CW structure of complex projective space

This article describes one possible structure of specific information about::complex projective space $$\mathbb{P}^n(\mathbb{C})$$ (which is a $$2n$$-dimensional real manifold) as a CW-complex.

Description of cells and attaching maps
There is one cell in dimension $$2k, 0 \le k \le n$$. Thus, there is a total of $$(n + 1)$$ different cells. Note that:


 * The $$2k$$-skeleton as well as the $$(2k + 1)$$-skeleton is homeomorphic to $$\mathbb{P}^k(\mathbb{C})$$, and in fact, the CW structure induced on this skeleton is the same as the CW structure we would have chosen for $$\mathbb{P}^k(\mathbb{C})$$ independently
 * The attaching map at stage $$2k + 2$$ is the map arising from the fiber bundle of sphere over projective space (complex case) $$S^{2k + 1} \to \mathbb{P}^k(\mathbb{C})$$.

A more concrete way of interpreting these cells and attaching maps is as follows. Choose a basis for $$\mathbb{C}^{n+1}$$. $$\mathbb{P}^n(\mathbb{C})$$ is the space of lines through the origin in $$\mathbb{C}^{n+1}$$. The $$2k$$-skeleton is the subspace comprising those lines that lie inside the subspace spanned by the first $$(k + 1)$$ 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 $$k^{th}$$ cellular chain group is $$\mathbb{Z}^d$$ where $$d$$ is the number of $$k$$-cells.

For the case of $$\mathbb{P}^n(\mathbb{C})$$, since there is one cell in dimension $$2k$$ for $$0 \le k \le n$$, the cellular chain groups are $$\mathbb{Z}$$ in dimensions $$2k$$ for $$0 \le k \le n$$ are $$0$$ elsewhere.

The cellular chain complex thus looks like:

$$\dots \to 0 \to 0 \to 0 \to \mathbb{Z} \to 0 \to \mathbb{Z} \to 0 \to \mathbb{Z} \to 0 \to \dots \to \mathbb{Z} \to 0 \to \mathbb{Z}$$

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, $$H_{2k}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}$$ for $$0 \le k \le n$$, and all other homology groups are zero.