On elementary transversely affine foliations. I (Q753212)

A codimension one transversely affine foliation of a closed manifold M is said to be elementary if the image of its holonomy homomorphism in the group of affine transformations of the real line is abelian. The main theorem says that a transversely oriented foliation F admits an elementary transversely affine structure if and only if (1) F is almost without holonomy, (2) for any connected component U of the complement of the union of all compact leaves, the transverse orientation of F is either inward or outward along the whole boundary of U, (3) there exists a spherical cohomology class A in SH(M,R) such that for any U as above, the inclusion of U into M maps A, up to sign, to the cohomology direction of F in U.