Contact homology and linearization without dga homotopies
Julian Chaidez
TL;DR
This work resolves the status of linearized contact homology within Pardon's contact DG-algebra framework by showing that the augmentation data carry canonical invariants: the set $Aug(Y,\xi)$ is bijectively determined, and $LCH_{[\epsilon]}(Y,\xi)$ is well-defined up to isomorphism for each augmentation class. The authors develop a self-contained homological-algebra toolkit for pointed CDGAs, including Sullivan models, path objects, and a main homological invertibility lemma in characteristic zero, and they connect linearization to this framework through $LC(A,\epsilon)$ and $LH(A,\epsilon)$. They then embed these results into the contact-homology context, showing that the Pardon-algebra $A(Y,\alpha)$ is Sullivan with an action filtration and that $LCH_{[\epsilon]}(Y,\xi)$ is determined by augmentation classes via $LH(A(Y,\xi),[\epsilon])$, independent of choices up to canonical isomorphism. The paper further develops a model-category perspective for CDGAs, proving factorization and replacement results that underpin invariance statements and applying them to ADNH and SADC classes to establish augmentation uniqueness and LCH stability, with applications to convex dividing sets and fillings. Overall, the results provide a robust, choice-free foundation for linearized contact homology as a genuine contact invariant under Pardon's foundations and standard cobordism maps, with concrete consequences for Reeb dynamics and rigidity phenomena.
Abstract
This article clarifies the status of linearized contact homology given the foundations of the contact dg-algebra established by Pardon. In particular, we prove that the set of isomorphism classes of linearized contact homologies of a closed contact manifold is a contact invariant.