Abstract
We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some high dimensional overtwisted spheres is not homotopically equivalent to a finite dimensional Lie group.
Highlights
Introduction and statements of the resultsLet (M, ξ) be a closed contact manifold
These short notes are concerned with the relationship between the topology of the connected component Diff0(M ) of the identity in the group of diffeomorphisms of M and its subgroup Diff0(M, ξ) consisting of contactomorphisms of (M, ξ)
The path components of the group of contactomorphisms of particular contact manifolds have been studied by several authors in the literature; see for instance Ding–Geiges [8], Dymara [9], Gironella [16, 17], Giroux [18], Giroux–Massot [19], Lanzat–Zapolsky [23], Massot– Niederkrüger [26], and Vogel [28]
Summary
Let (M , ξ) be a closed contact manifold These short notes are concerned with the relationship between the topology of the connected component Diff0(M ) of the identity in the group of diffeomorphisms of M and its subgroup Diff0(M , ξ) consisting of contactomorphisms of (M , ξ). Higher–order homotopy groups have been studied: for instance, Casals–Presas [6], Casals–Spácil [7] and Eliashberg [11] contain results for the case of the standard tight (2n + 1)–contact sphere. In this notes, we deal with the case of overtwisted spheres (cf Borman et al [2]). We point out that these methods could be applied to the case of any overtwisted contact manifold (M 2n+1, ξ) such that both the homotopy type of the space of almost contact structures on M and the diffeomorphism group of M can be (at least partially) understood
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