluble groups, Proc. Lond. Math. Soc. (3. 7 (1957), 29–62.Google Scholar
4. Hilton, P., Mislinand, G. Roitberg, J., Localizations of nilpotent groups + spaces, Mathematics Studies 15 (North Holland, 1975).Google Scholar
5. Hirsch, K. A., On infinite soluble groups III, Proc. Lond. Math. Soc. (2). 49 (1946), 184–194.Google Scholar
6. Shar, A. O., Localization, algebraic loops and H-spaces I, Can. J. Math. 31 (1979), 427–435.Google Scholar
7. Warhelcl, R. B., Jr., Nilpotent groups, Lecture Notes in Math. 513 (Springer, 1976).Google Scholar