2023/07/18 by Russell Avdek, Avdek, Russell · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2307.09068
Let ξ be a τ-invariant contact structure on N(W) = ℝτ × W for a closed, 2n-dimensional manifold W, so that each \τ\ × W is a convex hypersurface. When n=1, Giroux's criterion provides a simple means of determining exactly when ξ 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), ξ) has non-vanishing contact homology (CH) and computing CH(N(W), ξ) 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 ℚ. Our proof relies on the development of obstruction bundle gluing in the Kuranishi setting.