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

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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.

The inequality

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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 closed oriented surface   of positive genus g.

Theorem (Milnor, 1958)[1] Let   be a flat oriented linear circle bundle. Then the Euler number of the bundle satisfies  .

Theorem (Wood, 1971)[2] Let   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  , doubling the Euler number.

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

References

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  1. ^ J. Milnor. "On the existence of a connection of curvature zero". Comment. Math. Helv. 21 (1958): 215–223.
  2. ^ J. Wood (1971). "Bundles with totally disconnected structure group" (PDF). Comment. Math. Helv. 46 (1971): 257–273. doi:10.1007/BF02566843. S2CID 121003993.