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Localization, Algebraic Loops and H-Spaces I

Published online by Cambridge University Press:  20 November 2018

Albert O. Shar
Albert O. Shar*
Affiliation:
University of New Hampshire, Durham, New Hampshire
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If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 427 - 435
Copyright
Copyright © Canadian Mathematical Society 1979

References

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8. Shar, A. O., P-primary decomposition of maps into an H-space, Pacific Jour. Math. 59 (1975), 237–240.Google Scholar
9. Shar, A. O., Localization, algebraic loops and H-spaces, Can. J. Math., to appear.Google Scholar