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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">P</mi> </semantics> </math>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/1/86 |
| _version_ | 1811340035512860672 |
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| author | Dae-Woong Lee |
| author_facet | Dae-Woong Lee |
| author_sort | Dae-Woong Lee |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T18:35:20Z |
| format | Article |
| id | doaj.art-9eac2edebcd34a098e8c44d515209189 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-13T18:35:20Z |
| publishDate | 2020-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9eac2edebcd34a098e8c44d5152091892022-12-22T02:34:54ZengMDPI AGMathematics2227-73902020-01-01818610.3390/math8010086math8010086Comultiplications on the Localized Spheres and Moore SpacesDae-Woong Lee0Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, KoreaAny 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.https://www.mdpi.com/2227-7390/8/1/86comultiplicationslocalized spheresbasic whitehead productshilton formulamoore space |
| spellingShingle | Dae-Woong Lee Comultiplications on the Localized Spheres and Moore Spaces Mathematics comultiplications localized spheres basic whitehead products hilton formula moore space |
| title | Comultiplications on the Localized Spheres and Moore Spaces |
| title_full | Comultiplications on the Localized Spheres and Moore Spaces |
| title_fullStr | Comultiplications on the Localized Spheres and Moore Spaces |
| title_full_unstemmed | Comultiplications on the Localized Spheres and Moore Spaces |
| title_short | Comultiplications on the Localized Spheres and Moore Spaces |
| title_sort | comultiplications on the localized spheres and moore spaces |
| topic | comultiplications localized spheres basic whitehead products hilton formula moore space |
| url | https://www.mdpi.com/2227-7390/8/1/86 |
| work_keys_str_mv | AT daewoonglee comultiplicationsonthelocalizedspheresandmoorespaces |