Singular chain complex

Definition

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

• The $n^{th}$ member of this complex is the $n^{th}$ 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 $n^{th}$ chain group to the $(n-1{)}^{th}$ chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.