Cellular homology

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> X </math> is a CW-complex with n-skeleton <math> X_{n} </math>, the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex

:<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>

where <math> X_{-1} </math> is taken to be the empty set.

The group

:<math>

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

</math>

is free abelian, with generators that can be identified with the <math> n </math>-cells of <math> X </math>. Let <math> e_{n}^{\alpha} </math> be an <math> n </math>-cell of <math> X </math>, and let <math> \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>

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

The boundary map

:<math>

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

</math>

is then given by the formula

:<math>

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

</math>

where <math> \deg \left( \chi_{n}^{\alpha \beta} \right) </math> is the degree of <math> \chi_{n}^{\alpha \beta} </math> and the sum is taken over all <math> (n - 1) </math>-cells of <math> X </math>, considered as generators of <math> {H_{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>S^{n-1}</math> to the 0-cell. Since the generators of the cellular chain groups <math>{H_{k(S^n_{k},S^{n}_{k - 1})</math> can be identified with the k-cells of Sn, we have that <math>{H_{k(S^n_{k},S^{n}_{k - 1})=\Z</math> for <math>k = 0, n,</math> and is otherwise trivial.

Hence for <math>n>1</math>, the resulting chain complex is

:<math>\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\,}

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

When <math>n=1</math>, it is possible to verify that the boundary map <math>\partial_1</math> is zero, meaning the above formula holds for all positive <math>n</math>.

Genus g surface

Cellular homology can also be used to calculate the homology of the genus g surface <math>\Sigma_g</math>. The fundamental polygon of <math>\Sigma_g</math> is a <math>4g</math>-gon which gives <math>\Sigma_g</math> a CW-structure with one 2-cell, <math>2g</math> 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the <math>4g</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>S^0</math> to the 0-cell. Therefore, the resulting chain complex is

:<math>

\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>

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 (a non-orientable genus g surface) as a CW complex with one 0-cell, g 1-cells <math>\{a_1, \dotsm, a_g\} </math>, and one 2-cell which is attached along the word <math> a_1^1\dotsm a_g^2 </math>. Therefore, the resulting chain complex is:

<math>

\cdots \to 0 \xrightarrow{\partial_3} \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}^{g} \xrightarrow{\partial_1} \mathbb{Z} \to 0,

</math>

where the boundary maps are <math>\partial_3=\partial_1 =0</math> and <math>\partial_2(1)=2a_1+2a_2+\dotsm + 2a_g = 2(a_1+\dotsm+a_g)</math>.

Its homology groups are<math display="block">

H_k(N_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>(S^1)^n</math> can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell. The chain complex is <math display="block">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>n = 0, 1, 2, 3</math>, then see the pattern.

Thus, <math>H_k((S^1)^n) \simeq \Z^{\binom{n}{k</math> .

Complex projective space

If <math>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>H_n^{CW}(X)</math> is the free abelian group generated by its n-cells, for each <math>n</math>.

The complex projective space <math>\mathbb CP^n</math> is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus <math>H_k(\mathbb CP^n) = \Z</math> for <math>k = 0, 2, ..., 2n</math>, and zero otherwise.

Real projective space

The real projective space <math>\mathbb{R} P^n</math> admits a CW-structure with one <math>k</math>-cell <math>e_k</math> for all <math>k \in \{0, 1, \dots, n\}</math>.

The attaching map for these <math>k</math>-cells is given by the 2-fold covering map <math>\varphi_k \colon S^{k - 1} \to \mathbb{R} P^{k - 1}</math>.

(Observe that the <math>k</math>-skeleton <math>\mathbb{R} P^n_k \cong \mathbb{R} P^k</math> for all <math>k \in \{0, 1, \dots, n\}</math>.)

Note that in this case, <math>H_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \cong \mathbb{Z}</math> for all <math>k \in \{0, 1, \dots, n\}</math>.

To compute the boundary map

:<math>

\partial_k \colon H_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \to H_{k - 1}(\mathbb{R} P^n_{k - 1}, \mathbb{R} P^n_{k - 2}),

</math>

we must find the degree of the map

:<math>

\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>\varphi_k^{-1}(\mathbb{R} P^{k - 2}) = S^{k - 2} \subseteq S^{k - 1}</math>, and for each point <math>x \in \mathbb{R} P^{k - 1} \setminus \mathbb{R} P^{k - 2}</math>, we have that <math>\varphi^{-1}(\{x\})</math> consists of two points, one in each connected component (open hemisphere) of <math>S^{k - 1}\setminus S^{k - 2}</math>.

