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

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?

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...

• Principal circle bundles are classified by $H^2(S^2; \mathbb{Z})$, not torus bundles. Commented Feb 26, 2020 at 13:36
• Oh for the principal torus bundles then we should be using $(\mathbb{C}P^\infty)^2$? Let me think for a while...
– ah--
Commented Feb 26, 2020 at 14:32

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.