CW structure of complex projective space

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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 \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.