Contact homology and homotopy groups of the space of contact structures
On an orientable manifold \(M\) of odd dimension \(2n+1\), a contact structure is a smooth field \(\xi\) of tangent hyperplanes over \(M\), i.e. a \(2n\)-dimensional vector subbundle of the \((2n+1)\)-dimensional tangent bundle \(TM\) of \(M\), which is completely non-integrable. More precisely, \(\xi =\ker(\alpha)\) for some 1-form \(\alpha\) on \(M\) such that \(\alpha\wedge(d\alpha)^{n}\) is a (nowhere vanishing) volume form on \(M\). In this paper, the author uses contact homology to derive some topological results about the space \(\Xi(M)\) of contact structures on \(M\). First the author derives Geiges and Gonzalo's earlier results on 3-manifolds, for example, the theorem that for the 3-torus \(T^{3}\), the fundamental group \(\pi_{1}(\Xi(T^{3}),\xi_{n})\), based at any of the representative tight contact structures \(\xi_{n} :=\ker(\cos(n\theta)dx+\sin(n\theta)dy)\), \(n\in\mathbb{N}\), contains an infinite cyclic subgroup, and a similar statement for more general 3-manifolds which are \(T^{2}\)-bundles over \(S^{1}\). It is then shown that the usage of contact homology allows to study higher dimensional manifolds and higher order homotopy groups. Indeed, the author shows that \(\pi_{1}(\Xi(T^{5}),\xi)\) at a specific contact structure \(\xi\) contains a subgroup \(\mathbb{Z}^{3}\), \(\pi_{2n-1}(\Xi(T^{2n}\times S^{2n-1}),\xi_{0})\) based at the contact structure \(\xi_{0}:=\ker( \sum_{i=1}^{2n} y_{i}dx_{i}) \) contains an infinite cyclic subgroup, and a similar result for the unit cotangent bundle \(ST^{\ast}M\) of a \(2n\)-dimensional compact orientable Riemannian manifold \(M\) with negative sectional curvature and \(H_{1}(M,\mathbb{Z})\neq0\).
scientific article