The classifying space $BDiff(S^1 \times S^1)$ of the diffeomorphism group of the torus is a 2-type with $\pi\_1 = GL(2, \mathbb{Z})$ and $\pi\_2 = \mathbb{Z} \times \mathbb{Z}$ and all higher $\pi\_i$ = 0.


Presumably the action of $\pi\_1$ on $\pi\_2$ is the standard action of $GL(2, \mathbb{Z})$ on $\mathbb{Z} \times \mathbb{Z}$.


What is its first Postnikov invariant $k \in H^3\_{grp}(GL(2, \mathbb{Z}), \mathbb{Z} \times \mathbb{Z})$? 



What is the first Postnikov invariant of $BDiff(S^1 \times S^1)$?
It is zero, because $BDiff(S^1 \times S^1) \to BGL\_2(\mathbb{Z})$ is split by the standard action of $GL\_2(\mathbb{Z})$ on $S^1 \times S^1 = \mathbb{R}^2/\mathbb{Z}^2$. In other words
$$BDiff(S^1 \times S^1) \simeq B(S^1 \times S^1 \rtimes GL\_2(\mathbb{Z})).$$


