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Extra info for Algorithmic Geometry [auth. unkn.]
Crossbows. . The last wave front drawn (right of Figure 5) represents an exact Lagrangian immersion of the circle with two double points, which is regularly homotopic to the standard embedding (exactness meaning that the total area enclosed by this curve is zero). It appeared in Arnold’s papers on Lagrangian cobordisms : this is the generator of the cobordism group in dimension 1. Arnold calls it “the crossbow”. Which reminds me of something Stein is supposed to have told Remmert in 1953 when he learned the use Cartan and Serre made of sheaves and their cohomology to solve problems in complex analysis: “The French have tanks.
In contact geometry, this discrepancy does not generate any confusion thanks to the following lemma. It is a rather technical point but we discuss it here anyway because it doesn’t appear to be published anywhere else, although it is mentioned in [9, page 629]. Lemma 11 (Giroux). If two singular foliations on a surface have the same leaves and if their singularities have non-zero divergence then they are equal. The following proof can be safely skipped on ﬁrst reading. Proof. The statement is clear away from singularities and a partition of unity argument brings it down to a purely local statement.
It means we will construct the generating vector ﬁeld Topological Methods in 3-Dimensional Contact Geometry 37 Fig. 9. Proof of Gray’s theorem Xt rather than ϕt directly. The compactness assumption will guaranty that the ﬂow of Xt exists for all time. At any point p, if the plane ξt+ε coincides with ξt then we have nothing to do and set Xt = 0. Otherwise, these two planes intersect transversely along a line dt,ε . The natural way to bring ξt+ε back to ξt is to rotate it around dt,ε . Since we know from the proof of Theorem 2 that the ﬂow of Legendrian vector ﬁelds rotate the contact structure, we will choose Xt in the line dt := limε→0 dt,ε , see Figure 9.