| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">P</mi> </semantics> </math> </inline-formula> of prime numbers which may be empty and a localization <inline-formula> <math display="inline"> <semantics> <msub> <mi>X</mi> <mi mathvariant="script">P</mi> </msub> </semantics> </math> </inline-formula> of a nilpotent CW-space <i>X</i> at <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">P</mi> </semantics> </math> </inline-formula>, we let <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo>(</mo> <msub> <mi>X</mi> <mi mathvariant="script">P</mi> </msub> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics> </math> </inline-formula> be the cardinalities of the sets of all homotopy comultiplications on <i>X</i> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>X</mi> <mi mathvariant="script">P</mi> </msub> </semantics> </math> </inline-formula>, respectively. In this paper, we show that if <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics> </math> </inline-formula> is finite, then <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≥</mo> <mo stretchy="false">|</mo> <mi>C</mi> </mrow> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi mathvariant="script">P</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, and if <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics> </math> </inline-formula> is infinite, then <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi>C</mi> </mrow> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi mathvariant="script">P</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <i>X</i> is the <i>k</i>-fold wedge sum <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mo>⋁</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <msup> <mi mathvariant="double-struck">S</mi> <msub> <mi>n</mi> <mi>i</mi> </msub> </msup> </mrow> </semantics> </math> </inline-formula> or Moore spaces <inline-formula> <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Moreover, we provide examples to concretely determine the cardinality of homotopy comultiplications on the <i>k</i>-fold wedge sum of spheres, Moore spaces, and their localizations.
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