# 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.