Augmented Legendrian cobordism in $J^1S^1$

Abstract: We consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $\mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $\mathbb{F}_2$ this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $\dim M = 2$ realizing any prescribed monodromy representation, $\Phi:\pi_1(M,x_0) \rightarrow \mathit{GL}(\mathbf{n}, \mathbb{F})$.