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Difference between revisions of "Normal zero-dimensional cycle"

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''cycle of index zero''
 
''cycle of index zero''
  
A [[Cycle|cycle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067700/n0677001.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067700/n0677002.png" />. A cycle homologous to zero is always normal; the quotient group of the group of normal cycles by that of the cycles homologous to zero is called the reduced zero-dimensional homology group. For a connected complex the reduced group is zero, which is convenient in any kind of definition of acyclicity. A [[Proper cycle|proper cycle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067700/n0677003.png" /> is called normal if each of the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067700/n0677004.png" /> is normal.
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A [[Cycle|cycle]] $z=\sum_{i=1}^na_it_i^0$ such that $\sum_{i=1}^na_i=0$. A cycle homologous to zero is always normal; the quotient group of the group of normal cycles by that of the cycles homologous to zero is called the reduced zero-dimensional homology group. For a connected complex the reduced group is zero, which is convenient in any kind of definition of acyclicity. A [[Proper cycle|proper cycle]] $z^0=\{z_1^0,\dots,z_k^0,\dots\}$ is called normal if each of the cycles $z_k^0$ is normal.

Latest revision as of 10:42, 31 August 2014

cycle of index zero

A cycle $z=\sum_{i=1}^na_it_i^0$ such that $\sum_{i=1}^na_i=0$. A cycle homologous to zero is always normal; the quotient group of the group of normal cycles by that of the cycles homologous to zero is called the reduced zero-dimensional homology group. For a connected complex the reduced group is zero, which is convenient in any kind of definition of acyclicity. A proper cycle $z^0=\{z_1^0,\dots,z_k^0,\dots\}$ is called normal if each of the cycles $z_k^0$ is normal.

How to Cite This Entry:
Normal zero-dimensional cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_zero-dimensional_cycle&oldid=33220
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article