CW structure of real projective space

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

Description of cells and attaching maps

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

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

A more concrete way of interpreting these cells and attaching maps is as follows. Choose a basis for $R^{n + 1}$ . $P^{n} (R)$ is the space of lines through the origin in $R^{n + 1}$ . 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 $k^{t h}$ cellular chain group is $Z^{d}$ where $d$ is the number of $k$ -cells.

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

The cellular chain complex thus looks like:

$\dots \to 0 \to 0 \to 0 \to Z \to Z \to Z \to \dots \to Z \to Z \to Z$

The boundary maps are as follows: the map $H_{k} \to H_{k - 1}$ for $k$ even, $k \leq n$ 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.