CW structure of complex projective space

This article describes one possible structure of 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 $2k,0\leq k\leq n$. 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 $0\leq k\leq n$, the cellular chain groups are $\mathbb {Z}$ in dimensions $2k$ for $0\leq k\leq n$ 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 $0\leq k\leq n$, and all other homology groups are zero.