Homology of complex projective space - Revision history

Vipul at 16:59, 28 July 2011

2011-07-28T16:59:20Z

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	===Unreduced version with coefficients in integers===		===Unreduced version with coefficients in integers===

	<math>H_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>		<math>H_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.</math>

	===Reduced version with coefficients in integers===		===Reduced version with coefficients in integers===

	<math>\tilde{H}_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 2 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>		<math>\tilde{H}_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 2 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.</math>

	===Unreduced version with coefficients in an abelian group or module===		===Unreduced version with coefficients in an abelian group or module===
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	For coefficients in an abelian group <math>M</math>, the homology groups are:		For coefficients in an abelian group <math>M</math>, the homology groups are:

	<math>H_p(\mathbb{P}^n(\mathbb{C});M) = \left\lbrace\begin{array}{rl} M, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>		<math>H_p(\mathbb{P}^n(\mathbb{C});M) = \left\lbrace\begin{array}{rl} M, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.</math>

	==Homology groups with integer coefficients in tabular form==		==Homology groups with integer coefficients in tabular form==

Vipul at 16:58, 28 July 2011

2011-07-28T16:58:44Z

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	==Statement==		==Statement==

	~~The homology of complex projective space is given as follows:~~		===Unreduced version with coefficients in integers===

	<math>H_p(\mathbb{C} P^n) = \Z \qquad p = 0, ~~2, 4,~~ \~~ldots, 2n~~</math>		<math>H_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>

	~~and zero otherwise.~~		===Reduced version with coefficients in integers===

	==~~Related invariants~~==		<math>\tilde{H}_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 2 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>

			===Unreduced version with coefficients in an abelian group or module===

			For coefficients in an abelian group <math>M</math>, the homology groups are:

			<math>H_p(\mathbb{P}^n(\mathbb{C});M) = \left\lbrace\begin{array}{rl} M, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}</math>

			==Homology groups with integer coefficients in tabular form==

	~~===Betti numbers===~~		We illustrate how the homology groups work for small values of <math>n</math> (whereby the dimension of the corresponding complex projective space is <math>2n</math>). Note that for <math>p > 2n</math>, all homology groups are zero, so we omit those cells for visual clarity.

	~~The~~ [[~~Betti number~~]]~~s are~~ <math>1</math> ~~for~~ <math>0~~,2,~~4,\~~ldots,2n~~</math> ~~and~~ <math>0</math> ~~elsewhere.~~		{\| class="sortable" border="1"
			! <math>n</math> !! Complex projective space <math>\mathbb{C}\mathbb{P}^n</math> !! <math>H_0</math> !! <math>H_1</math> !! <math>H_2</math> !! <math>H_3</math> !! <math>H_4</math> !! <math>H_5</math> !! <math>H_6</math> !! <math>H_7</math> !! <math>H_8</math>
			\|-
			\| 1 \|\| [[2-sphere]] \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math>
			\|-
			\| 2 \|\| [[complex projective plane]] \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math>
			\|-
			\| 3 \|\| [[CP^3]] \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math>
			\|-
			\| 4 \|\| [[CP^4]] \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math> \|\| 0 \|\| <math>\mathbb{Z}</math>
			\|}

	==~~=Poincare polynomial=~~==		==Related invariants==

	~~The [[Poincare polynomial]] is given by:~~		These are all invariants that can be computed in terms of the homology groups.

	<math>PX = 1 + x^2 + x^4 + \~~ldots~~ + x^{2n} = \frac{x^{2n + 2} - 1}{x^2 - 1}</math>		{\| class="sortable" border="1"
			! Invariant !! General description !! Description of value for complex projective space <math>\mathbb{P}^n(\mathbb{C})</math>
			\|-
			\| [[Betti number]]s \|\| The <math>k^{th}</math> Betti number <math>b_k</math> is the rank of the torsion-free part of the <math>k^{th}</math> homology group. \|\| <math>b_0 = b_2 = b_4 = \dots = b_{2n} = 1</math>, all other <math>b_k</math> values are zero.
			\|-
			\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>1 + x^2 + x^4 + \dots + x^{2n} = \frac{x^{2n + 2} - 1}{x^2 - 1}</math>
			\|-
			\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>n + 1</math>
			\|}

	==~~=Euler characteristic=~~==		==Facts used==

	~~The Euler characteristic is <math>n+1</math>.~~		# [[uses::CW structure of complex projective space]]

	==Proof==		==Proof==

	~~{{further\|[[CW structure of complex projective space]]}}~~

	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>.		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>.

Vipul at 18:08, 31 December 2010

2010-12-31T18:08:59Z

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	{{homology of\|complex projective space}}		{{homotopy invariant computation\|
			invariant = homology group\|
			space = complex projective space}}

	==Statement==		==Statement==

Vipul at 17:55, 31 December 2010

2010-12-31T17:55:41Z

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	The Euler characteristic is <math>n+1</math>.		The Euler characteristic is <math>n+1</math>.

	==~~Cohomology ring=~~==		==Proof==

	{{further\|[[~~Cohomology ring~~ of complex projective space]]}}		{{further\|[[CW structure of complex projective space]]}}
	~~==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>.		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>.

Vipul: 5 revisions

2008-05-11T19:46:18Z

5 revisions

← Older revision	Revision as of 19:46, 11 May 2008
(No difference)

Vipul at 22:43, 1 December 2007

2007-12-01T22:43:47Z

← Older revision		Revision as of 22:43, 1 December 2007
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	The Euler characteristic is <math>n+1</math>.		The Euler characteristic is <math>n+1</math>.

			==Cohomology ring===

			{{further\|[[Cohomology ring of complex projective space]]}}
	==Proof==		==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>.		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>.

Vipul at 21:53, 9 November 2007

2007-11-09T21:53:03Z

← Older revision		Revision as of 21:53, 9 November 2007
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	{{homology of ~~collection of spaces~~}}		{{homology of\|complex projective space}}

	==Statement==		==Statement==

Vipul at 21:42, 3 November 2007

2007-11-03T21:42:41Z

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 and zero otherwise.
 ==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>.

Vipul: /* Statement */

2007-11-03T20:25:26Z

Statement

← Older revision		Revision as of 20:25, 3 November 2007
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	The homology of complex projective space is given as follows:		The homology of complex projective space is given as follows:

	<math>H_p(\C P^n) = \Z \qquad p = 0, 2, 4, \ldots, 2n</math>		<math>H_p(\mathbb{C} P^n) = \Z \qquad p = 0, 2, 4, \ldots, 2n</math>

	and zero otherwise.		and zero otherwise.

Vipul at 20:25, 3 November 2007

2007-11-03T20:25:08Z

New page

{{homology of collection of spaces}}

==Statement==

The homology of complex projective space is given as follows:

<math>H_p(\C P^n) = \Z \qquad p = 0, 2, 4, \ldots, 2n</math>

and zero otherwise.

==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>.