Freudenthal suspension theorem: Difference between revisions

Latest revision as of 19:45, 11 May 2008

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 $q=2n-1$ .

Revision as of 20:41, 27 October 2007 (view source) Vipul (talk \| contribs) No edit summary		Latest revision as of 19:45, 11 May 2008 (view source) Vipul (talk \| contribs) m (2 revisions)
(One intermediate revision by the same user not shown)
Line 3:		Line 3:
	==Statement==		==Statement==

	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>.