invariant $c(\xi_{n})\in HF^+(-Y_{-1/n}(K))$ is not zero.

2.  Fit $HF^+(-Y_{-1/n}(K))$ into an exact triangle of modules over ${{\mathbb F}}[U]$

    \(m) \[matrix of math nodes, row sep=1em,column sep=1em\] [ HF\^+(-Y\_[-1/n]{}(K) ; [[F]{}]{})& & [HF]{}\^+(-Y; [[F]{}]{})\
    & &\
    & \^+(-Y\_0(K); [[F]{}]{}\[C\_n\])&\
    ]{};

    (m-1-1) edge\[-&gt;\] (m-1-3) (m-1-3) edge\[-&gt;\] node\[below\] [$\ \ \ \ G$]{}(m-3-2) (m-3-2) edge\[-&gt;\] node\[below\] [$F\ \ \ \ $]{} (m-1-1);

    where the bottom term is a twisted version of the Floer homology for zero surgery on the binding, with coefficients in the group algebra of the cyclic group $C_n={{\mathbb Z}}/n{{\mathbb Z}}$.

3.  \[item3\] Show that non-triviality of $F$, restricted to a particular subgroup $$\underline{HF}^+(-Y_0(K),{\mathfrak{s}}_{1-g}; {{\mathbb F}}[C_n]),$$ implies $c(\xi_n)\ne 0$. Here ${\mathfrak{s}}_{1-g}$ is the ${{\mbox{spin$^c$} }}$ structure on $Y_0(K)$ whose Chern class evaluates to $2-2g$ on the fiber and which is cobordant through the surgery cobordism to the ${{\mbox{spin$^c$} }}$ structure associated to $\xi$.

4.  Show that the subgroup from Step $(3)$ is isomorphic to ${{\mathbb F}}[C_n]$, as an ${{\mathbb F}}[U]$–module where $U$ acts as zero. In particular, this group is a vector space of dimension $n$ over ${{\mathbb F}}$.

5.  Conclude, by Item \[item3\] and exactness at $-Y_0$, that $c(\xi_{n})\ne 0$ provided that $$n> \dim_{{\mathbb F}}\ {\mbox{\rm coker}}\ U: HF^+(-Y)\rightarrow HF^+(-Y),$$ 