The Milnor-Wood inequality, introduced in two landmark papers by John Milnor (1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat ${\mathbb{S}}^1$-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with “linear” structures, which Gromov later generalized using bounded cohomology.

Wood extended Milnor’s result to “non-linear” flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented ${\mathbb{S}}^3$-bundles. In this talk, we will also discuss the surprising ways in which inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.