An algebraic generalization of Giroux's criterion

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.