Freudenthal suspension theorem: Difference between revisions

Revision as of 22:29, 24 December 2007

This fact is related to: homotopy groups

Statement

Let $X$ be a $(n-1)$ -connected space having a nondegenerate basepoint $x_{0}$ . Then the suspension homomorphism from $\pi _{q}(X)\to \pi _{q+1}(\Sigma X)$ is an isomorphism for $q\leq 2n-2$ and is surjective for $q=2n-1$ .

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	Let <math>X</math> be a <math>(n-1)</math>-[[multiply connected space\|connected space]] having a [[nondegenerate point\|nondegenerate]] basepoint <math>x_0</math>. Then the [[suspension homomorphism]] from <math>\pi_q(X) \to \pi_{q+1}(X)</math> is an isomorphism for <math>q \le 2n - 2</math> and is surjective for <math>q = 2n - 1</math>.		Let <math>X</math> be a <math>(n-1)</math>-[[multiply connected space\|connected space]] having a [[nondegenerate point\|nondegenerate]] basepoint <math>x_0</math>. Then the [[suspension homomorphism]] from <math>\pi_q(X) \to \pi_{q+1}(\Sigma X)</math> is an isomorphism for <math>q \le 2n - 2</math> and is surjective for <math>q = 2n - 1</math>.