We saw that gravitational instantons of type $A_{k}$ can be constructed from Dirac monopoles on $\mathbb{R}^{3}$ with $k+1$ singularities via the Gibbons–Hawking ansatz. These are usually called multi-Taub–NUT metrics. The case $k=0$ is the Taub–NUT metric on $\mathbb{R}^{4}$ and $k=-1$ is $\mathbb{R}^{3}\times\mathbb{S}^{1}$ with its flat metric. Minerbe [35, Theorem 0.2] has shown that every ALF space of cyclic type must be isometric to a multi-Taub–NUT metric.

From their explicit description one can easily compute basic information about cyclic ALF spaces: the fundamental group $\pi_{1}(M)$, the second Betti number $b_{2}(M)$, the Euler characteristic and the dimension of the moduli space $\mathcal{M}$ of $A_{k}$ metrics:

$k$	$\pi_{1}(M)$	$b_{2}(M)$	$\chi(M)$	$\text{dim}(\mathcal{M})$
$-1$	$\mathbb{Z}$	$0$	$0$	$0$
$k>-1$	$1$	$k$	$k+1$	$3k$

Here we assume that the asymptotic length of the circle fibre is normalised to be $1$ so that $\text{dim}(\mathcal{M})$ does not include rescalings.