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Some function spaces of CW type
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| Zeitschriftentitel: | Proceedings of the American Mathematical Society |
|---|---|
| Personen und Körperschaften: | |
| In: | Proceedings of the American Mathematical Society, 90, 1984, 4, S. 599-607 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
American Mathematical Society (AMS)
|
| Schlagwörter: |
| author_facet |
Kahn, Peter J. Kahn, Peter J. |
|---|---|
| author |
Kahn, Peter J. |
| spellingShingle |
Kahn, Peter J. Proceedings of the American Mathematical Society Some function spaces of CW type Applied Mathematics General Mathematics |
| author_sort |
kahn, peter j. |
| spelling |
Kahn, Peter J. 0002-9939 1088-6826 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9939-1984-0733413-x <p>J. Milnor’s result on the CW type of certain function spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m a p left-parenthesis upper X comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>map</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {map}}\left ( {X,Y} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is extended to allow the case in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-skeleton and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript i Baseline upper Y equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _i}Y = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i greater-than k"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i > k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis pi comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>π<!-- π --></mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K\left ( {\pi ,1} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold has contractible components when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finitely-generated.</p> Some function spaces of CW type Proceedings of the American Mathematical Society |
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10.1090/s0002-9939-1984-0733413-x |
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Online Free |
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Mathematik |
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ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTM5LTE5ODQtMDczMzQxMy14 |
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DE-L229 DE-D275 DE-Bn3 DE-Brt1 DE-Zwi2 DE-D161 DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 DE-Ch1 |
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American Mathematical Society (AMS), 1984 |
| imprint_str_mv |
American Mathematical Society (AMS), 1984 |
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1088-6826 0002-9939 |
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1088-6826 0002-9939 |
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kahn1984somefunctionspacesofcwtype |
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1984 |
| publisher |
American Mathematical Society (AMS) |
| recordtype |
ai |
| record_format |
ai |
| series |
Proceedings of the American Mathematical Society |
| source_id |
49 |
| title |
Some function spaces of CW type |
| title_unstemmed |
Some function spaces of CW type |
| title_full |
Some function spaces of CW type |
| title_fullStr |
Some function spaces of CW type |
| title_full_unstemmed |
Some function spaces of CW type |
| title_short |
Some function spaces of CW type |
| title_sort |
some function spaces of cw type |
| topic |
Applied Mathematics General Mathematics |
| url |
http://dx.doi.org/10.1090/s0002-9939-1984-0733413-x |
| publishDate |
1984 |
| physical |
599-607 |
| description |
<p>J. Milnor’s result on the CW type of certain function spaces <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m a p left-parenthesis upper X comma upper Y right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>map</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>X</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>Y</mml:mi>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\operatorname {map}}\left ( {X,Y} \right )</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is extended to allow the case in which <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
<mml:semantics>
<mml:mi>X</mml:mi>
<mml:annotation encoding="application/x-tex">X</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> has a finite <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">
<mml:semantics>
<mml:mi>k</mml:mi>
<mml:annotation encoding="application/x-tex">k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-skeleton and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript i Baseline upper Y equals 0">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>π<!-- π --></mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mi>Y</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\pi _i}Y = 0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i greater-than k">
<mml:semantics>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>></mml:mo>
<mml:mi>k</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">i > k</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis pi comma 1 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>π<!-- π --></mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">K\left ( {\pi ,1} \right )</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> manifold has contractible components when <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi">
<mml:semantics>
<mml:mi>π<!-- π --></mml:mi>
<mml:annotation encoding="application/x-tex">\pi</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is finitely-generated.</p> |
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| description | <p>J. Milnor’s result on the CW type of certain function spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m a p left-parenthesis upper X comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>map</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {map}}\left ( {X,Y} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is extended to allow the case in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-skeleton and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript i Baseline upper Y equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _i}Y = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i greater-than k"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i > k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis pi comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>π<!-- π --></mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K\left ( {\pi ,1} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold has contractible components when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finitely-generated.</p> |
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| id | ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTM5LTE5ODQtMDczMzQxMy14 |
| imprint | American Mathematical Society (AMS), 1984 |
| imprint_str_mv | American Mathematical Society (AMS), 1984 |
| institution | DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161, DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14, DE-Ch1 |
| issn | 1088-6826, 0002-9939 |
| issn_str_mv | 1088-6826, 0002-9939 |
| language | English |
| last_indexed | 2024-03-01T11:08:17.156Z |
| match_str | kahn1984somefunctionspacesofcwtype |
| mega_collection | American Mathematical Society (AMS) (CrossRef) |
| physical | 599-607 |
| publishDate | 1984 |
| publishDateSort | 1984 |
| publisher | American Mathematical Society (AMS) |
| record_format | ai |
| recordtype | ai |
| series | Proceedings of the American Mathematical Society |
| source_id | 49 |
| spelling | Kahn, Peter J. 0002-9939 1088-6826 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9939-1984-0733413-x <p>J. Milnor’s result on the CW type of certain function spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m a p left-parenthesis upper X comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>map</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {map}}\left ( {X,Y} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is extended to allow the case in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-skeleton and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript i Baseline upper Y equals 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _i}Y = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i greater-than k"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i > k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis pi comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>π<!-- π --></mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K\left ( {\pi ,1} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold has contractible components when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finitely-generated.</p> Some function spaces of CW type Proceedings of the American Mathematical Society |
| spellingShingle | Kahn, Peter J., Proceedings of the American Mathematical Society, Some function spaces of CW type, Applied Mathematics, General Mathematics |
| title | Some function spaces of CW type |
| title_full | Some function spaces of CW type |
| title_fullStr | Some function spaces of CW type |
| title_full_unstemmed | Some function spaces of CW type |
| title_short | Some function spaces of CW type |
| title_sort | some function spaces of cw type |
| title_unstemmed | Some function spaces of CW type |
| topic | Applied Mathematics, General Mathematics |
| url | http://dx.doi.org/10.1090/s0002-9939-1984-0733413-x |