build the twisted symplectic instanton homology groups $HSI(Y,c)$ within this framework. In section \[sec:firstprop\] we give their first properties: we prove in section \[sec:compusplitting\] that the definition agrees with the one we outline in the next paragraph, and prove Theorem \[sommecnx\] in section \[sec:connectedsum\]. Section \[sec:surgery\] is devoted to Dehn surgery, we prove Theorem \[trianglechir\] in section \[sec:proofsurgeryexactseq\], and provide applications in section \[sec:applic\].

Outline of the construction of $HSI(Y,c)$, and sketch of the proofs {#sec:outlineproofs}
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We first outline the construction of the twisted version from a Heegaard splitting, without using Wehrheim and Woodward’s theory. Let $Y = H_0 \cup_\Sigma H_1 $ be a Heegaard splitting, and $C_0$, $C_1$ two knots inside $H_0$ and $H_1$ respectively, such that the class of their union in $H_1(Y;{{\mathbb{Z}}_{2}})$ equals $c$. Let $\Sigma'$ be the surface with boundary obtained by removing a disc to $\Sigma$, and $*\in \partial\Sigma'$ a base point. The open part of Huebschmann and Jeffrey’s extended moduli space ${\mathscr{N}}(\Sigma')$ that will be involved in the construction admits the following description: $${\mathscr{N}}(\Sigma') = \lbrace  \rho \in \mathrm{Hom}(\pi_1(\Sigma',*) , SU(2) ):  \rho(\partial \Sigma')   \neq -I \rbrace.$$ By choosing a base of the free group $\pi_1(\Sigma',*)$, this space can be realized as an 