An introduction to contact topology by Hansjörg Geiges

By Hansjörg Geiges

This article on touch topology is the 1st entire creation to the topic, together with contemporary notable functions in geometric and differential topology: Eliashberg's evidence of Cerf's theorem through the class of tight touch buildings at the 3-sphere, and the Kronheimer-Mrowka evidence of estate P for knots through symplectic fillings of touch 3-manifolds. beginning with the elemental differential topology of touch manifolds, all points of third-dimensional touch manifolds are handled during this publication. One extraordinary function is an in depth exposition of Eliashberg's type of overtwisted touch constructions. Later chapters additionally care for higher-dimensional touch topology. right here the point of interest is on touch surgical procedure, yet different structures of touch manifolds are defined, equivalent to open books or fibre attached sums. This publication serves either as a self-contained advent to the topic for complicated graduate scholars and as a reference for researchers.

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0. This means that φω (v) = 0 ∈ V ∗ . In other words, φω is injective, and hence ω symplectic. 1) it follows that an isotropic subspace of a 2n–dimensional symplectic vector space has dimension less than or equal to n; Lagrangian subspaces are precisely the isotropic subspaces of dimension n. 7 Observe that an equivalent condition for ξ ⊂ T M to be a contact structure on the (2n + 1)–dimensional manifold M is that for any (local) 1–form α with ξ = ker α we have that (dα)n |ξ = 0, in other words, that (ξp , dα|ξ p ) be a symplectic vector space for all points p ∈ M .

Indeed, if M has dimension 2n − 1, then ω n = (et (dt ∧ α + dα))n = nen t dt ∧ α ∧ (dα)n −1 = 0. One easily sees that the vector field ∂t is a Liouville vector field for this symplectic form ω. The manifold (R × M, ω) is called the symplectisation of (M, α). The orientation of R×M induced by the volume form ω n coincides with the product orientation on R × M , where M is oriented by α ∧ (dα)n −1 . 8 Consider W = R4 with its standard symplectic form ω = dx1 ∧ dy1 + dx2 ∧ dy2 . The radial vector field 1 1 Y = r ∂r = (x1 ∂x 1 + y1 ∂y 1 + x2 ∂x 2 + y2 ∂y 2 ) 2 2 (where r2 = x21 + y12 + x22 + y22 ) is a Liouville vector field for ω.

3. (c) The map Φ: RP 2 (q × (RP 2 )∗ , p) −→ RP 2 −→ (ϕ∗ (p) × (RP 2 )∗ , ϕ(q)) restricts to a contact transformation of M . Proof (a) Consider a coordinate transformation q = Bq of RP 2 , with B an invertible 3 × 3 matrix. e. with respect to the new coordinates (with tildes) the conic is described by the matrix A = B t AB. Moreover, we have pq = 0 if and only if pBq = 0, which implies that the corresponding coordinate transformation of (RP 2 )∗ is given by p = pB. With respect to the old coordinates, we had ϕ(q) = qt A = qt B t A; with respect to the new coordinates we have ϕ(q) = qt A = qt B t AB = ϕ(q)B.

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