By Y. Eliashberg

This e-book provides the 1st steps of a idea of confoliations designed to hyperlink geometry and topology of three-d touch constructions with the geometry and topology of codimension-one foliations on three-d manifolds. constructing virtually independently, those theories firstly look belonged to 2 diversified worlds: the speculation of foliations is a part of topology and dynamical structures, whereas touch geometry is the odd-dimensional 'brother' of symplectic geometry. notwithstanding, either theories have built a couple of awesome similarities. Confoliations - which interpolate among touch constructions and codimension-one foliations - will help us to appreciate larger hyperlinks among the 2 theories. those hyperlinks offer instruments for transporting effects from one box to the other.It's gains contain: a unified method of the topology of codimension-one foliations and make contact with geometry; perception at the geometric nature of integrability; and, new effects, specifically at the perturbation of confoliations into touch buildings

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**Sample text**

9. (4) The ﬁbers of g can be completely described. Presumably, the C ∞ stable map g : M = CP 2 → R3 thus constructed coincides with Kobayashi’s example presented in [26, 27]. By choosing an appropriate orientation for M , we may assume that it is orientation preservingly diﬀeomorphic to CP 2 . By our construction, it is easy to see that g −1 (Y ) is diﬀeomorphic to D4 . Hence f −1 (R3 Int Y ) is diﬀeomorphic to CP 2 − Int D4 . Let us determine the diﬀeomorphism type of f −1 (Y ). Take a properly embedded 2-disk D32 in Y as in Fig.

We may assume that N (A1j ) ∼ = D2 × [0, 1] is attached to N (A01 ) along D2 × 0 {0, 1} and that N (A1 ) ∪ N (A11 ) ∪ N (A12 ) ∪ N (A13 ) is a regular neighborhood of A01 ∪ A11 ∪ A12 ∪ A13 in R3 . Similarly, we construct N (A2j ), j = 1, 2, . . , 7, and N (A3j ), j = 1, 2, . . , 5, so that the family of closed sets {N (Aij )}0≤i≤3 50 6 Examples of Stable Maps of 4-Manifolds R 2 T(3) 2 Fig. 3. A 2-parameter deformation of Morse functions on T(3) covers R3 and that distinct members intersect only along their boundaries.

Let f : M → N be a C ∞ stable map of a closed orientable 4-manifold into a 3-manifold. Set 0o (f ) = {y ∈ N f (S(f )) : b0 (f −1 (y)) ≡ 1 (mod 2)}, 0e (f ) = {y ∈ N f (S(f )) : b0 (f −1 (y)) ≡ 0 (mod 2)}. It is easy to see that they are disjoint open sets of N . Furthermore, since M is compact, the closure 0o (f ) of 0o (f ) is compact. Let y and y be points in N belonging to adjacent regions of N f (S(f )). Since M is orientable, the diﬀerence between the numbers of components of the ﬁbers over y and y is always equal to one.