On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy $\operatorname{Sp}(n)$

We discuss aspects of the global topology of moduli spaces of hyperkähler metrics.
If the second Betti number is larger than $4$, we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\
An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer $K3$ surface.\\
We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than $3$. For a compact simply connected manifold $N$ we show that the moduli space of Ricci flat metrics on $N\times T^k$ splits homeomorphically into a product of the moduli space of Ricci flat metrics on $N$ and the moduli of sectional curvature flat metrics on the torus $T^k$.