open subset of $SU(2)^{2g}$, where $g$ is the genus of $\Sigma$. It also admits a natural symplectic structure, for which the following sub-manifolds are Lagrangian: $$\begin{aligned}
 L_0 &= \lbrace  \rho \circ i_{0,*}: \rho  \in \mathrm{Hom}(\pi_1(H_0\setminus C_0,*), SU(2) ) ,  \rho(\mu_0)  = -I \rbrace  \\
 L_1 &= \lbrace  \rho \circ i_{1,*}: \rho  \in \mathrm{Hom}(\pi_1(H_1\setminus C_1,*), SU(2) ) , \rho(\mu_1)  = -I \rbrace ,\end{aligned}$$ with $i_{0,*}$ and $i_{1,*}$ induced by the inclusions, and $\mu_0$, $\mu_1$ meridians of $C_0$ and $C_1$ respectively.

The group $HSI(Y,c)$ can then be defined as the Lagrangian Floer homology $HF(L_0,L_1)$.

However, since ${\mathscr{N}}(\Sigma')$ is noncompact, the fact that Lagrangian Floer homology can be defined inside it is not immediate at all. Manolescu and Woodward manage to do so after a quite involved construction. They first compactify ${\mathscr{N}}(\Sigma')$ by a symplectic cutting, and obtain a compact moduli space ${\mathscr{N}^c}(\Sigma') = {\mathscr{N}}(\Sigma')\cup R$, with a symplectic hypersurface $R$ and a monotone 2-form which degenerates at $R$. By performing another symplectic cutting, they find a symplectic form on ${\mathscr{N}^c}(\Sigma')$ which is not monotone anymore, and after some interplay with both forms, they define Floer homology relatively to $R$, which corresponds to Floer homology inside ${\mathscr{N}}(\Sigma')$.

In order to be able to 