CW structure of spheres
Description of cells and attaching maps
The space here is , a discrete two-point space. There is an obvious and unique CW-structure for this space: two 0-cells, and no other cells.
The sphere has one 0-cell and one -cell. The The 0-cell gives a point. The attaching map for the boundary of the -cell sends its boundary (homeomorphic to ) to the 0-cell point. Specifically, we are using the fact that .
Cellular chain complex and cellular homology
Any CW structure on a topological space provides a cellular filtration relative to the empty space. The corresponding cellular chain complex is described below. By excision, the cellular chain group is where is the number of -cells.
In this case, the zeroth chain group is and all higher chain groups are zero. All boundary maps for this chain complex are zero, so the homology groups are the same as the chain groups. In particular, and all higher homology groups are zero.
In this case, the zeroth and chain groups are and all other chain groups are zero. All boundary maps for this chain complex are zero, so the homology groups are the same as the chain groups. In particular, and all other homology groups are zero.
Note that for , the fact that all the chain maps are zero is forced just by looking at the chain groups. This is not the case for , but a careful look at the boundary map from the first to the zeroth chain group reveals that it is zero.