in $S^{4}$, which we call $R'$ (the image of $R$ under the antipodal map, in a standard construction of $R$). The interiors of the disks $D_{i}$ are the fibers of this retraction. So $S^{4}\sminus
    \Psi_{n}$ has the homotopy type of $R'$ with $n$ punctures. In particular, the fundamental group is $\Z/2$ for $n=0$ and $\Z *
    \Z$ for $n=2$. For $i=0$ and $2$, the smallest possible action is clearly $\kappa=0$, and we are therefore looking at representations of $\Z/2$ or $\Z * \Z$ in $\SO(3)$ sending the standard generators to involutions. In the second case, the two involutions must be distinct and commuting because of the presence of the tetrahedral point where the disks meet. So the flat connections are as described in the lemma. For these bifold connections $A$, we can read off $H^{0}_{A}$ and $H^{1}_{A}$ in the deformation complex by elementary means, and conclude that $H^{2}_{A}=0$ from the dimension formula.

For the case $n=1$, the corresponding bifold admits a double-cover, branched along $R$, for which the total space is $-\CP^{2}$ containing a complex line $L$ as orbifold locus with cone-angle $\pi$. According to [@Abreu; @Bryant], there exists a conformally anti-self-dual bifold metric on $-\CP^{2}$ with cone-angle $\pi$ along 