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Difference between revisions of "Fundamental cocycle"

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''of a cellular space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421701.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421702.png" />-skeleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421703.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421704.png" />''
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The cochain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421705.png" /> whose value on the cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421706.png" /> is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421707.png" /> corresponding to the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421708.png" />. The cohomology class of the fundamental cocycle is the [[Fundamental class|fundamental class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042170/f0421709.png" />.
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''of a cellular space  $  X $
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whose  $  ( n - 1) $-
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skeleton  $  X ^ {n - 1 } $
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is a point  $  x _ {0} $''
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The cochain in $  C  ^ {n} ( X, \pi _ {n} ( X)) $
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whose value on the cell $  e _ {i}  ^ {n} $
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is the element of $  \pi _ {n} ( X, x _ {0} ) $
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corresponding to the closure $  \overline{ {e _ {i}  ^ {n} }}\; $.  
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The cohomology class of the fundamental cocycle is the [[Fundamental class|fundamental class]] of $  X $.

Latest revision as of 19:40, 5 June 2020


of a cellular space $ X $ whose $ ( n - 1) $- skeleton $ X ^ {n - 1 } $ is a point $ x _ {0} $

The cochain in $ C ^ {n} ( X, \pi _ {n} ( X)) $ whose value on the cell $ e _ {i} ^ {n} $ is the element of $ \pi _ {n} ( X, x _ {0} ) $ corresponding to the closure $ \overline{ {e _ {i} ^ {n} }}\; $. The cohomology class of the fundamental cocycle is the fundamental class of $ X $.

How to Cite This Entry:
Fundamental cocycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_cocycle&oldid=47022
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article