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cellular homology (Definition)

If $X$ is a cell space, then let $(\mc{C}_*(X),\fr{d})$ be the cell complex where the $n$ -th group $\mc{C}_n(X)$ is the free abelian group on the cells of dimension $n$ , and the boundary map is as follows: If $e^n$ is an $n$ -cell, then we can define a map $\vp_f:\partial e^n\to f^{n-1}$ , where $f^{n-1}$ is any cell of dimension $n-1$ by the following rule: let $\vp:e^n\to\mathrm{sk}_{n-1} X$ be the attaching map for $e^n$ , where $\sk_{n-1}X$ is the $(n-1)$ -skeleton of $X$ . Then let $\pi_f$ be the natural projection $$\pi_f:\mathrm{sk}_{n-1}X\to \mathrm{sk}_{n-1} X/(\mathrm{sk}_{n-1} X-f)\cong f/\partial f.$$ Let $\vp_f=\pi_f\circ\vp$ . Now, $f/\partial f$ is a (n-1)-sphere, so the map $\vp_f$ has a degree $\deg f$ which we use to define the boundary operator: $$\fr{d}([e^n])=\sum_{\dim f=n-1}(\deg \vp_f)[f^{n-1}].$$

The resulting chain complex is called the cellular chain complex.

Theorem 1   The homology of the cellular complex is the same as the singular homology of the space. That is $$H_*(\mc{C},\fr d)=H_*(C,\partial).$$

Cellular homology is tremendously useful for computations because the groups involved are finitely generated.




"cellular homology" is owned by bwebste.
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Cross-references: finitely generated, complex, homology, chain complex, boundary operator, degree, projection, attaching map, map, boundary map, dimension, free abelian group, group, cell complex, cell
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This is version 3 of cellular homology, born on 2002-12-10, modified 2002-12-10.
Object id is 3725, canonical name is CellularHomology.
Accessed 5013 times total.

Classification:
AMS MSC55N10 (Algebraic topology :: Homology and cohomology theories :: Singular theory)

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