A conjecture in arithmetic theory of differential equations by Katz N.M.

By Katz N.M.

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For any flow J 7 in a compact manifold M , catj(M, T) = 2. 22 HELLEN COLMAN 2. For any 2-dimensional Seifert fibration T, cat t (M, T) = 3. 3. Any 2-dimensional foliation in the 3-sphere has tangential category 3. 2 Transverse category An open subset U of M is transversely categorical if the inclusion map {U,Tu) *-> (M, T) factors through a leaf up to foliated homotopy. 6 The transverse category cat^ (M, T) of a foliated manifold (M, T) is the least number of transversely categorical open sets required to cover M.

One consequence of the Heitsch-Hurder reformulation of Dummy's methods was that it is possible to renormalize the transverse volume form using 36 STEVEN HURDER a leafwise smooth, transversally measurable change of scale function. If such a change of scale function can be chosen so that the new transverse volume form has uniform arbitrarily slow growth, then the modular form r\ is uniformly estimated arbitrarily close to 0, hence the Godbillon measure must vanish. This is just a measurable version of the original idea from Herman [53]!

8. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier, 26 (1976), 265-282. 9. E. Fadell, The equivariant Lusternik-Schnirelmann method for invariant functionals and relative cohomological index theories, in Methodes Topologiques en Analyse Non-Lineaire, ed. A. Granas, Montreal, 1985. 10. A. Haefiiger, Varietes feuilletes, Ann. Scuola Norm. Pisa, (3) 16 (1962), 367-397. 11. G. Hector and U. Hirsch, Introduction to the Geometry of foliations, A and B, Vieweg, 1981-83. 12. L. Heitsch, A cohomology for foliated manifolds, Comment.

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