A generalization of milnor's inequality concerning affine foliations and affine manifolds

Abstract

Fixed the Euler number takes on infinitely many values. In dimension two the Euler number is a complete invariant of the isomorphism class of the bundle. In higher dimensions, the Euler class ~1> and the characteristic classes of Pontryagin determine the isomorphism class up to finite number of possibilities. Now suppose the family is endowed with a continuous system of isomorphisms between the vector spaces Vx and Vx, over nearby points x and x' in the manifold M. We assume the natural compatibility of isomorphisms for three nearby points x, x', and x". This system of isomorphisms can be pictured as a foliation of the total space of the bundle transverse to the fibres with the zero section a leaf and the holonomy linear. We will construct a finite upper bound kM depending only on the topology of the manifold M for the absolute value of the Euler number of a bundle admitting one of these affine (or linear) foliations. If M is surface of genus g, we recover the inequality of Milnor [M], Euler number of a 2-plane bundle over M with affine foliation < g. In dimension 2 Milnor showed that this condition is also sufficient for a 2-plane bundle to admit a linear foliation transverse to the fibres. In higher dimensions we are far from such a sufficient condition. Our argument which is quite direct and geometric, clarifies John Wood's generalization of Milnor's work to foliated Sl-bundles over surfaces, [W]. This aspect of the paper was worked out with John Morgan.