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

If [math]\displaystyle{ X }[/math] is a CW-complex with n-skeleton [math]\displaystyle{ X_{n} }[/math], the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex

[math]\displaystyle{ \cdots \to {C_{n + 1}}(X_{n + 1},X_{n}) \to {C_{n}}(X_{n},X_{n - 1}) \to {C_{n - 1}}(X_{n - 1},X_{n - 2}) \to \cdots, }[/math]

where [math]\displaystyle{ X_{-1} }[/math] is taken to be the empty set.

The group

[math]\displaystyle{ {C_{n}}(X_{n},X_{n - 1}) }[/math]

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

[math]\displaystyle{ \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}, }[/math]

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

The boundary map

[math]\displaystyle{ \partial_{n}: {C_{n}}(X_{n},X_{n - 1}) \to {C_{n - 1}}(X_{n - 1},X_{n - 2}) }[/math]

is then given by the formula

[math]\displaystyle{ {\partial_{n}}(e_{n}^{\alpha}) = \sum_{\beta} \deg \left( \chi_{n}^{\alpha \beta} \right) e_{n - 1}^{\beta}, }[/math]

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

Examples

The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

The n-sphere

The n-dimensional sphere Sn admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from [math]\displaystyle{ S^{n-1} }[/math] to 0-cell. Since the generators of the cellular chain groups [math]\displaystyle{ {C_{k}}(S^n_{k},S^{n}_{k - 1}) }[/math] can be identified with the k-cells of Sn, we have that [math]\displaystyle{ {C_{k}}(S^n_{k},S^{n}_{k - 1})=\Z }[/math] for [math]\displaystyle{ k = 0, n, }[/math] and is otherwise trivial.

Hence for [math]\displaystyle{ n\gt 1 }[/math], the resulting chain complex is

[math]\displaystyle{ \dotsb\overset{\partial_{n+2}}{\longrightarrow\,}0 \overset{\partial_{n+1}}{\longrightarrow\,}\Z \overset{\partial_n}{\longrightarrow\,}0 \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} 0 \overset{\partial_1}{\longrightarrow\,} \Z {\longrightarrow\,} 0, }[/math]

but then as all the boundary maps are either to or from trivial groups, they must all be zero, meaning that the cellular homology groups are equal to

[math]\displaystyle{ H_k(S^n) = \begin{cases} \mathbb Z & k=0, n \\ \{0\} & \text{otherwise.} \end{cases} }[/math]

When [math]\displaystyle{ n=1 }[/math], it is possible to verify that the boundary map [math]\displaystyle{ \partial_1 }[/math] is zero, meaning the above formula holds for all positive [math]\displaystyle{ n }[/math].

Genus g surface

Cellular homology can also be used to calculate the homology of the genus g surface [math]\displaystyle{ \Sigma_g }[/math]. The fundamental polygon of [math]\displaystyle{ \Sigma_g }[/math] is a [math]\displaystyle{ 4n }[/math]-gon which gives [math]\displaystyle{ \Sigma_g }[/math] a CW-structure with one 2-cell, [math]\displaystyle{ 2n }[/math] 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the [math]\displaystyle{ 4n }[/math]-gon, which contains every 1-cell twice, once forwards and once backwards. This means the attaching map is zero, since the forwards and backwards directions of each 1-cell cancel out. Similarly, the attaching map for each 1-cell is also zero, since it is the constant mapping from [math]\displaystyle{ S^0 }[/math] to the 0-cell. Therefore, the resulting chain complex is

[math]\displaystyle{ \cdots \to 0 \xrightarrow{\partial_3} \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}^{2g} \xrightarrow{\partial_1} \mathbb{Z} \to 0, }[/math]

where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by

[math]\displaystyle{ H_k(\Sigma_g) = \begin{cases} \mathbb{Z} & k = 0,2 \\ \mathbb{Z}^{2g} & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} }[/math]

