View source for Milnor–Wood inequality

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. It is named after [[John Milnor]] and John W. Wood.

== Flat bundles ==
For [[vector bundle|linear bundles]], [[Flat vector bundle|flatness]] is defined as the vanishing of the curvature form of an associated [[Connection (vector bundle)|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.

== The inequality ==
The Milnor–Wood inequality is named after two separate results that were proven by [[John Milnor]] and John W. Wood. Both of them deal with orientable circle bundles over a [[Surface (topology)|closed oriented surface]] <math> \Sigma_g </math> of positive genus ''g''.

'''Theorem (Milnor, 1958)<ref>{{cite journal|title=On the existence of a connection of curvature zero|author=J. Milnor|journal=[[Comment. Math. Helv.]]| volume = 21|pages=215–223|issue = 1958|url=https://eudml.org/doc/139154}}</ref> ''' Let <math>\pi\colon E \to \Sigma_g </math> be a flat oriented linear circle bundle. Then the [[Euler class|Euler number]] of the bundle satisfies <math> |e(\pi)| \leq g -1</math>.

'''Theorem (Wood, 1971)<ref>{{cite journal|author=J. Wood|title=Bundles with totally disconnected structure group|journal=[[Comment. Math. Helv.]]|year=1971 |volume=46|pages=257–273|issue = 1971|url = https://link.springer.com/content/pdf/10.1007%2FBF02566843.pdf|doi=10.1007/BF02566843|s2cid=121003993 }}</ref> ''' Let <math> \pi\colon E \to \Sigma_g </math> be a flat oriented topological circle bundle. Then the [[Euler class|Euler number]] of the bundle satisfies <math> |e(\pi)| \leq 2g -2</math>.

Wood's theorem implies Milnor's older result, as the homomorphism <Math>\pi_1:\Sigma\to SL(2,\R)</Math> classifying the linear flat circle bundle gives rise to a topological circle bundle via the 2-fold [[covering map]] <Math>SL(2,\R)\to PSL(2,\R) \subset \operatorname{Homeo}^+(S^1)</Math>, doubling the Euler number.

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

== References ==
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[[Category:Differential geometry]]
[[Category:Geometric topology]]