TITLE: Homotopy groups of a wedge sum
QUESTION [15 upvotes]: Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book,
$$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$
Is it true in general that
$$\pi_{n}(\bigvee_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha})?$$

REPLY [16 votes]: No. In general, homotopy groups behave nicely under homotopy pull-backs (e.g., fibrations and products), but not homotopy push-outs (e.g., cofibrations and wedges). Homology is the opposite. 
For a specific example, consider the case of the fundamental group. The Seifert-Van Kampen theorem implies that $\pi_1(A\vee B)$ is isomorphic to the free product $\pi_1(A)*\pi_1(B)$, not the product.