Homotopy of complex projective space - Revision history

Vipul: /* Case of higher n */

2016-04-15T19:49:47Z

Case of higher n

← Older revision		Revision as of 19:49, 15 April 2016
Line 36:		Line 36:
	* Case <math>2 < k < 2n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C}))</math> is the trivial group.		* Case <math>2 < k < 2n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C}))</math> is the trivial group.
	* Case <math>k = 2n + 1</math>: We get that <math>\pi_{2n+1}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>.		* Case <math>k = 2n + 1</math>: We get that <math>\pi_{2n+1}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>.
	* Case <math>k = 4n + 1</math>: We get that <math>\pi_{4n+1}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>.
	* Case <math>2n + 1 < k, k \ne 4n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C})) \cong \pi_k(S^{2n + 1})</math>.		* Case <math>2n + 1 < k, k \ne 4n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C})) \cong \pi_k(S^{2n + 1})</math>.

Vipul: /* Statement */

2011-04-02T20:39:19Z

Statement

← Older revision		Revision as of 20:39, 2 April 2011
Line 5:		Line 5:
	==Statement==		==Statement==

	This article describes the [[homotopy group]]s, including the [[set of path ~~component~~]]s <math>\pi_0</math>, the [[fundamental group]] <math>\pi_1</math>, and the higher [[homotopy group]]s <math>\pi_k</math> of <math>\mathbb{P}^n(\mathbb{C})</math>.		This article describes the [[homotopy group]]s, including the [[set of path components]] <math>\pi_0</math>, the [[fundamental group]] <math>\pi_1</math>, and the higher [[homotopy group]]s <math>\pi_k</math> of <math>\mathbb{P}^n(\mathbb{C})</math>.

	===Case <math>n = 0</math>===		===Case <math>n = 0</math>===

Vipul at 18:10, 31 December 2010

2010-12-31T18:10:07Z

← Older revision		Revision as of 18:10, 31 December 2010
Line 1:		Line 1:
			{{homotopy invariant computation\|
			invariant = homotopy group\|
			space = complex projective space}}

	==Statement==		==Statement==

Vipul: Created page with '==Statement== This article describes the homotopy groups, including the set of path components $\pi_0$ , the fundamental group $\pi_1$ , and the ...'

2010-12-31T17:00:18Z

Created page with '==Statement== This article describes the homotopy groups, including the set of path components <math>\pi_0</math>, the fundamental group <math>\pi_1</math>, and the ...'

New page

==Statement==

This article describes the [[homotopy group]]s, including the [[set of path component]]s <math>\pi_0</math>, the [[fundamental group]] <math>\pi_1</math>, and the higher [[homotopy group]]s <math>\pi_k</math> of <math>\mathbb{P}^n(\mathbb{C})</math>.

===Case <math>n = 0</math>===

For <math>n = 0</math>, <math>\mathbb{P}^n(\mathbb{C})</math> is the one-point set. Hence, all its homotopy groups are the trivial group. The set of path components <math>\pi_0</math> is a one-point set and can be considered the trivial group.

===Case <math>n = 1</math>===

For <math>n = 1</math>, <math>\mathbb{P}^1(\mathbb{C}) \cong S^2</math> (a homeomorphism), i.e., it is the [[2-sphere]]. Its homotopy groups are hence the same as those of the 2-sphere. Specifically, they are as follows:

* <math>\pi_0(\mathbb{P}^1(\mathbb{C}))</math> is a one-point set.
* <math>\pi_1(\mathbb{P}^1(\mathbb{C}))</math> is the trivial group.
* <math>\pi_2(\mathbb{P}^1(\mathbb{C})) \cong \mathbb{Z}</math>, i.e., it is isomorphic to the group of integers, with the identity map being the generator.
* <math>\pi_3(\mathbb{P}^1(\mathbb{C})) \cong \mathbb{Z}</math>, i.e., it is isomorphic to the group of integers, with the map being the [[Hopf fibration]].
* <math>\pi_4(\mathbb{P}^1(\mathbb{C})) \cong \mathbb{Z}/2\mathbb{Z}</math>.

Higher homotopy groups are the same as those of the 2-sphere.

===Case of higher <math>n</math>===

For this case, we use the [[fiber bundle of sphere over projective space]] <math>S^{2n + 1} \to \mathbb{P}^n(\mathbb{C})</math> with fiber <math>S^1</math>. We get the following [[long exact sequence of homotopy of a Serre fibration]]:

<math> \dots \to \pi_k(S^1) \to \pi_k(S^{2n + 1}) \to \pi_k(\mathbb{P}^n(\mathbb{C})) \to \pi_{k-1}(S^1) \to \dots</math>

For <math>k \ge 2</math>, <math>\pi_k(S^1)</math> is trivial. Thus we get the following:

* Case <math>k = 0</math>: <math>\pi_0(\mathbb{P}^n(\mathbb{C}))</math> is a one-point space.
* Case <math>k = 1</math>: We get <math>\pi_1(\mathbb{P}^n(\mathbb{C}))</math> is trivial.
* Case <math>k = 2</math>: We get <math>\pi_2(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>..
* Case <math>2 < k < 2n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C}))</math> is the trivial group.
* Case <math>k = 2n + 1</math>: We get that <math>\pi_{2n+1}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>.
* Case <math>k = 4n + 1</math>: We get that <math>\pi_{4n+1}(\mathbb{P}^n(\mathbb{C})) \cong \mathbb{Z}</math>.
* Case <math>2n + 1 < k, k \ne 4n + 1</math>: We get that <math>\pi_k(\mathbb{P}^n(\mathbb{C})) \cong \pi_k(S^{2n + 1})</math>.

Homotopy of complex projective space - Revision history

Vipul: /* Case of higher n */

Vipul: /* Statement */

Vipul at 18:10, 31 December 2010

Vipul: Created page with '==Statement== This article describes the homotopy groups, including the set of path components \pi_0, the fundamental group \pi_1, and the ...'

Vipul: Created page with '==Statement== This article describes the homotopy groups, including the set of path components $\pi_0$ , the fundamental group $\pi_1$ , and the ...'