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

Abstract

We consider Legendrian links and tangles in $J^1{S}^1$ 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 $J^{1}M$ with $\dim M=2$ realizing any prescribed monodromy representation, $\Phi :{\pi }_1(M,x_0)\to \mathrm{GL}(\mathbf{n},\mathbb{F})$.