Contact homology and homotopy groups of the space of contact structures
type: Item
instance of: scholarly article
title: Contact homology and homotopy groups of the space of contact structures
zbMATH Open document ID: 1099.57023
author: Frédéric Bourgeois
published in: Mathematical Research Letters
publication date: 2006-03-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0407531
review text: 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 \(H_{1}(M,\mathbb{Z})\neq0\).
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D35
zbMATH DE Number: 5014794
zbMATH Keywords: contact manifold
zbMATH Keywords: contact structure
zbMATH Keywords: fundamental group
zbMATH Keywords: homotopy group
zbMATH Keywords: contact homology
zbMATH Keywords: Reeb orbit
zbMATH Keywords: 3-manifold
zbMATH Keywords: Riemannian manifold
reviewed by: Albert Sheu
MaRDI profile type: Publication
arXiv ID: math/0407531
DOI: 10.4310/MRL.2006.V13.N1.A6
description: scientific article