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$n$-chain (Definition)

Let $ X$ be a topological space and let $ K$ be a simplicial approximation to $ X$. An $ n$-chain on $ X$ is a finite formal sum of oriented $ n$-simplices in $ K$. The group of such chains is denoted by $ C_n(X)$ and is called the $ n$th chain group of $ X$. In other words, $ C_n(X)$ is the free abelian group generated by the oriented $ n$-simplices in $ K$.

We have defined chain groups for simplicial homology. Their definition is similar in singular homology and the homology of CW complexes. For example, if $ Y$ is a CW complex, then its $ n$th chain group is the free abelian group on the cells of $ Y^n$, the $ n$-skeleton of $ Y$.

The formal boundary of an oriented $ n$-simplex $ \sigma=(v_0,\dots,v_n)$ is given by the alternating sum of the oriented $ n$-simplices forming the topological boundary of $ \sigma$, that is,

$\displaystyle \partial_n(\sigma) = \sum_{j=0}^n (-1)^j (v_0,\dots, v_{j-1},v_{j+1},\dots, v_n). $
The boundary of a 0-simplex is 0.

Since $ n$-simplices form a basis for the chain group $ C_n(X)$, this extends to give a group homomorphism $ \partial_n\colon C_n(X)\to C_{n-1}(X)$, called the boundary map. An $ n$-chain is closed if its boundary is 0 and exact if it is the boundary of some $ (n+1)$-chain. Closed $ n$-chains are also called cycles. Every exact $ n$-chain is also closed. This implies that the sequence

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \cdots \ar[r] & C_{n+1}(X)\ar[r]... ...artial_{n+1}} & C_n(X)\ar[r]^{\partial_n} & C_{n-1}\ar[r] & \cdots } } \end{xy}$
is a complex of free abelian groups. This complex is usually called the chain complex of $ X$ corresponding to the simplicial complex $ K$. Note that while the chain groups $ C_n(X)$ depend on the choice of simplicial approximation $ K$, the resulting homology groups
$\displaystyle H_n(X) = \frac{\ker\partial_n}{{\mathrm{im}}\,\partial_{n+1}} $
do not.



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"$n$-chain" is owned by mps. [ full author list (3) | owner history (1) ]
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See Also: long exact sequence (of homology groups)

Other names:  chain
Also defines:  closed n-chain, exact n-chain, boundary map
Keywords:  chain complex, cell
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Cross-references: homology groups, simplicial complex, chain complex, complex, sequence, implies, group homomorphism, basis, topological boundary, alternating sum, cells, CW complexes, singular homology, similar, simplicial homology, generated by, free abelian group, group, oriented, sum, finite, simplicial approximation, topological space
There are 12 references to this entry.

This is version 8 of $n$-chain, born on 2003-07-18, modified 2007-03-09.
Object id is 4478, canonical name is NChain.
Accessed 4640 times total.

Classification:
AMS MSC: 16E05 (Associative rings and algebras :: Homological methods :: Syzygies, resolutions, complexes)
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