Generating families and constructible sheaves

Abstract

Let $Λ$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $Λ$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $Λ$, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this sheaf. That is, the theory of generating family homology embeds in sheaf theory, and more specifically in the category studied in [STZ]. When $X = \mathbb{R}$, i.e., for the theory of Legendrian knots and links in the standard contact $\mathbb{R}^3$, we use ideas from the proof of the h-cobordism theorem to show this embedding is an equivalence. Combined with the results of [NRSSZ], this implies in particular that the generating family homologies of a knot are the same as its linearized Legendrian contact homologies.

As pointed out to me by Sylvain Courte, the main argument fails to account for higher stable homotopy obstructions to simplifying Morse functions. Sylvain and I are preparing an account of what is actually true. (The arguments showing that sheaves compute generating family invariants are correct, but also can be found in other works.)