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I am studying now a homotopy theory and I have a question.

Suppose that we have a connected space $X$, and a $\pi_{1}(X)$ action on $\pi_{2}(X)$ is trivial.

In this case, is it true that

$\pi_{1}(\Omega X) \simeq \pi_{2}(X)$ ?

If it is true, is there a reference for this result?

If not, is there any relation?

Thank you.

share|cite|improve this question
This is always true, in fact. It should be proved in any good algebraic topology textbook. – Mariano Suárez-Alvarez Mar 29 '13 at 5:42
($\Omega X $ is called the loop space of $ X $; the path space is something else. – Mariano Suárez-Alvarez Mar 29 '13 at 5:44
This is a special case of the adjunction between smash products and mapping spaces. – Dylan Wilson Apr 3 '13 at 1:43

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