Complex projective plane - Revision history

Vipul: /* Homotopy groups */

2011-07-29T14:40:13Z

Homotopy groups

← Older revision		Revision as of 14:40, 29 July 2011
Line 32:		Line 32:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Value of <math>k</math> !! General name for homotopy group/set <math>\pi_k</math> !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^n</math> for generic <math>n \ge 2</math>? !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^2)</math>?		! Value of <math>k</math> !! General name for homotopy group/set <math>\pi_k</math> !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^n</math> for generic <math>n \ge 2</math>?) !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^2)</math>?
	\|-		\|-
	\| 0 \|\| [[set of path components]] \|\| one-point set \|\| one-point set, so <math>\mathbb{C}\mathbb{P}^2</math> is a [[path-connected space]]		\| 0 \|\| [[set of path components]] \|\| one-point set \|\| one-point set, so <math>\mathbb{C}\mathbb{P}^2</math> is a [[path-connected space]]

Vipul: /* Homotopy groups */

2011-07-21T23:32:00Z

Homotopy groups

← Older revision		Revision as of 23:32, 21 July 2011
Line 29:		Line 29:
	{{further\|[[homotopy of complex projective space]]}}		{{further\|[[homotopy of complex projective space]]}}

	{{~~fillin~~}}		The homotopy groups are as follows:

			{\| class="sortable" border="1"
			! Value of <math>k</math> !! General name for homotopy group/set <math>\pi_k</math> !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^n</math> for generic <math>n \ge 2</math>? !! What is <math>\pi_k(\mathbb{C}\mathbb{P}^2)</math>?
			\|-
			\| 0 \|\| [[set of path components]] \|\| one-point set \|\| one-point set, so <math>\mathbb{C}\mathbb{P}^2</math> is a [[path-connected space]]
			\|-
			\| 1 \|\| [[fundamental group]] \|\| trivial group \|\| trivial group, so <math>\mathbb{C}\mathbb{P}^2</math> is a [[simply connected space]].
			\|-
			\| 2 \|\| [[second homotopy group]] \|\| <math>\mathbb{Z}</math> \|\| <math>\mathbb{Z}</math>
			\|-
			\| 3 \|\| third homotopy group \|\| trivial group \|\| trivial group
			\|-
			\| 4 \|\| fourth homotopy group \|\| trivial group \|\| trivial group
			\|-
			\| 5 \|\| fifth homotopy group \|\| <math>\mathbb{Z}</math> if <math>n = 2</math>, zero otherwise \|\| <math>\mathbb{Z}</math>
			\|-
			\| <math>k \ge 6</math> \|\| \|\| \|\| Same as <math>\pi_k(S^5)</math>
			\|}

Vipul: Created page with '{{particular topospace}} ==Definition== The '''complex projective plane''' is the complex projective space of complex dimension 2. As a manifold over the reals, it has ...'

2010-12-31T04:25:57Z

Created page with '{{particular topospace}} ==Definition== The '''complex projective plane''' is the complex projective space of complex dimension 2. As a manifold over the reals, it has ...'

New page

{{particular topospace}}

==Definition==

The '''complex projective plane''' is the [[complex projective space]] of complex dimension 2. As a [[manifold]] over the reals, it has dimension 4. It is denoted <math>\mathbb{C}\mathbb{P}^2</math> or <math>\mathbb{P}^2(\mathbb{C})</math>.

Alternatively, it can be viewed as the quotient of the space <math>S^2(\mathbb{C}) \cong S^5</math> under the action of <math>S^0(\mathbb{C}) \cong S^1</math> by multiplication. In particular, there is a fibration <math>S^1 \to S^5 \to \mathbb{C}\mathbb{P}^2</math>.

==Algebraic topology==

===Homology groups===

{{further|[[homology of complex projective space]]}}

The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(\mathbb{C}\mathbb{P}^2) \cong H_2(\mathbb{C}\mathbb{P}^2) \cong H_4(\mathbb{C}\mathbb{P}^2) \cong \mathbb{Z}</math>, and all other homology groups are zero.

More generally, the homology group with coefficients in a module <math>M</math> over a commutative unital ring <math>R</math> are as follows: <math>H^0(\mathbb{C}\mathbb{P}^2;M) \cong H^2(\mathbb{C}\mathbb{P}^2;M) \cong H^4(\mathbb{C}\mathbb{P}^2;M) \cong M</math>, and all other homology groups are zero.

===Cohomology groups===

{{further|[[cohomology of complex projective space]]}}

The cohomology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H^0(\mathbb{C}\mathbb{P}^2) \cong H^2(\mathbb{C}\mathbb{P}^2) \cong H^4(\mathbb{C}\mathbb{P}^2) \cong \mathbb{Z}</math>, and all other cohomology groups are zero. The cohomology ring is <math>\mathbb{Z}[x]/(x^3)</math> where <math>x</math> is an additive generator for the second cohomology group.

More generally, the cohomology group with coefficients in a commutative unital ring <math>R</math> are as follows: <math>H^0(\mathbb{C}\mathbb{P}^2;R) \cong H^2(\mathbb{C}\mathbb{P}^2;R) \cong H^4(\mathbb{C}\mathbb{P}^2;R) \cong R</math>, and all other cohomology groups are zero. The cohomology ring is <math>R[x]/(x^3)</math> where <math>x</math> is a <math>R</math>-module generator for the second cohomology module.

===Homotopy groups===

{{further|[[homotopy of complex projective space]]}}

{{fillin}}