mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, more specifically in

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

and

geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated i ...

, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

and John W. Wood.

Flat bundles

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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

is flat if it can be endowed with a

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...

of codimension d that is transverse to the fibers.

The inequality

The Milnor–Wood inequality is named after two separate results that were proven by

and John W. Wood. Both of them deal with orientable circle bundles over a closed oriented surface

\Sigma_g

of positive genus ''g''. Theorem (Milnor, 1958) Let

\pi\colon E \to \Sigma_g

be a flat oriented linear circle bundle. Then the

Euler number In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion :\frac = \frac = \sum_^\infty \frac \cdot t^n, where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...

of the bundle satisfies

, e(\pi),  \leq g -1

. Theorem (Wood, 1971) Let

\pi\colon E \to \Sigma_g

be a flat oriented topological circle bundle. Then the

of the bundle satisfies

, e(\pi),  \leq 2g -2

. Wood's theorem implies Milnor's older result, as the homomorphism

\pi_1:\Sigma\to SL(2,\R)

classifying the linear flat circle bundle gives rise to a topological circle bundle via the 2-fold

covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...

SL(2,\R)\to PSL(2,\R) \subset \operatorname^+(S^1)

, doubling the Euler number. Either of these two statements can be meant by referring to the Milnor–Wood inequality.

References

{{DEFAULTSORT:Milnor-Wood inequality Differential geometry Geometric topology