Semi-global Kuranishi charts and the definition of contact homology
This fundamental work presents important applications of co-called Kuranishi perturbation theory. The authors define the full contact homology differential graded algebra (DGA) for an arbitrary closed contact manifold in any dimension and show that its isomorphism class is an invariant of the contact manifold. Namely, for a closed cooriented contact manifold $(M,\xi)$ and some auxiliary data $D$ which includes a nondegenerate contact form $\alpha$ for $\xi$, a direct limit DGA $A(D)$ is defined. Its homology is called the full contact homology algebra $HC(D)$. The most important results of the work are the following two Theorems. Theorem 1. If $D_1$ and $D_2$ are two choices of auxiliary data for the manifold $(M,\xi)$, then the algebras $HC(D_1)$ and $HC(D_2)$ are isomorphic. That is why the isomorphism class of the algebra $HC(D)$ is an invariant of the manifold $(M,\xi)$. Theorem 2. Let $(W,\alpha)$ be a compact Liouville cobordism which restricts to $(M_+, \xi_+=\mathrm{Ker}\alpha|M_+)$ at the positive boundary and to $(M_-, \xi_-=\mathrm{Ker}\alpha|M_-)$ at the negative boundary. Then there is an algebra homomorphism \[ \Phi:HC(\xi_+)\to HC(\xi_-). \] In the case when $\xi_+=\xi_-$ and $(W,\alpha)$ is the trivial cobordisms the homomorphism $\Phi$ is identical. Here $HC(\xi)$ is the isomorphism class of the full contact homology algebra $HC(D)$.
scientific article; zbMATH DE number 7656377
scientific article