Difference between revisions of "Normal zero-dimensional cycle"
From Encyclopedia of Mathematics
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| − | A [[Cycle|cycle]] | + | 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=16391
Normal zero-dimensional cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_zero-dimensional_cycle&oldid=16391
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