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Cellular homology : ウィキペディア英語版
Cellular homology
In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.
== Definition ==

If X is a CW-complex with n-skeleton X_ , the cellular-homology modules are defined as the homology groups of the cellular chain complex
:
\cdots \to ,X_) \to ,X_) \to ,X_) \to \cdots,

where X_ is taken to be the empty set.
The group
:
,X_)

is free abelian, with generators that can be identified with the n -cells of X . Let e_^ be an n -cell of X , and let \chi_^: \partial e_^ \cong \mathbb^ \to X_ be the attaching map. Then consider the composition
:
\chi_^:
\mathbb^ \, \stackrel \,
\partial e_^ \, \stackrel} \,
X_ \, \stackrel \,
X_ / \left( X_ \setminus e_^ \right) \, \stackrel \,
\mathbb^,

where the first map identifies \mathbb^ with \partial e_^ via the characteristic map \Phi_^ of e_^ , the object e_^ is an (n - 1) -cell of ''X'', the third map q is the quotient map that collapses X_ \setminus e_^ to a point (thus wrapping e_^ into a sphere \mathbb^ ), and the last map identifies X_ / \left( X_ \setminus e_^ \right) with \mathbb^ via the characteristic map \Phi_^ of e_^ .
The boundary map
:
d_: ,X_) \to ,X_)

is then given by the formula
:
^) = \sum_ \deg \left( \chi_^ \right) e_^,

where \deg \left( \chi_^ \right) is the degree of \chi_^ and the sum is taken over all (n - 1) -cells of X , considered as generators of ,X_) .

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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