N-sphere is (n-1)-connected: Difference between revisions

Latest revision as of 01:16, 20 December 2010

Statement

Suppose $n$ is a natural number (i.e., $n \geq 1$ ). Then, the $n$ -Sphere (?) is $(n - 1)$ -connected. In other words:

For $n = 1$ , the $n$ -sphere, better known as the circle, is a path-connected space.
For $n \geq 2$ , the $n$ -sphere is a Path-connected space (?) and Simply connected space (?) and all its homotopy groups up to the $(n - 1)^{t h}$ homotopy group are trivial. In other words, $π_{1} (S^{n}), π_{2} (S^{n}), \dots, π_{n - 1} (S^{n})$ are all trivial.

By the Hurewicz theorem, this is equivalent (for $n \geq 2$ ) to the assertion that $S^{n}$ is simply connected and the first $n - 1$ homology groups are trivial.

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	* For <math>n \ge 2</math>, the <math>n</math>-sphere is a [[fact about::path-connected space]] and [[fact about::simply connected space]] and all its [[homotopy group]]s up to the <math>(n - 1)^{th}</math> homotopy group are trivial. In other words, <math>\pi_1(S^n), \pi_2(S^n), \dots, \pi_{n-1}(S^n)</math> are all trivial.		* For <math>n \ge 2</math>, the <math>n</math>-sphere is a [[fact about::path-connected space]] and [[fact about::simply connected space]] and all its [[homotopy group]]s up to the <math>(n - 1)^{th}</math> homotopy group are trivial. In other words, <math>\pi_1(S^n), \pi_2(S^n), \dots, \pi_{n-1}(S^n)</math> are all trivial.

	By the ~~(what~~ theorem?), this is equivalent (for <math>n \ge 2</math>) to the assertion that <math>S^n</math> is simply connected and the first <math>n - 1</math> homology groups are trivial.		By the [[Hurewicz theorem]], this is equivalent (for <math>n \ge 2</math>) to the assertion that <math>S^n</math> is simply connected and the first <math>n - 1</math> homology groups are trivial.