Similarly, one can construct the genus g surface with a crosscap attached as a CW complex with 1 0-cell, g 1-cells, and 1 2-cell. Its homology groups are[math]\displaystyle{ H_k(\Sigma_g) = \begin{cases} \mathbb{Z} & k = 0 \\ \mathbb{Z}^{g-1} \oplus \Z_2 & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} }[/math]

Torus

The n-torus [math]\displaystyle{ (S^1)^n }[/math] can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is [math]\displaystyle{ 0\to \Z^{\binom{n}{n}} \to \Z^{\binom{n}{n-1}} \to \cdots \to \Z^{\binom{n}{1}} \to \Z^{\binom{n}{0}} \to 0 }[/math] and all the boundary maps are zero. This can be understood by explicitly constructing the cases for [math]\displaystyle{ n = 0, 1, 2, 3 }[/math], then see the pattern.

Thus, [math]\displaystyle{ H_k((S^1)^n) \simeq \Z^{\binom{n}{k}} }[/math] .

Complex projective space

If [math]\displaystyle{ X }[/math] has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then [math]\displaystyle{ H_n^{CW}(X) }[/math] is the free abelian group generated by its n-cells, for each [math]\displaystyle{ n }[/math].

The complex projective space [math]\displaystyle{ P^n\mathbb C }[/math] is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus [math]\displaystyle{ H_k(P^n\mathbb C) = \Z }[/math] for [math]\displaystyle{ k = 0, 2, ..., 2n }[/math], and zero otherwise.

Real projective space

The real projective space [math]\displaystyle{ \mathbb{R} P^n }[/math] admits a CW-structure with one [math]\displaystyle{ k }[/math]-cell [math]\displaystyle{ e_k }[/math] for all [math]\displaystyle{ k \in \{0, 1, \dots, n\} }[/math]. The attaching map for these [math]\displaystyle{ k }[/math]-cells is given by the 2-fold covering map [math]\displaystyle{ \varphi_k \colon S^{k - 1} \to \mathbb{R} P^{k - 1} }[/math]. (Observe that the [math]\displaystyle{ k }[/math]-skeleton [math]\displaystyle{ \mathbb{R} P^n_k \cong \mathbb{R} P^k }[/math] for all [math]\displaystyle{ k \in \{0, 1, \dots, n\} }[/math].) Note that in this case, [math]\displaystyle{ C_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \cong \mathbb{Z} }[/math] for all [math]\displaystyle{ k \in \{0, 1, \dots, n\} }[/math].

To compute the boundary map

[math]\displaystyle{ \partial_k \colon C_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \to C_{k - 1}(\mathbb{R} P^n_{k - 1}, \mathbb{R} P^n_{k - 2}), }[/math]

we must find the degree of the map

[math]\displaystyle{ \chi_k \colon S^{k - 1} \overset{\varphi_k}{\longrightarrow} \mathbb{R} P^{k - 1} \overset{q_k}{\longrightarrow} \mathbb{R} P^{k - 1}/\mathbb{R} P^{k - 2} \cong S^{k - 1}. }[/math]

Now, note that [math]\displaystyle{ \varphi_k^{-1}(\mathbb{R} P^{k - 2}) = S^{k - 2} \subseteq S^{k - 1} }[/math], and for each point [math]\displaystyle{ x \in \mathbb{R} P^{k - 1} \setminus \mathbb{R} P^{k - 2} }[/math], we have that [math]\displaystyle{ \varphi^{-1}(\{x\}) }[/math] consists of two points, one in each connected component (open hemisphere) of [math]\displaystyle{ S^{k - 1}\setminus S^{k - 2} }[/math]. Thus, in order to find the degree of the map [math]\displaystyle{ \chi_k }[/math], it is sufficient to find the local degrees of [math]\displaystyle{ \chi_k }[/math] on each of these open hemispheres. For ease of notation, we let [math]\displaystyle{ B_k }[/math] and [math]\displaystyle{ \tilde B_k }[/math] denote the connected components of [math]\displaystyle{ S^{k - 1}\setminus S^{k - 2} }[/math]. Then [math]\displaystyle{ \chi_k|_{B_k} }[/math] and [math]\displaystyle{ \chi_k|_{\tilde B_k} }[/math] are homeomorphisms, and [math]\displaystyle{ \chi_k|_{\tilde B_k} = \chi_k|_{B_k} \circ A }[/math], where [math]\displaystyle{ A }[/math] is the antipodal map. Now, the degree of the antipodal map on [math]\displaystyle{ S^{k - 1} }[/math] is [math]\displaystyle{ (-1)^k }[/math]. Hence, without loss of generality, we have that the local degree of [math]\displaystyle{ \chi_k }[/math] on [math]\displaystyle{ B_k }[/math] is [math]\displaystyle{ 1 }[/math] and the local degree of [math]\displaystyle{ \chi_k }[/math] on [math]\displaystyle{ \tilde B_k }[/math] is [math]\displaystyle{ (-1)^k }[/math]. Adding the local degrees, we have that

