From Harvard qualification exam, 1990. Consider the space $X=\mathbb{S}^{1}\wedge \mathbb{S}^{2}$, alternatively viewing it as a sphere with north and south poles connected. I was asked to:
1): the relationship between $\pi_{2}X$ and $\pi_{2}\overline{X}$, where $\overline{X}$ is $X$'s universal cover.
2): Calculate $\pi_{2}(X)$.
I think $\overline{X}$ is just $\mathbb{R}^{2}\wedge \mathbb{R}^{1}$, and its second homotopy group should be $0$. But I do not see any nontrivial relationship between $\pi_{2}X$ and $\pi_{2}\overline{X}$, for the deck transformation argument does not extend to spheres(as opposed to loops). For the second question, my guess is $\pi_{2}(X)=\mathbb{Z}$; again I need a proof. Does the relationship $\pi_{2}(X\wedge Y)=\pi_{2}(X)\oplus \pi_{2}(Y)$ hold as fundamental groups?
I do not see a nontrivial fibration from $\overline{X}\rightarrow X$ that can make me use the long exact sequence of homotopy groups. So I ask in here.