In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes.
, the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex where
The group is free abelian, with generators that can be identified with the
Then consider the composition where the first map identifies
The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.
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
Since the generators of the cellular chain groups
, the resulting chain complex is 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 When
is zero, meaning the above formula holds for all positive
This means the attaching map is zero, since the forwards and backwards directions of each 1-cell cancel out.
Therefore, the resulting chain complex is where all the boundary maps are zero.
Therefore, this means the cellular homology of the genus g surface is given by Similarly, one can construct the genus g surface with a crosscap attached as a CW complex with one 0-cell, g 1-cells, and one 2-cell.
can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell.
This can be understood by explicitly constructing the cases for
is the free abelian group generated by its n-cells, for each
consists of two points, one in each connected component (open hemisphere) of
Thus, in order to find the degree of the map
, it is sufficient to find the local degrees of
Hence, without loss of generality, we have that the local degree of
Adding the local degrees, we have that The boundary map
are the following: One sees from the cellular chain complex that the
-skeleton determines all lower-dimensional homology modules: for
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
, and 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.
is then defined by The Euler characteristic is a homotopy invariant.
In fact, in terms of the Betti numbers of
Consider the long exact sequence of relative homology for the triple
: Chasing exactness through the sequence gives The same calculation applies to the triples