Cellular homology

If X is a CW-complex with n-skeleton X_{n} , the cellular-homology modules are defined as the homology groups Hi 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 : \partial_{n}: {H_{n}}(X_{n},X_{n - 1}) \to {H_{n - 1}}(X_{n - 1},X_{n - 2}) is then given by the formula : {\partial_{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}) . == 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 S^{n-1} to the 0-cell. Since the generators of the cellular chain groups {H_{k}}(S^n_{k},S^{n}_{k - 1}) can be identified with the k-cells of Sn, we have that {H_{k}}(S^n_{k},S^{n}_{k - 1})=\Z for k = 0, n, and is otherwise trivial. Hence for n>1, the resulting chain complex is :\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, 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 :H_k(S^n) = \begin{cases} \mathbb Z & k=0, n \\ \{0\} & \text{otherwise.} \end{cases} When n=1, it is possible to verify that the boundary map \partial_1 is zero, meaning the above formula holds for all positive n. Genus g surface Cellular homology can also be used to calculate the homology of the genus g surface \Sigma_g. The fundamental polygon of \Sigma_g is a 4n-gon which gives \Sigma_g a CW-structure with one 2-cell, 2n 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the 4n-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 S^0 to the 0-cell. Therefore, the resulting chain complex is : \cdots \to 0 \xrightarrow{\partial_3} \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}^{2g} \xrightarrow{\partial_1} \mathbb{Z} \to 0, where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by : H_k(\Sigma_g) = \begin{cases} \mathbb{Z} & k = 0,2 \\ \mathbb{Z}^{2g} & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} 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 \{a_1, \dotsm, a_g\} , and one 2-cell which is attached along the word a_1^1\dotsm a_g^2 . Therefore, the resulting chain complex is: \cdots \to 0 \xrightarrow{\partial_3} \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}^{g} \xrightarrow{\partial_1} \mathbb{Z} \to 0, where the boundary maps are \partial_3=\partial_1 =0 and \partial_2(1)=2a_1+2a_2+\dotsm + 2a_g = 2(a_1+\dotsm+a_g). Its homology groups are H_k(N_g) = \begin{cases} \mathbb{Z} & k = 0 \\ \mathbb{Z}^{g-1} \oplus \Z_2 & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} Torus The n-torus (S^1)^n can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell. The chain complex is 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 and all the boundary maps are zero. This can be understood by explicitly constructing the cases for n = 0, 1, 2, 3, then see the pattern. Thus, H_k((S^1)^n) \simeq \Z^{\binom{n}{k}} . Complex projective space If X has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then H_n^{CW}(X) is the free abelian group generated by its n-cells, for each n. The complex projective space \mathbb CP^n is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus H_k(\mathbb CP^n) = \Z for k = 0, 2, ..., 2n, and zero otherwise. Real projective space The real projective space \mathbb{R} P^n admits a CW-structure with one k-cell e_k for all k \in \{0, 1, \dots, n\}. The attaching map for these k-cells is given by the 2-fold covering map \varphi_k \colon S^{k - 1} \to \mathbb{R} P^{k - 1}. (Observe that the k-skeleton \mathbb{R} P^n_k \cong \mathbb{R} P^k for all k \in \{0, 1, \dots, n\}.) Note that in this case, H_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \cong \mathbb{Z} for all k \in \{0, 1, \dots, n\}. To compute the boundary map : \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}), we must find the degree of the map : \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}. Now, note that \varphi_k^{-1}(\mathbb{R} P^{k - 2}) = S^{k - 2} \subseteq S^{k - 1}, and for each point x \in \mathbb{R} P^{k - 1} \setminus \mathbb{R} P^{k - 2}, we have that \varphi^{-1}(\{x\}) consists of two points, one in each connected component (open hemisphere) of S^{k - 1}\setminus S^{k - 2}. Thus, in order to find the degree of the map \chi_k, it is sufficient to find the local degrees of \chi_k on each of these open hemispheres. For ease of notation, we let B_k and \tilde B_k denote the connected components of S^{k - 1}\setminus S^{k - 2}. Then \chi_k|_{B_k} and \chi_k|_{\tilde B_k} are homeomorphisms, and \chi_k|_{\tilde B_k} = \chi_k|_{B_k} \circ A, where A is the antipodal map. Now, the degree of the antipodal map on S^{k - 1} is (-1)^k. Hence, without loss of generality, we have that the local degree of \chi_k on B_k is 1 and the local degree of \chi_k on \tilde B_k is (-1)^k. Adding the local degrees, we have that : \deg(\chi_k) = 1 + (-1)^k = \begin{cases} 2 & \text{if } k \text{ is even,} \\ 0 & \text{if } k \text{ is odd.} \end{cases} The boundary map \partial_k is then given by \deg(\chi_k). We thus have that the CW-structure on \mathbb{R} P^n gives rise to the following chain complex: : 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, where \partial_n = 2 if n is even and \partial_n = 0 if n is odd. Hence, the cellular homology groups for \mathbb{R} P^n are the following: : 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 == Functoriality ==

Cellular homology is a functor from the category of CW complexes with cellular maps to the category of abelian groups. A cellular map f:X\to Y gives a map of pairs f:(X_n,X_{n-1})\to(Y_n,Y_{n-1}) for all n, and thus induces a map f_*:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1}) between the cellular chain groups of X and Y. That f_* is a chain map follows from the naturality of the long exact sequence of a pair. Hence f_* is a map between the cellular homology groups of X and Y. The formula presented below allows one to compute the chain map f_* in terms of the degrees of certain maps, similarly to the formula above for the boundary map in the cellular chain complex. Let e_n^\alpha be an n-cell of X and e_n^\beta be an n-cell of Y. Consider the composition : 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, where \bar\Phi_n^\alpha is the quotient map obtained from the characteristic map of e_n^\alpha, and \bar{f} is the quotient map induced by the composition X_n \, \stackrel{f}{\longrightarrow} \, Y_n\to Y_n/(Y_n\setminus e_n^\beta). The last map comes from the characteristic map of e_n^\beta. Then the chain map f_*:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1}) is determined by the formula : f_*(e_n^\alpha)=\sum_\beta \deg(f_\beta^\alpha)e_n^\beta, where the summation takes place over all n-cells of Y. == Other properties ==

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 . 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 ==

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 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==