Cellular homology

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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[edit]

If  X is a CW-complex with n-skeleton  X_{n} , the cellular-homology modules are defined as the homology groups of the cellular chain complex


\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,

where  X_{-1} is taken to be the empty set.

The group


{H_{n}}(X_{n},X_{n - 1})

is free abelian, with generators that can be identified with the  n -cells of  X . Let  e_{n}^{\alpha} be an  n -cell of  X , and let  \chi_{n}^{\alpha}: \partial e_{n}^{\alpha} \cong \mathbb{S}^{n - 1} \to X_{n-1} be the attaching map. Then consider the composition


\chi_{n}^{\alpha \beta}:
\mathbb{S}^{n - 1}                                               \, \stackrel{\cong}{\longrightarrow} \,
\partial e_{n}^{\alpha}                                          \, \stackrel{\chi_{n}^{\alpha}}{\longrightarrow} \,
X_{n - 1}                                                        \, \stackrel{q}{\longrightarrow} \,
X_{n - 1} / \left( X_{n - 1} \setminus e_{n - 1}^{\beta} \right) \, \stackrel{\cong}{\longrightarrow} \,
\mathbb{S}^{n - 1},

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

The boundary map


d_{n}: {H_{n}}(X_{n},X_{n - 1}) \to {H_{n - 1}}(X_{n - 1},X_{n - 2})

is then given by the formula


{d_{n}}(e_{n}^{\alpha}) = \sum_{\beta} \deg \left( \chi_{n}^{\alpha \beta} \right) e_{n - 1}^{\beta},

where  \deg \left( \chi_{n}^{\alpha \beta} \right) is the degree of  \chi_{n}^{\alpha \beta} and the sum is taken over all  (n - 1) -cells of  X , considered as generators of  {H_{n - 1}}(X_{n - 1},X_{n - 2}) .

Other Properties[edit]

One sees from the cellular-chain complex that the  n -skeleton determines all lower-dimensional homology modules:


{H_{k}}(X) \cong {H_{k}}(X_{n})

for  k < n .

An important consequence of this cellular perspective is that if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space  \mathbb{CP}^{n} has a cell structure with one cell in each even dimension; it follows that for  0 \leq k \leq n ,


{H_{2 k}}(\mathbb{CP}^{n};\mathbb{Z}) \cong \mathbb{Z}

and


{H_{2 k + 1}}(\mathbb{CP}^{n};\mathbb{Z}) = 0.

Generalization[edit]

The Atiyah-Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

Euler Characteristic[edit]

For a cellular complex  X , let  X_{j} be its  j -th skeleton, and  c_{j} be the number of  j -cells, i.e., the rank of the free module  {H_{j}}(X_{j},X_{j - 1}) . The Euler characteristic of  X is then defined by


\chi(X) = \sum_{j = 0}^{n} (-1)^{j} c_{j}.

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of  X ,


\chi(X) = \sum_{j = 0}^{n} (-1)^{j} \operatorname{Rank}({H_{j}}(X)).

This can be justified as follows. Consider the long exact sequence of relative homology for the triple  (X_{n},X_{n - 1},\varnothing) :


\cdots \to {H_{i}}(X_{n - 1},\varnothing) \to {H_{i}}(X_{n},\varnothing) \to {H_{i}}(X_{n},X_{n - 1}) \to \cdots.

Chasing exactness through the sequence gives


  \sum_{i = 0}^{n} (-1)^{i} \operatorname{Rank}({H_{i}}(X_{n},\varnothing)
= \sum_{i = 0}^{n} (-1)^{i} \operatorname{Rank}({H_{i}}(X_{n},X_{n - 1})) +
  \sum_{i = 0}^{n} (-1)^{i} \operatorname{Rank}({H_{i}}(X_{n - 1},\varnothing)).

The same calculation applies to the triples  (X_{n - 1},X_{n - 2},\varnothing) ,  (X_{n - 2},X_{n - 3},\varnothing) , etc. By induction,


  \sum_{i = 0}^{n} (-1)^{i} \; \operatorname{Rank}({H_{i}}(X_{n},\varnothing))
= \sum_{j = 0}^{n} \sum_{i = 0}^{j} (-1)^{i} \operatorname{Rank}({H_{i}}(X_{j},X_{j - 1}))
= \sum_{j = 0}^{n} (-1)^{j} c_{j}.

References[edit]