# When are fiber bundles reversible?

My question, in its most general form is this:

Given a fiber bundle $F\rightarrow E\rightarrow B$, when is there a fiber bundle $B\rightarrow E\rightarrow F$?

Here, F,E, and B can lie in whichever category you wish, but I'm mostly interested in the case where all 3 are smooth closed manifolds.

Now, I realize that the initial answer is "unless E is a product, essentially never", so here is a more focused question (with background).

I've been studying a certain class of free actions of the 3-torus $T^3$ on $S^3\times S^3\times S^3 = (S^3)^3$. For each of these actions, by quotienting out by various subtori, I can show that the orbit space $E=(S^3)^3/T^3$ simultaneously fits into 2 fiber bundles:

$$S^2\rightarrow E \rightarrow S^2\times S^2$$ and $$S^2\times S^2\rightarrow E\rightarrow S^2$$ where the structure group for both bundles is $S^1$.

(In fact, the class of actions also gives rise to examples where either $S^2\times S^2$ can independently be replaced with $\mathbb{C}P^2\sharp -\mathbb{C}P^2$, the unique nontrivial $S^2$ bundle over $S^2$.)

By computing characteristic classes for (the tangent bundle to) E, I know that for an infinite sublcass of the actions I'm looking at, E is not homotopy equivalent to $S^2\times S^2\times S^2$, and each of the E are pairwise nondiffeomorphic.

I suspect the reason I could find so many E which fit into "reversible" fiber bundles is strongly related with the fact that the fiber and base are so closely related.

For fixed manifold M, what is the relationship between bundles $X\rightarrow E\rightarrow M$ and $M\rightarrow E'\rightarrow X$ where $X$ is some $M$ bundle over $M$?

And just in case there is no general relationship,

Is there a reason I should have expected there to be a relationship in my examples, even though in general there isn't?

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If $G$ acts freely on $M_1\times M_2$ and $G=G_1\times G_2$ where $G_i$ preserves $M_i$ then $(M_1\times M_2)/G$ can be thought of as a fibration with base $M_1/G_1$ and fibre $M_2/G_2$ or vice-versa. Is this the case with your examples? –  Somnath Basu Apr 11 '10 at 2:21
@Somnath: It doesn't QUITE seem to be of that type, though it's close: the first circle factor of $T^3$ acts only on the first $S^3$ factor, the second circle factor acts on the first two $S^3$ factors, and the last circle factor acts on all 3 $S^3$ factors. Or maybe I'm missing something obvious... –  Jason DeVito Apr 11 '10 at 2:36

In terms of Seifert fiber spaces, there are two examples when you consider torus bundles over $S^1$ among the ones which use periodic mapping classes: These are $(No,1|(1,0))$ and $(No,1|(1,1))$ which are respectively $K\times S^1$ and $K\times_{\tau}S^1$, where $\tau$ is the unique Dehn's twist on the Klein bottle.
Those fibrations are not unique because also $K\times S^1=(NnI,2|(1,0))$ and $K\times_{\tau}S^1=(NnI,2|(1,1))$. Curiously, if $T=S^1\times S^1$ is the 2-torus, then $T\hookrightarrow K\times S^1\to S^1$ is a "non-trivial" fibering.