2 dimensional torus bundles over $\mathbb{S}^2$

Question

With my limited knowledge of bundles, it seems that the isomorphism classes of principal torus bundles are in one-to-one correspondence with the homotopy classes of maps $[\mathbb{S}^2,\mathbb{C}P^\infty]\cong H^2(\mathbb{S}^2,\mathbb{Z}^2)\cong\mathbb{Z}^2$. On the other hand, with the clutching construction, we have the isomorphism classes of all bundles over $\mathbb{S}^2$ to be $[\mathbb{S}^1,\mathbb{T}^2]\cong\pi_1(\mathbb{T}^2)\cong\mathbb{Z}^2$, if I take the structure group just to be $\mathbb{T}^2$ (I am also wondering what $Homeo(\mathbb{T}^2)$ is...).

Now, I am wondering what these bundles are. For instance, which principal torus bundle does, say, $(1,1)\in\mathbb{Z}^2$ correspond to? And what does it correspond to if one does clutching?

Thanks in advance!!

Edited: I corrected above mistakes, as far as I recognized. And okay, I guess I was just being silly. The $G$-bundle with structure group $G$ is of course the principal bundle... But still the clutching construction seems more intuitive for me to see what is going on there, but with the classifying map, I have no idea. Finally, I guess $Homeo(\mathbb{T}^2)\cong\mathbb{T}^2\rtimes GL(2,\mathbb{Z})$ but I found no references...

Answer 1

I might suggest a partial answer.

As noted in the comments, $\mathbb{C}\mathbb{P}^{\infty}$ is a classifying space for $S^1$, and not of $S^1\times S^1$.

You were goint the right direction, when you mentioned the clutching functions. Indeed, on $D^2$ our bundles are trivial, so we need to decide on how to glue on the intersection $S^1$, so we need a (based) map $S^1 \to Homeo(T^2)$ up to (based) isotopy. I don't know how to compute $\pi_1(Homeo(T^2))$...

Definitely, there are a lot of different examples of torus bundles besides trivial $T^2\times S^2$. Given any $S^1$-bundles, say, the Hopf bundle $f:S^3\to S^2$, or in general, Lens spaces $g:L(p,1)\to S^2$ you can multiply the fiber by $S^1$ and get $f\times id : S^3\times S^1\to S^2$, and similarly for Lens spaces. And all of these examples are different because the total spaces have different fundamental groups. But I don't believe that these exaust all examples.

Side remark: If we allow singular fiber bundles(Lefschetz fibration), then we know all possibilities for total spaces, namely, it must be $\mathbb{C}\mathbb{P}^2$, $\mathbb{C}\mathbb{P}^1\times \mathbb{C}\mathbb{P}^1$ or $E(n)$, or blow-up of these. I don't know if this result covers nonsingular case.