Introduction to the Geometry of Foliations, Part B: by Dr. Gilbert Hector, Dr. Ulrich Hirsch (auth.)

By Dr. Gilbert Hector, Dr. Ulrich Hirsch (auth.)

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The quotient under this action is a closed orientable 3-manifold iii) Let h G ~ Z L - 36 - be a homomorphism. Since phic to Z2 ~ Z2' h factors through we conclude that h G/[G,G], which is isomor- must be trivial. This implies that I H (L;lR) = HI (L;lR) = O. iv) For fig. 7 so that I = fog [-1,1] we let -I gof, g E Diff7(I) be as indicated in f: lR .... lR is the flip. We define where a representation by H(I,I) (M,F) The suspension Since H(y) a leaf of of = g and preserves the origin for any F. On the other hand, for This shows that all leaves of = t Dm(SI) f.

1jJ. 1. 1. 1jJ. 1. in W.. (When 1. defined only on a neighbourhood of ~ L. x [0, I) 1. On quotient Lx[o,I). L. x [o,I)/TlIL 1. is trivial. We may there- acts 1. TlIL er is L- L. is not compact 1. x {o} in ~ L. x [0, I)) • 1. via the holonomy of Li' and the isomorphic to (a subpseudobundle of) Thus we get a commutative diagram of maps [0, I) Lx [0,I) where Tl i and fi are quotient maps. 1. [W. 1. 1. 1. Co. we use special be an open neighbourhood 1. denotes the universal covering then of Fm constructed as folIows.

10. 2. iv) Instead of on 51 Fm R we could have used any for the construction of er Reeb foliation Of course this would have lead to a foliation which is (in general only) homeomorphic to (one of the two possibilities of) Fe. - 51 - In what follows we mean by a turbulized foliation more generally any foliation which is obtained in this way. v) Sometimes we shall refer to the converse process as a deturbulization of Fe. - is of class Co, subject to the possible exception mentio- ned in 5. 5. 4 it not available here.

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