CW Complex - Homology and Cohomology of CW-complexes | Homology Cohomology CW-complexes

Homology and Cohomology of CW-complexes

Singular homology and 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 extraordinary (co)homology theory for a CW-complex, the Atiyah-Hirzebruch spectral sequence is the analogue of cellular homology.

Some examples:

For the sphere, take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology chain complex and homology are given by:

$C_k = \left\{ \begin{array}{lr} \mathbb Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{array} \right.$ $H_k = \left\{ \begin{array}{lr} \mathbb Z & k \in \{0,n\} \\ 0 & k \notin \{0,n\} \end{array} \right.$ since all the differentials are zero.

Alternatively, if we use the equatorial decomposition with two cells in every dimension $C_k = \left\{ \begin{array}{lr} \mathbb Z^2 & 0 \leq k \leq n \\ 0 & \text{otherwise} \end{array} \right.$ and the differentials are matrices of the form . This gives the same homology computation above, as the chain complex is exact at all terms except and .

For we get similarly

$H^k(\mathbb{P}^n\mathbb{C}) = \begin{cases} \mathbb{Z} \quad\text{for } 0\le k\le 2n,\text{even}\\ 0 \quad\text{otherwise}. \end{cases}$

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.