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cellular homology (Definition)

If $ X$ is a cell space, then let $ (\mathcal{C}_*(X),\mathfrak{d})$ be the cell complex where the $ n$-th group $ \mathcal{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 $ \varphi _f:\partial e^n\to f^{n-1}$, where $ f^{n-1}$ is any cell of dimension $ n-1$ by the following rule: let $ \varphi :e^n\to\mathrm{sk}_{n-1} X$ be the attaching map for $ e^n$, where $ \mathrm{sk}_{n-1}X$ is the $ (n-1)$-skeleton of $ X$. Then let $ \pi_f$ be the natural projection

$\displaystyle \pi_f:\mathrm{sk}_{n-1}X\to \mathrm{sk}_{n-1} X/(\mathrm{sk}_{n-1} X-f)\cong f/\partial f.$
Let $ \varphi _f=\pi_f\circ\varphi $. Now, $ f/\partial f$ is a (n-1)-sphere, so the map $ \varphi _f$ has a degree $ \deg f$ which we use to define the boundary operator:
$\displaystyle \mathfrak{d}([e^n])=\sum_{\dim f=n-1}(\deg \varphi _f)[f^{n-1}].$

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
$\displaystyle H_*(\mathcal{C},\mathfrak{d})=H_*(C,\partial).$

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
There are 2 references to this entry.

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 3608 times total.

Classification:
AMS MSC: 55N10 (Algebraic topology :: Homology and cohomology theories :: Singular theory)
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