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Comultiplications on the Localized Spheres and Moore Spaces

Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, Korea
Mathematics 2020, 8(1), 86; https://doi.org/10.3390/math8010086
Received: 2 December 2019 / Revised: 27 December 2019 / Accepted: 30 December 2019 / Published: 5 January 2020
Any nilpotent CW-space can be localized at primes in a similar way to the localization of a ring at a prime number. For a collection P of prime numbers which may be empty and a localization X P of a nilpotent CW-space X at P , we let | C ( X ) | and | C ( X P ) | be the cardinalities of the sets of all homotopy comultiplications on X and X P , respectively. In this paper, we show that if | C ( X ) | is finite, then | C ( X ) | | C ( X P ) | , and if | C ( X ) | is infinite, then | C ( X ) | = | C ( X P ) | , where X is the k-fold wedge sum i = 1 k S n i or Moore spaces M ( G , n ) . Moreover, we provide examples to concretely determine the cardinality of homotopy comultiplications on the k-fold wedge sum of spheres, Moore spaces, and their localizations. View Full-Text
Keywords: comultiplications; localized spheres; basic Whitehead products; Hilton formula; Moore space comultiplications; localized spheres; basic Whitehead products; Hilton formula; Moore space
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Lee, D.-W. Comultiplications on the Localized Spheres and Moore Spaces. Mathematics 2020, 8, 86.

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