In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure.
For linear bundles, flatness is defined as the vanishing of the curvature form of an associated connection.
An arbitrary smooth (or topological) d-dimensional fiber bundle is flat if it can be endowed with a foliation of codimension d that is transverse to the fibers.
of positive genus g. Theorem (Milnor, 1958)[1] Let
be a flat oriented linear circle bundle.
Then the Euler number of the bundle satisfies
be a flat oriented topological circle bundle.
Then the Euler number of the bundle satisfies
Wood's theorem implies Milnor's older result, as the homomorphism
classifying the linear flat circle bundle gives rise to a topological circle bundle via the 2-fold covering map
Either of these two statements can be meant by referring to the Milnor–Wood inequality.