Let be a topological space and let be a simplicial approximation to . An -chain on is a finite formal sum of oriented -simplices in . The group of such chains is denoted by and is called the th chain group of . In other words, is the free abelian group generated by the oriented -simplices in .
We have defined chain groups for simplicial homology. Their definition is similar in singular homology and the homology of CW complexes. For example, if is a CW complex, then its th chain group is the free abelian group on the cells of , the -skeleton of .
The formal boundary of an oriented -simplex is given by the alternating sum of the oriented -simplices forming the topological boundary of , that is,
The boundary of a -simplex is .
Since -simplices form a basis for the chain group , this extends to give a group homomorphism , called the boundary map. An -chain is closed if its boundary is 0 and exact if it is the boundary of some -chain. Closed -chains are also called cycles. Every exact -chain is also closed. This implies that the sequence
is a complex of free abelian groups. This complex is usually called the chain complex of corresponding to the simplicial complex . Note that while the chain groups depend on the choice of simplicial approximation , the resulting homology groups
|Date of creation||2013-03-22 13:46:20|
|Last modified on||2013-03-22 13:46:20|
|Last modified by||mps (409)|