and relate $\dim_{{\mathbb F}}{\mbox{\rm coker}}\ U$ to $N_Y$.

There are two main technical issues involved in implementing this strategy. The first pertains to Steps $(2)$ and $(3)$. The issue is that while the surgery exact triangle used for Step $(2)$ appears in various places in the literature, neither the definition nor the geometric content of the maps in the triangle is totally clear. For Step $(3)$ we need to prove a naturality result for the contact invariant under the map $F$. We achieve this by first relating the maps in the exact triangle to maps on twisted Floer homology groups associated to $2$-handle cobordisms, and then relying on a naturality result for the contact submodule in twisted Floer homology under these latter maps. In order to achieve this, we establish a general exact triangle satisfied by the (twisted) Floer homologies of certain triples of Dehn filled manifolds using a well-known “exact triangle detection lemma". The above surgery triangle, and indeed all previously known exact triangles satisfied by Heegaard Floer modules of closed three manifolds, can be viewed as specializations. So as not to disrupt the flow of the argument, this discussion is postponed to Section \[trianglesec\].

The other technical issue 