TITLE: Classify sphere bundles over a sphere
QUESTION [10 upvotes]: Problem
(1) Classify all $S^1$ bundles over the base manifold $S^2$.
(2) Do the same question for $S^2$ bundles.

Moreover, does there exist a universal method to solve this kind of problem? What mathematical tool and concept should be required? And, could someone suggest some reference books related to the problem?
(I'm not sure my tags are correct)

REPLY [9 votes]: The key to answering this question is the clutching construction. Let $p_1, p_2$ be antipodal points on $S^2$ and define $U_i = S^2\setminus\{p_i\}$. Note that $U_i$ is homeomorphic to $\mathbb{R}^2$ (via stereographic projection) and therefore contractible.
Suppose $E$ is an $S^1$-bundle over $S^2$, then as $U_1$, $U_2$ are contractible, $E|_{U_1}$ and $E|_{U_2}$ are trivial; i.e. there are bundle isomorphisms $\psi_i : E|_{U_i} \to U_i\times S^1$. Therefore, the bundle $E$ is completely determined by the transition map $\phi_{12} : U_1\cap U_2 \to \operatorname{Homeo}(S^1)$ which satisfies 
$$\psi_2|_{U_1\cap U_2} = (\operatorname{id}_{U_1\cap U_2}, \phi_{12})\circ\psi_1|_{U_1\cap U_2}.$$
It turns out that isomorphism classes of such bundles correspond to homotopy classes of such maps. That is, isomorphism classes of $S^1$-bundles over $S^2$ are in one-to-one correspondence with homotopy classes of maps $U_1\cap U_2 \to \operatorname{Homeo}(S^1)$, i.e. $[U_1\cap U_2, \operatorname{Homeo}(S^1)]$. Note that $U_1\cap U_2$ deformation retracts onto $S^1$ (the equator of $S^2$), so 
$$[U_1\cap U_2, \operatorname{Homeo}(S^1)] = [S^1, \operatorname{Homeo}(S^1)] = \pi_1(\operatorname{Homeo}(S^1)).$$
In order to compute this, we use the fact that $\operatorname{Homeo}(S^1)$ deformation retracts onto $O(2)$ and therefore $\pi_1(\operatorname{Homeo}(S^1)) = \pi_1(O(2)) = \mathbb{Z}$.

I see that Mike Miller has now posted an answer, so I will stop here as his answer addresses the general picture.