Homology of complex projective space: Difference between revisions

Revision as of 21:42, 3 November 2007

Template:Homology of collection of spaces

Statement

The homology of complex projective space is given as follows:

$H_{p} (C P^{n}) = Z p = 0, 2, 4, \dots, 2 n$

and zero otherwise.

Related invariants

Betti numbers

The Betti numbers are $1$ for $0, 2, 4, \dots, 2 n$ and $0$ elsewhere.

Poincare polynomial

The Poincare polynomial is given by:

$P X = 1 + x^{2} + x^{4} + \dots + x^{2 n} = \frac{x^{2 n + 2} - 1}{x^{2} - 1}$

Euler characteristic

The Euler characteristic is $n + 1$ .

Proof

We use the CW-complex structure on complex projective space that has exactly one cell in every even dimension till $2 n$ . The cellular chain complex of this thus has $Z$ s in all the even positions till $2 n$ , and hence its homology is $Z$ in all even dimensions till $2 n$ .

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 and zero otherwise.
+==Related invariants==
+===Betti numbers===
+The [[Betti number]]s are <math>1</math> for <math>0,2,4,\ldots,2n</math> and <math>0</math> elsewhere.
+===Poincare polynomial===
+The [[Poincare polynomial]] is given by:
+<math>PX = 1 + x^2 + x^4 + \ldots + x^{2n} = \frac{x^{2n + 2} - 1}{x^2 - 1}</math>
+===Euler characteristic===
+The Euler characteristic is <math>n+1</math>.
 ==Proof==
 We use the CW-complex structure on complex projective space that has exactly one cell in every even dimension till <math>2n</math>. The cellular chain complex of this thus has <math>\Z</math>s in all the even positions till <math>2n</math>, and hence its homology is <math>\Z</math> in all even dimensions till <math>2n</math>.