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.