Decompose $M_{\epsilon}$ into the union of a piece $P/\tilde{\tau}$, an $A_{k_{i}-1}$ ALF space for each $i=1,\dots,n$ and a $D_{m_{j}}$ ALF space for each $j=1,\dots,8$. These pieces are identified along their common boundaries, which are homology spheres. Since all components have vanishing first Betti number, the reduced Mayer–Vietoris sequence yields $b_{1}(M_{\epsilon})=0$. The Euler characteristic is also easily calculated:

by the balancing condition (4.1).

It remains to calculate the signature $\tau(M_{\epsilon})$. Below we will construct a definite triple $\bm{\underline{\omega}}_{\epsilon}$ on $M_{\epsilon}$ which is close to define a hyperkähler structure. By changing basis of $\Lambda^{+}T^{\ast}M_{\epsilon}$ one can always deform this triple to a genuine $SU(2)$–structure (without requiring any differential constraint). In particular, $M_{\epsilon}$ can be endowed with an almost complex structure $J$ with $c_{1}(M_{\epsilon},J)=0$. Since $c_{2}(M_{\epsilon},J)=\chi(M_{\epsilon})=24$, Hirzebruch’s Signature Theorem and the equality of characteristic classes $p_{1}=c_{1}^{2}-2c_{2}$ yield $\tau(M_{\epsilon})=-16$. ∎