Let $M$ be an instantaneously analytic, closed, $C^\infty,$ $2-$manifold, with genus, $g=0,$ endowed with a metric that restricts contractibility. Contractibility is restricted to avoid the deformation of $M$ into $S^2.$
An important problem is to
Determine the set of all the isomorphism classes of differentiable $M$-bundles
$$ \pi:E\to X $$
over $X:=\Bbb T^3.$
What I know is that this problem can be translated into a problem in homotopy theory. Let Diff$(M)$ be the diffeomorphism group of $M$ equipped with the $C^\infty$ topology and let $B$ Diff $(M)$ be its classifying space. Then we have a natural identification
$\{$isomorphism classses of smooth M-bundles over X$\} \cong \ [X,B$ Diff $(M)$$].$
where the RHS denotes the set of all the homotopy classes of continuous mappings from $X$ to $B$ Diff $(M).$ So in essence I have to figure out how to compute the homotopy groups $\pi_i($Diff $(M)).$