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| In [[mathematics]], '''cellular homology''' in [[algebraic topology]] is a [[homology theory]] for [[CW-complex]]es. It agrees with [[singular homology]], and can provide an effective means of computing homology modules.
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| == Definition ==
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| If ''X'' is a CW-complex with [[n-skeleton]] ''X<sub>n</sub>'', the cellular homology modules are defined as the [[homology group]]s of the cellular [[chain complex]]
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| :<math> \cdots \to H_{n+1}( X_{n+1}, X_n ) \to H_n( X_n, X_{n-1} ) \to H_{n-1}( X_{n-1}, X_{n-2} ) \to \cdots . </math>
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| [<math>X_{-1}</math> is the empty set]
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| The group
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| :<math>H_n( X_n, X_{n-1} ) \,</math>
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| is [[Free module|free]], with generators which can be identified with the ''n''-cells of ''X''. Let <math>e_n^{\alpha}</math> be an ''n''-cell of ''X'', let <math>\chi_n^{\alpha} : \partial e_n^{\alpha}\cong S^{n-1} \to X_{n-1}</math> be the attaching map, and consider the composite maps
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| :<math>\chi_n^{\alpha\beta}:S^{n-1} \to X_{n-1} \to X_{n-1}/(X_{n-1}-e_{n-1}^{\beta})\cong S^{n-1}</math>
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| where <math>e_{n-1}^{\beta}</math> is an <math>(n-1)</math>-cell of ''X'' and the second map is the quotient map identifying <math>(X_{n-1}-e_{n-1}^{\beta})</math> to a point.
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| The [[boundary map]]
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| :<math>d_n:H_n(X_n,X_{n-1}) \to H_{n-1}(X_{n-1},X_{n-2}) \,</math> | |
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| is then given by the formula
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| :<math>d_n(e_n^{\alpha})=\sum_{\beta}\deg(\chi_n^{\alpha\beta})e_{n-1}^{\beta}\, </math>
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| where <math>deg(\chi_n^{\alpha\beta})</math> is the [[Degree of a continuous mapping|degree]] of <math>\chi_n^{\alpha\beta}</math> and the sum is taken over all <math>(n-1)</math>-cells of ''X'', considered as generators of <math>H_{n-1}(X_{n-1},X_{n-2})\,</math>.
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| == Other properties ==
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| One sees from the cellular chain complex that the ''n''-skeleton determines all lower-dimensional homology:
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| :<math>H_k(X) \cong H_k(X_n) </math>
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| for ''k'' < ''n''.
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| An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example, [[complex projective space]] '''CP'''<sup>''n''</sup> has a cell structure with one cell in each even dimension; it follows that for 0 ≤ ''k'' ≤ ''n'',
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| :<math> H_{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} </math>
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| and
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| :<math> H_{2k+1}(\mathbb{CP}^n) = 0 .</math>
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| == Generalization ==
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| The [[Atiyah–Hirzebruch spectral sequence|Atiyah-Hirzebruch spectral sequence]] is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary [[Extraordinary homology theory|extraordinary (co)homology theory]].
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| == Euler characteristic ==
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| For a cellular complex ''X'', let ''X<sub>j</sub>'' be its ''j''-th skeleton, and ''c<sub>j</sub>'' be the number of ''j''-cells, i.e. the rank of the free module ''H<sub>j</sub>''(''X<sub>j</sub>'', ''X''<sub>''j''-1</sub>). The [[Euler characteristic]] of ''X'' is defined by
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| :<math>\chi (X) = \sum _0 ^n (-1)^j c_j.</math>
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| The Euler characteristic is a homotopy invariant. In fact, in terms of the [[Betti number]]s of ''X'',
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| :<math>\chi (X) = \sum _0 ^n (-1)^j \; \mbox{rank} \; H_j (X). </math>
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| This can be justified as follows. Consider the long exact sequence of [[relative homology]] for the triple (''X<sub>n</sub>'', ''X''<sub>''n'' - 1 </sub>, ∅):
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| :<math> \cdots \to H_i( X_{n-1}, \empty) \to H_i( X_n, \empty) \to H_i( X_{n}, X_{n-1} ) \to \cdots . </math>
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| Chasing exactness through the sequence gives
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| :<math>
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| \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty)
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| = \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, X_{n-1}) \; + \; \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_{n-1}, \empty).</math>
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| The same calculation applies to the triple (''X''<sub>''n'' - 1</sub>, ''X''<sub>''n'' - 2</sub>, ∅), etc. By induction,
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| :<math>
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| \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty)
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| = \sum_{j = 0} ^n \; \sum_{i = 0} ^j (-1)^i \; \mbox{rank} \; H_i (X_j, X_{j-1})
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| = \sum_{j = 0} ^n (-1)^j c_j.</math>
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| ==References==
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| * A. Dold: ''Lectures on Algebraic Topology'', Springer ISBN 3-540-58660-1.
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| * A. Hatcher: ''Algebraic Topology'', Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the [http://www.math.cornell.edu/~hatcher/ author's homepage].
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| [[Category:Homology theory]]
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