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 $M_{\epsilon}$.

We are now going to define an approximately hyperkähler triple $\bm{\underline{\omega}}_{\epsilon}$ on the $4$–manifold $M_{\epsilon}$.

For each $j=1,\dots,8$ denote by $\bm{\underline{\omega}}_{q_{j},\epsilon}$ the hyperkähler triple obtained from the Gibbons–Hawking ansatz (3.3) using the harmonic function

By abuse of notation we think of $\bm{\underline{\omega}}_{q_{j},\epsilon}$ as defined both on the circle bundle $H^{2m_{j}-4}\rightarrow\mathbb{R}^{3}\setminus B_{8\epsilon}(0)$ as well as on its quotient by the involution that acts as the simultaneous standard involution on $\mathbb{R}^{3}$ and the circle fibres.

Similarly, for each $i=1,\dots,n$ let $\bm{\underline{\omega}}_{p_{i},\epsilon}$ be the hyperkähler triple obtained from the Gibbons–Hawking ansatz using the harmonic function

Here $\lambda_{j},\lambda_{i}$ and $\ell_{i}$ are the constants and linear functions appearing in Lemma 4.7. We will assume that $\epsilon$ is small enough to guarantee that $\tfrac{1}{2}<\lambda_{i},\lambda_{j}<\tfrac{3}{2}$.

For all $j=1,\dots,8$ let $(M_{j},\bm{\underline{\omega}}_{M_{j}})$ be a complete $D_{m_{j}}$ ALF space. By Definition 3.6 there exists a compact set $K\subset M_{j}$, $R_{0}>0$ and a diffeomorphism $M_{j}\setminus K\simeq H^{2m_{j}-4}/\mathbb{Z}_{2}$ such that

for $\rho>\epsilon R_{0}$ with $\bm{\underline{\eta}}_{q_{j},\epsilon}=O(\epsilon^{3}\rho^{-3})$ and similar estimates on the derivatives.

For each $i=1,\dots,n$ let $(N_{i},\bm{\underline{\omega}}_{N_{i}})$ be a complete $A_{k_{i}-1}$ ALF space. By the classification of ALF spaces of cyclic type [35] the hyperkähler structure on $N_{i}$ is explicitly given via the Gibbons–Hawking ansatz starting from a harmonic function on $\mathbb{R}^{3}$ with $k_{i}$ singularities. We can add to this function the smooth harmonic function $\epsilon^{2}\ell_{i}$. Over a ball in $\mathbb{R}^{3}$ of radius much smaller than $\epsilon^{-2}$ we can regard the resulting hyperkähler structure as a small perturbation of the ALF $A_{k_{i}-1}$ hyperkähler structure $\bm{\underline{\omega}}_{N_{i}}$. By abuse of notation we denote this perturbed hyperkähler structure with the same symbol $\bm{\underline{\omega}}_{N_{i}}$. The advantage of this modification is that now $\epsilon^{2}\bm{\underline{\omega}}_{N_{i}}$ approaches $\bm{\underline{\omega}}_{p_{i},\epsilon}$ with a smaller error: by Definition 3.6 there exists a compact set $K\subset N_{i}$, $R_{0}>0$ and a diffeomorphism $N_{i}\setminus K\simeq H^{k_{i}}|_{\mathbb{R}^{3}\setminus B_{R_{0}}}$ such that

with (by scaling) $\bm{\underline{\eta}}_{p_{i},\epsilon}=O(\epsilon^{3}\rho^{-3})$ and similar estimates on the derivatives.

We will assume that $\epsilon$ is chosen so small as to make sure that $\epsilon R_{0}\ll\rho_{0}$, where $\rho_{0}>0$ was fixed in Lemma 4.7. By choosing $R_{0}$ larger if necessary we can assume that the harmonic functions $h_{q_{j}}$ and $h_{p_{i}}$ in (5.2) and (5.3) are as close to constant functions as we please for $\epsilon R_{0}\leq\rho\leq\rho_{0}$. Then the hyperkähler triples $\bm{\underline{\omega}}_{q_{j},\epsilon}$ and $\bm{\underline{\omega}}_{p_{i},\epsilon}$ define metrics $g_{q_{j},\epsilon}$ and $g_{p_{i},\epsilon}$ which are uniformly equivalent to $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ in the regions $\epsilon R_{0}\leq\rho_{j},\rho_{i}\leq 2\rho_{0}$. In the rest of the section all norms and covariant derivatives will be computed with respect to this metric.

A crucial observation is that we can take $\bm{\underline{\eta}}_{p_{i},\epsilon}$ and $\bm{\underline{\eta}}_{q_{j},\epsilon}$ to be exact.

By a similar radial integration, Lemma 4.7.(i) implies that in the regions $\epsilon R_{0}\leq\rho_{j}\leq 2\rho_{0}$ and $\epsilon R_{0}\leq\rho_{i}\leq 2\rho_{0}$, respectively, we can write

for $k=0,1,2,3$.

Now, let $\chi_{q_{j}}$ and $\chi_{p_{i}}$ be cut-off functions with the following properties:

We finally define a triple of closed $2$–forms on $M_{\epsilon}$ by