An algebraic generalization of Giroux's criterion

TL;DR

This paper extends Giroux's criterion from dimension two to higher dimensions by providing an algebraic method to determine when a contact structure has non-vanishing contact homology, using advanced algebraic and geometric tools.

Contribution

It introduces an algebraic generalization of Giroux's criterion for higher dimensions, characterizing contact structures via contact homology and developing new algebraic invariants.

Findings

Determines when contact homology is non-zero for higher-dimensional contact structures.

Provides explicit computations of contact homology in the non-vanishing case.

Defines bilinearized homology theories for DGAs over .

Abstract

Let $\xi$ be a $\tau$-invariant contact structure on $N(W) = \mathbb{R}_{\tau} \times W$ for a closed, $2n$-dimensional manifold $W$, so that each $\{\tau\} \times W$ is a convex hypersurface. When $n=1$, Giroux's criterion provides a simple means of determining exactly when $\xi$ is tight. It is an open problem to find a generalization applicable for $n>1$. This article solves an algebraic version of the problem, determining exactly when $(N(W), \xi)$ has non-vanishing contact homology ($CH$) and computing $CH(N(W), \xi)$ when it is non-zero. The result can be expressed in terms of homotopy equivalence of augmentations of the chain level $CH$ algebra of the dividing set or in terms of bilinearized homology theories, which we define for free, commutative DGAs over $\mathbb{Q}$. Our proof relies on the development of obstruction bundle gluing in the Kuranishi setting.