$$i =\diag(1,-1,-1) ,\; \; j=\diag(-1,1,-1) , 
        \;  \; k=\diag(-1,-1,1).$$ Up to conjugacy, the monodromy of $A_{V}$ around the link of the disks $D_{1}, D_{2}\subset \Psi^{-}_{2}$ is $i$. The monodromies around the links of the two faces $R \setminus (D_{1} \cup D_{2})$ are $j$ and $k$. The $1$-dimensional vector space $H^{2}(A_{V})$ is spanned a by an $E$-valued form $\omega$ whose values lie in the summand $L_{0}\subset E$ (as follows from the fact that the branched double cover of $S^{4}$ branched over $R$ has $b^{+}=1$). When we form connected sum at a point $f_{2}$ in a face of $\Psi_{2}^{-}$, the obstruction bundle $h$ will be non-trivial on $S^{1}$ if and only if the monodromy of the link at $f_{2}$ acts non-trivially on the summand $L_{0}$ in which $\omega$ lies. Thus $h$ is non-trivial if the monodromy at $f_{2}$ is $j$ or $k$, but $h$ is trivial if the monodromy is $i$. Under the hypotheses of the proposition, $f_{2}$ belongs to one of the faces where the monodromy if $j$ or $k$, so the result follows.

There is a variant of Proposition \[prop:sum-with-Psi-2-minus\] which we will apply in section \[sec:deducing-other\]. Consider $S^{4}$ as the union of two standard balls $B^{4}_{+}\cup
B^{4}_{-}$. Let $M\subset 