Let $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ be the hyperkähler triple defined in (4.6). By Lemma 4.9 for $\epsilon>0$ small enough we think of $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ as defined on $M^{\textup{gh}}_{\epsilon}$, a smooth manifold with boundary obtained by restricting the line bundle $P$ to the complement of (arbitrarily) small balls centred at the punctures and then taking the quotient by the involution $\tilde{\tau}$. The boundary of $M^{\textup{gh}}_{\epsilon}$ has $n+8$ components, each of which has a collar neighbourhood diffeomorphic either to $H^{2m_{j}-4}/\mathbb{Z}_{2}$ for $j=1,\dots,8$, or $H^{k_{i}}$ for $i=1,\dots,n$.

For each $j=1,\dots,8$ let $M_{j}$ be the smooth $4$–manifold underlying a $D_{m_{j}}$ ALF space and for each $i=1,\dots,n$ let $N_{i}$ be the smooth manifold underlying an $A_{k_{i}-2}$ ALF space. We construct a smooth $4$–manifold $M_{\epsilon}$ by cutting the ends of $M_{j}$ and $N_{i}$ and gluing the resulting manifolds with boundary to $M^{\textup{gh}}_{\epsilon}$ in a neighbourhood of $q_{j}$ or $\pm p_{i}$, respectively.

We will construct an approximate hyperkähler structure on $M_{\epsilon}$ in the next subsection. Here we pause for a moment to determine the Betti numbers of $M_{\epsilon}$. While we will not use this result in an essential way in the rest of the paper, it is interesting to note how the balancing condition (4.1) appears naturally in the calculation of the Euler characteristic of $M_{\epsilon}$.