Singular chain complex: Difference between revisions

Revision as of 23:25, 30 September 2007

The singular complex associated with a topological space is defined as the following chain complex of Abelian groups:

The $n^{t h}$ member of this complex is the $n^{t h}$ chain group, or the group of singular n-chains. This is essentially the free Abelian group on the set of all singular $n$ -simplices.

The boundary map goes from the $n^{t h}$ chain group to the $(n - 1)^{t h}$ chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.

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 The '''singular complex''' associated with a topological space is defined as the following chain complex of Abelian groups:
-* The <math>n^{th}</math> member of this complex is the <math>n^{th}</math> chain group, or the group of [[singular chain|singular <math>n</math>-chains]]. This is essentially the free Abelian group on the set of all [[singular simplex|singular <math>n</math>-simplices]].
+* The <math>n^{th}</math> member of this complex is the <math>n^{th}</math> chain group, or the group of [[singular chain|singular n-chains]]. This is essentially the free Abelian group on the set of all [[singular simplex|singular <math>n</math>-simplices]].
 * The '''boundary map''' goes from the <math>n^{th}</math> chain group to the <math>(n-1)^{th}</math> chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.