A friendly introduction to the bordered contact invariant

Abstract

We give a short introduction to the contact invariant in bordered Floer homology defined by Földvári, Hendricks, and the authors. We survey the contact geometry required to understand the new invariant but assume some familiarity with bordered Heegaard Floer invariants. The input for the construction is a special class of foliated open books, which are introduced carefully and with multiple examples. We discuss how a foliated open book may be constructed from an open book for a closed manifold, and how it may be modified to ensure compatibility with the contact bordered invariant. As an application of these techniques, we give a "local proof" of the vanishing of the contact invariant for overtwisted structures in the form of an explicit bordered computation.

Morse foliated open books and right-veering monodromies

“…Beginning with the simplest open book foliation on a transverse overtwisted disc, we first construct something that is almost a foliated book by thickening the disc and taken the thickened leaves of the foliation as pages. This general approach for constructing foliated open books described in Section 4.2 of [11], and this specific case is explored in detail as Example 4.7 of [2]. In this construction, the function to S 1 is induced from the S 1 function on the foliation, but when we thicken the surface, each of the two critical pages constructed thus has a pair of critical points.…”