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

Augmentations and immersed Lagrangian fillings

“…(3) As Legendrians that admit augmentations exhibit significantly more rigid behavior than general Legendrians, we do not see any a priori reason to expect that cobordism classes of 𝜌-graded augmented Legendrians should closely match the classical cobordism classes of Legendrians. In fact, in the case of 𝐽 1 𝑆 1 we will show in [31] that there are many examples of non-cobordant augmented Legendrians that become cobordant if one ignores the augmentations.…”

“…We expect that a version of this isomorphism should extend to a more general coefficient ring, for example, to the setting of the LCH DGA with fully non-commutative ℤ[𝜋 1 (Λ)] coefficients which is appropriate for considering augmentations to general rings. Assuming this point, the proof of Theorem 1.2 should be extendable to apply to augmentations with values in an arbitrary field; see also the discussion on coefficients in [31,Section 3.3]. The proof breaks down in the case of augmentations to a general ring, as the ability to take multiplicative inverses of handleslide coefficients is crucial in extending the arguments in Section 7.…”

“…In a further article [31], we give a complete classification of cobordism classes of augmented Legendrians in 𝐽 1 𝑆 1 . Interestingly, in 𝐽 1 𝑆 1 it is often the case that cobordant Legendrians may become non-cobordant once they are equipped with augmentations.…”