CW structure of real projective space

This article discusses a common choice of CW structure for real projective space $$\mathbb{P}^n(\R)$$, i.e., a CW-complex having this as its underlying topological space.

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


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

A more concrete way of interpreting these cells and attaching maps is as follows. Choose a basis for $$\mathbb{R}^{n+1}$$. $$\mathbb{P}^n(\mathbb{R})$$ is the space of lines through the origin in $$\mathbb{R}^{n+1}$$. The $$k$$-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:

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

The boundary maps are as follows: the map $$H_k \to H_{k-1}$$ for $$k$$ even, $$k \le n$$ is the doubling map. For $$k$$ odd, the map is the zero map.

This gives the homology groups as described in homology of real projective space.