CW structure of complex projective space: Difference between revisions

Latest revision as of 18:32, 31 December 2010

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.

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 * The <math>2k</math>-skeleton as well as the <math>(2k + 1)</math>-skeleton is homeomorphic to <math>\mathbb{P}^k(\mathbb{C})</math>, and in fact, the CW structure induced on this skeleton is the same as the CW structure we would have chosen for <math>\mathbb{P}^k(\mathbb{C})</math> independently
-* The attaching map at stage <math>2k + 2</math> is the map arising from the [[fiber bundle of sphere over projective space]] (complex case) <mah>S^{2k + 1} \to \mathbb{P}^k(\mathbb{C})</math>.
+* The attaching map at stage <math>2k + 2</math> is the map arising from the [[fiber bundle of sphere over projective space]] (complex case) <math>S^{2k + 1} \to \mathbb{P}^k(\mathbb{C})</math>.
 A more concrete way of interpreting these cells and attaching maps is as follows. Choose a basis for <math>\mathbb{C}^{n+1}</math>. <math>\mathbb{P}^n(\mathbb{C})</math> is the space of lines through the origin in <math>\mathbb{C}^{n+1}</math>. The <math>2k</math>-skeleton is the subspace comprising those lines that lie inside the subspace spanned by the first <math>(k + 1)</math> basis vectors. Each time we add a new cell, we are allowing directions that lie in the span of one more basis vector.