An introduction to contact topology by Hansjörg Geiges

By Hansjörg Geiges

This article on touch topology is the 1st complete advent to the topic, together with contemporary outstanding purposes in geometric and differential topology: Eliashberg's facts of Cerf's theorem through the category of tight touch constructions at the 3-sphere, and the Kronheimer-Mrowka facts of estate P for knots through symplectic fillings of touch 3-manifolds. beginning with the elemental differential topology of touch manifolds, all facets of third-dimensional touch manifolds are taken care of during this e-book. One amazing characteristic is an in depth exposition of Eliashberg's class of overtwisted touch buildings. Later chapters additionally take care of higher-dimensional touch topology. right here the focal point is on touch surgical procedure, yet different buildings of touch manifolds are defined, equivalent to open books or fibre attached sums. This booklet serves either as a self-contained advent to the topic for complicated graduate scholars and as a reference for researchers.

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7 Applications of contact geometry to topology 41 boundary, we have a natural restriction homomorphism −→ Diff (S n −1 ) −→ f |S n −1 . Diff (Dn ) f ρn : The group Γn is defined as Γn = Diff (S n −1 )/im ρn . In order to show that this is indeed a group, we need to prove that im ρn is a normal subgroup in Diff (S n −1 ). We begin with two lemmata. Write Diff 0 (S n −1 ) for the group of diffeomorphisms of S n −1 that are isotopic to the identity. 1 With the commutator [f, g] of two diffeomorphisms f, g defined by [f, g] = f ◦ g ◦ f −1 ◦ g −1 we have [Diff (S n −1 ), Diff (S n −1 )] ⊂ Diff 0 (S n −1 ).

Then dim L ≤ n. 5 The geodesic flow and Huygens’ principle 35 Proof Write i for the inclusion of L in M and let α be an (at least locally defined) contact form defining ξ. Then the condition for L to be isotropic becomes i∗ α ≡ 0. It follows that i∗ dα ≡ 0. In particular, Tp L ⊂ ξp is an isotropic subspace of the 2n–dimensional symplectic vector space (ξp , dα|ξ p ). 6 we have dim Tp L ≤ (dim ξp )/2 = n. 13 An isotropic submanifold L ⊂ (M 2n +1 , ξ) of maximal possible dimension n is called a Legendrian submanifold.

If B comes equipped with a Riemannian metric g, the space of cooriented contact elements can be naturally identified with either of ST B and ST ∗ B: the identification with ST B is given by associating to (b, V ) the vector X ∈ Tb B positively orthonormal to the cooriented hyperplane V ⊂ Tb B with respect to the inner product gb ; the corresponding element in ST ∗ B is Ψb (X) = gb (X, −). Under this identification, the Liouville form λ on ST ∗ B defines the natural contact structure on the space of cooriented contact elements.

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