The Gauss-Bonnet Theorem, which was generalized by Shiing-Shen Chern in 1944 to all oriented closed even-dimensional smooth manifolds, correlates the curvature of the Levi-Civita connection of a Riemannian manifold with its Euler characteristic. This result provides a strong restriction on the kind of geometry such a manifold can support. For instance, let us consider Euclidean manifolds, i.e. manifolds which admit an atlas whose coordinate change functions are isometries of $\mathbb R^n$. Since the curvature of the Levi-Civita connection they inherit from $\mathbb R^n$ vanishes, the Euler characteristic represents an obstruction to the existence of Euclidean structures: indeed if $\chi(M) \neq 0$ then $M$ cannot support a flat metric.
On the other hand, let us consider affine manifolds, i.e. manifolds which admit an atlas whose coordinate change functions are affine isomorphisms of $\mathbb R^n$. These can be characterized as those manifolds whose tangent bundle supports flat and symmetric connections. Although they may seem to be a mild generalization of Euclidean manifolds, the attempt to generalize the above result to affine manifolds resulted in the formulation of a long standing open conjecture:
\textbf{Conjecture 1:} The Euler characteristic of a closed oriented affine manifold vanishes.
The key point is that the Euler characteristic of a manifold cannot be computed from the curvature of an arbitrary linear connection $\nabla$, because it is essential for $\nabla$ to be compatible with a Riemannian metric. Now, although Conjecture 1 was shown to hold true for complete affine manifolds by Bertram Kostant and Dennis Sullivan, the non-complete case is much more difficult. There are known examples, due to John Smillie, of manifods with non-zero Euler characteristic and flat tangent bundle in every even dimension greater than 2. However, as William Goldman points out, since the torsion of these connections seems hard to control they do not disprove Conjecture 1. None of Smillie’s manifolds is aspherical, and indeed another open conjecture is:
\textbf{Conjecture 1:} The Euler characteristic of a closed oriented aspherical manifold whose tangent bundle is flat vanishes.
The first important breakthrough was made by John Milnor in 1958, when he proved both conjectures for closed oriented surfaces. He exploited the fact that the existence of a flat connection on a rank-$m$ vector bundle $\pi : E \rightarrow M$ is equivalent to the existence of a holonomy representation $\rho : \pi_1(M,x_0) \rightarrow \mathrm{GL}^+(m,\mathbb R)$ which induces the bundle. The study of all possible holonomy representations for closed oriented surfaces enabled him to establish a much more detailed result: he managed to characterize all flat oriented plane bundles over closed oriented surfaces by means of their Euler class, that is a cohomology class in the cohomology ring of the base space which generalizes the Euler characteristic. What happens is that the Euler class of flat bundles over a fixed surface $\Sigma$ is bounded, that is, just a finite number (up to isomorphism) of oriented plane bundles over $\Sigma$ can support flat connections. In particular, none of these is the tangent bundle if $\Sigma$ is not the torus. This remarkable result is now known as Milnor-Wood inequality (the name celebrates John Wood's generalization to $S^1$-bundles).
While it has been proven that the boundedness of the Euler class of flat bundles generalizes to all dimensions, Conjectures 1 and 2 remain elusive. Indeed one needs explicit inequalities in order to determine whether the tangent bundle can be ruled out from the flat ones or not. Since Milnor's work very little progress has been made until very recently. In 2011 Michelle Bucher and Tsachik Gelander published a generalization of Milnor-Wood inequality to closed oriented manifolds whose universal cover is isometric to $(\mathbb H^2)^n$, thus confirming both conjectures for all manifolds which are locally isometric to a product of surfaces of constant curvature. Their work, which takes up the largest part of our exposition, uses the theory of bounded cohomology developed by Mikha\"{i}l Gromov in 1982 and some deep results about the super-rigidity of lattices in semisimple Lie groups due to Gregori Margulis.
