Editing CW complex (section)

==Homology and cohomology of CW complexes==

[[Singular homology]] and [[singular cohomology|cohomology]] of CW complexes is readily computable via [[cellular homology]].  Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a [[homology theory]]. To compute an [[cohomology#Generalized cohomology theories|extraordinary (co)homology theory]] for a CW complex, the [[Atiyah–Hirzebruch spectral sequence]] is the analogue of cellular homology.

Some examples:

* For the sphere, <math>S^n,</math> take the cell decomposition with two cells: a single 0-cell and a single ''n''-cell. The cellular homology [[chain complex]] <math>C_*</math> and homology are given by:
::<math>C_k = \begin{cases} \Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{cases} \quad H_k =  \begin{cases} \Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{cases}</math> 
:since all the differentials are zero.

:Alternatively, if we use the equatorial decomposition with two cells in every dimension 
::<math>C_k = \begin{cases} \Z^2 & 0 \leqslant k \leqslant n \\ 0 & \text{otherwise} \end{cases}</math> 
:and the differentials are matrices of the form <math>\left ( \begin{smallmatrix} 1 & -1 \\ 1 & -1\end{smallmatrix} \right ).</math> This gives the same homology computation above, as the chain complex is exact at all terms except <math>C_0</math> and <math>C_n.</math>

* For <math>\mathbb{P}^n(\Complex)</math> we get similarly
::<math>H^k \left (\mathbb{P}^n(\Complex) \right ) = \begin{cases} \Z & 0\leqslant k\leqslant 2n, \text{ even}\\ 0  & \text{otherwise}\end{cases}</math>

Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.: the cellular attaching maps have no role in these computations.  This is a very special phenomenon and is not indicative of the general case.