E8-plumbings and Exotic Contact Structures on Spheres

The standard contact structure ξst on the unit sphere S 2n−1 = ∂D2n ⊂ Cn can be defined as the hyperplane field of complex tangencies. In other words, if we write J0 for the complex structure on (the tangent bundle of) C n, then ξst(p) = TpS 2n−1 ∩ J0(TpS 2n−1) for all p ∈ S2n−1. Conversely, given any contact structure ξ = kerα on S2n−1 (with α a 1-form such that α ∧ (dα)n−1 is a volume form defining the standard orientation), there is a homotopically unique complex bundle structure J on ξ such that dα(., J.) is a J-invariant Riemannian metric on ξ. This extends to a complex structure J on TR|S2n−1 by requiring that J send the outer normal of S2n−1 to a vector field X on S2n−1 with α(X) > 0. If this J extends as an almost complex structure over the disc D2n, then ξ is called homotopically trivial. The complex structure J |ξ and the trivial line bundle spanned by the vector field X can be interpreted as a reduction of the structure group of TS2n−1 to Un−1 × 1. Such a reduction is called an almost contact structure. Homotopy classes of almost contact structures on S2n−1 are classified by