Thus, in order to find the degree of the map <math>\chi_k</math>, it is sufficient to find the local degrees of <math>\chi_k</math> on each of these open hemispheres.

For ease of notation, we let <math>B_k</math> and <math>\tilde B_k</math> denote the connected components of <math>S^{k - 1}\setminus S^{k - 2}</math>.

Then <math>\chi_k|_{B_k}</math> and <math>\chi_k|_{\tilde B_k}</math> are homeomorphisms, and <math>\chi_k|_{\tilde B_k} = \chi_k|_{B_k} \circ A</math>, where <math>A</math> is the antipodal map.

Now, the degree of the antipodal map on <math>S^{k - 1}</math> is <math>(-1)^k</math>.

Hence, without loss of generality, we have that the local degree of <math>\chi_k</math> on <math>B_k</math> is <math>1</math> and the local degree of <math>\chi_k</math> on <math>\tilde B_k</math> is <math>(-1)^k</math>.

Adding the local degrees, we have that

:<math>

\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>\partial_k</math> is then given by <math>\deg(\chi_k)</math>.

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

:<math>

\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

</math>

where <math>\partial_n = 2</math> if <math>n</math> is even and <math>\partial_n = 0</math> if <math>n</math> is odd.

Hence, the cellular homology groups for <math>\mathbb{R} P^n</math> are the following:

:<math>

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 < k < n \text{ odd,}

0 & \text{otherwise.}

\end{cases}

</math>

Functoriality

Cellular homology is a functor from the category of CW complexes with cellular maps to the category of abelian groups. A cellular map <math>f:X\to Y</math> gives a map of pairs <math>f:(X_n,X_{n-1})\to(Y_n,Y_{n-1})</math> for all <math>n</math>, and thus induces a map <math>f_*:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1})</math> between the cellular chain groups of <math>X</math> and <math>Y</math>. That <math>f_*</math> is a chain map follows from the naturality of the long exact sequence of a pair. Hence <math>f_*</math> is a map between the cellular homology groups of <math>X</math> and <math>Y</math>.

The formula presented below allows one to compute the chain map <math>f_*</math> in terms of the degrees of certain maps, similarly to the formula above for the boundary map in the cellular chain complex. Let <math>e_n^\alpha</math> be an <math>n</math>-cell of <math>X</math> and <math>e_n^\beta</math> be an <math>n</math>-cell of <math>Y</math>.

Consider the composition

:<math>

f_\beta^\alpha:

S^n \,

\stackrel{\cong}{\longrightarrow} \,

D^n/S^{n-1} \,

\stackrel{\bar\Phi_n^\alpha}{\longrightarrow} \,

X_n/X_{n-1} \,

\stackrel{\bar{f{\longrightarrow} \,

Y_n/(Y_n\setminus e_n^\beta) \,

\stackrel{\cong}{\longrightarrow} \,

S^n,

</math>

where <math>\bar\Phi_n^\alpha</math> is the quotient map obtained from the characteristic map of <math>e_n^\alpha</math>, and <math>\bar{f}</math> is the quotient map induced by the composition <math>X_n \, \stackrel{f}{\longrightarrow} \, Y_n\to Y_n/(Y_n\setminus e_n^\beta)</math>. The last map comes from the characteristic map of <math>e_n^\beta</math>.

Then the chain map <math>f_*:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1})</math> is determined by the formula

:<math>

f_*(e_n^\alpha)=\sum_\beta \deg(f_\beta^\alpha)e_n^\beta,

</math>

where the summation takes place over all <math>n</math>-cells of <math>Y</math>.

Other properties

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

:<math>

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

</math>

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

:<math>

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

</math>

and

:<math>

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

:<math>

\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> X </math>,

:<math>

\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> (X_{n},X_{n - 1},\varnothing) </math>:

:<math>

\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>

\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> (X_{n - 1},X_{n - 2},\varnothing) </math>, <math> (X_{n - 2},X_{n - 3},\varnothing) </math>, etc. By induction,

:<math>

\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

Notes

General References

Albrecht Dold: Lectures on Algebraic Topology, Springer .
Allen Hatcher: Algebraic Topology, Cambridge University Press . A free electronic version is available on the author's homepage.