Comultiplications on the Localized Spheres and Moore Spaces

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.