Homotopy of spheres - Revision history

Vipul: /* For higher n */

2016-04-15T19:53:32Z

For higher n

← Older revision		Revision as of 19:53, 15 April 2016
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	* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.		* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.
	* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.		* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.
	* <math>\pi_{2n - 1}(S^n)</math> is an infinite group for even <math>n</math>.
	* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.		* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.

Vipul: /* For higher n */

2016-04-15T19:52:49Z

For higher n

← Older revision		Revision as of 19:52, 15 April 2016
Line 22:		Line 22:
	* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.		* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.
	* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.		* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.
	* <math>\pi_{2n - 1}(S^n)</math> is an infinite group.		* <math>\pi_{2n - 1}(S^n)</math> is an infinite group for even <math>n</math>.
	* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.		* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.

Vipul: /* For higher n */

2016-04-15T19:47:46Z

For higher n

← Older revision		Revision as of 19:47, 15 April 2016
Line 22:		Line 22:
	* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.		* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.
	* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.		* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.
	* <math>\pi_{2n - 1}(S^n)</math> is an infinite group ~~for <math>n \ge 2</math>~~.		* <math>\pi_{2n - 1}(S^n)</math> is an infinite group.
	* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.		* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.

Vipul: /* For higher n */

2016-04-15T19:47:22Z

For higher n

← Older revision		Revision as of 19:47, 15 April 2016
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	* <math>\pi_k(S^n)</math> is the trivial group for <math>0 < k < n</math>.		* <math>\pi_k(S^n)</math> is the trivial group for <math>0 < k < n</math>.
	* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.		* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.
	* <math>\pi_{2n - 1}(S^n) \~~cong \mathbb{Z}~~</math>.		* <math>\pi_3(S^2) \cong \mathbb{Z}</math>.
			* <math>\pi_{2n - 1}(S^n)</math> is an infinite group for <math>n \ge 2</math>.
	* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.		* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.

Vipul at 18:07, 31 December 2010

2010-12-31T18:07:48Z

← Older revision		Revision as of 18:07, 31 December 2010
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			{{homotopy invariant computation\|
			invariant = homotopy group\|
			space = sphere}}
	This article gives the key facts about the [[computation of::homotopy group]]s of [[specific information about::sphere]]s.		This article gives the key facts about the [[computation of::homotopy group]]s of [[specific information about::sphere]]s.

Vipul at 17:40, 31 December 2010

2010-12-31T17:40:37Z

← Older revision		Revision as of 17:40, 31 December 2010
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	This article gives the key facts about the [[~~fact about~~::homotopy group]]s of [[~~fact~~ about::sphere]]s.		This article gives the key facts about the [[computation of::homotopy group]]s of [[specific information about::sphere]]s.

	==Statement==		==Statement==

Vipul: moved Homotopy computation for spheres to Homotopy of spheres

2010-12-31T04:26:40Z

moved Homotopy computation for spheres to Homotopy of spheres

← Older revision	Revision as of 04:26, 31 December 2010
(No difference)

Vipul: Created page with 'This article gives the key facts about the fact about::homotopy groups of fact about::spheres. ==Statement== ===For $n = 0$ === $\pi_0(S^0)$ is a ...'

2010-12-24T04:24:24Z

Created page with 'This article gives the key facts about the fact about::homotopy groups of fact about::spheres. ==Statement== ===For <math>n = 0</math>=== <math>\pi_0(S^0)</math> is a ...'

New page

This article gives the key facts about the [[fact about::homotopy group]]s of [[fact about::sphere]]s.

==Statement==

===For <math>n = 0</math>===

<math>\pi_0(S^0)</math> is a two-point set. If we think of <math>S^0</math> as a group, <math>\pi_0(S^0)</math> gets the same group structure, namely the structure of the cyclic group of order two. For all <math>k > 0</math>, <math>\pi_k(S^0)</math> is the trivial group.

===For <math>n = 1</math>===

<math>\pi_0(S^1)</math> is a one-point set (or trivial group, if we choose to use <math>S^1</math>'s group structure to induce a group structure on it). <math>\pi_1(S^1) \cong \mathbb{Z}</math>, and <math>\pi_k(S^1)</math> is trivial for all <math>k \ge 2</math>.

===For higher <math>n</math>===

We have that:

* <math>\pi_0(S^n)</math> is a one-point set (which we can interpret as a trivial group in some cases).
* <math>\pi_k(S^n)</math> is the trivial group for <math>0 < k < n</math>.
* <math>\pi_n(S^n) \cong \mathbb{Z}</math>.
* <math>\pi_{2n - 1}(S^n) \cong \mathbb{Z}</math>.
* <math>\pi_k(S^n)</math> is a [[finite group]] for <math>k > n</math> and <math>k \ne 2n - 1</math>.

==Proof==

===The case <math>n = 0</math>===

{{fillin}}

===The case <math>n = 1</math>===

{{fillin}}

===For higher <math>n</math>===

{{fillin}}

Homotopy of spheres - Revision history

Vipul: /* For higher n */

Vipul: /* For higher n */

Vipul: /* For higher n */

Vipul: /* For higher n */

Vipul at 18:07, 31 December 2010

Vipul at 17:40, 31 December 2010

Vipul: moved Homotopy computation for spheres to Homotopy of spheres

Vipul: Created page with 'This article gives the key facts about the fact about::homotopy groups of fact about::spheres. ==Statement== ===For n = 0=== \pi_0(S^0) is a ...'

Vipul: Created page with 'This article gives the key facts about the fact about::homotopy groups of fact about::spheres. ==Statement== ===For $n = 0$ === $\pi_0(S^0)$ is a ...'