[math]\displaystyle{ \deg(\chi_k) = 1 + (-1)^k = \begin{cases} 2 & \text{if } k \text{ is even,} \\ 0 & \text{if } k \text{ is odd.} \end{cases} }[/math]

The boundary map [math]\displaystyle{ \partial_k }[/math] is then given by [math]\displaystyle{ \deg(\chi_k) }[/math].

We thus have that the CW-structure on [math]\displaystyle{ \mathbb{R} P^n }[/math] gives rise to the following chain complex:

[math]\displaystyle{ 0 \longrightarrow \mathbb{Z} \overset{\partial_n}{\longrightarrow} \cdots \overset{2}{\longrightarrow} \mathbb{Z} \overset{0}{\longrightarrow} \mathbb{Z} \overset{2}{\longrightarrow} \mathbb{Z} \overset{0}{\longrightarrow} \mathbb{Z} \longrightarrow 0, }[/math]

where [math]\displaystyle{ \partial_n = 2 }[/math] if [math]\displaystyle{ n }[/math] is even and [math]\displaystyle{ \partial_n = 0 }[/math] if [math]\displaystyle{ n }[/math] is odd. Hence, the cellular homology groups for [math]\displaystyle{ \mathbb{R} P^n }[/math] are the following:

[math]\displaystyle{ H_k(\mathbb{R} P^n) = \begin{cases} \mathbb{Z} & \text{if } k = 0 \text{ and } k=n \text{ odd}, \\ \mathbb{Z}/2\mathbb{Z} & \text{if } 0 \lt k \lt n \text{ odd,} \\ 0 & \text{otherwise.} \end{cases} }[/math]

Other properties

One sees from the cellular chain complex that the [math]\displaystyle{ n }[/math]-skeleton determines all lower-dimensional homology modules:

[math]\displaystyle{ {H_{k}}(X) \cong {H_{k}}(X_{n}) }[/math]

for [math]\displaystyle{ k \lt n }[/math].

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 [math]\displaystyle{ \mathbb{CP}^{n} }[/math] has a cell structure with one cell in each even dimension; it follows that for [math]\displaystyle{ 0 \leq k \leq n }[/math],

[math]\displaystyle{ {H_{2 k}}(\mathbb{CP}^{n};\mathbb{Z}) \cong \mathbb{Z} }[/math]

and

[math]\displaystyle{ {H_{2 k + 1}}(\mathbb{CP}^{n};\mathbb{Z}) = 0. }[/math]

Generalization

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

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

[math]\displaystyle{ \chi(X) = \sum_{j = 0}^{n} (-1)^{j} c_{j}. }[/math]

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of [math]\displaystyle{ X }[/math],

[math]\displaystyle{ \chi(X) = \sum_{j = 0}^{n} (-1)^{j} \operatorname{Rank}({H_{j}}(X)). }[/math]

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

[math]\displaystyle{ \cdots \to {H_{i}}(X_{n - 1},\varnothing) \to {H_{i}}(X_{n},\varnothing) \to {H_{i}}(X_{n},X_{n - 1}) \to \cdots. }[/math]

Chasing exactness through the sequence gives

[math]\displaystyle{ \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)). }[/math]

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

[math]\displaystyle{ \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}. }[/math]

References



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