CW structure of complex projective space
Description of cells and attaching maps
There is one cell in dimension . Thus, there is a total of different cells. Note that:
- The -skeleton as well as the -skeleton is homeomorphic to , and in fact, the CW structure induced on this skeleton is the same as the CW structure we would have chosen for independently
- The attaching map at stage is the map arising from the fiber bundle of sphere over projective space (complex case) .
A more concrete way of interpreting these cells and attaching maps is as follows. Choose a basis for . is the space of lines through the origin in . The -skeleton is the subspace comprising those lines that lie inside the subspace spanned by the first basis vectors. Each time we add a new cell, we are allowing directions that lie in the span of one more basis vector.
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.
For the case of , since there is one cell in dimension for , the cellular chain groups are in dimensions for are elsewhere.
The cellular chain complex thus looks like:
In particular, since there are no two adjacent nonzero cellular chain groups, all the boundary maps are zero, so the homology groups are the same as the chain groups. Thus, for , and all other homology groups are zero.