We establish a parametric extension$h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.

We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of$g$copies of$S$^{$n$}×$S$^{$n$}, in the limit $${g \to \infty}$$ . Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.

In the moduli space $$ \mathcal{M} $$ _{$g$}of genus-$g$Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field$F$of degree$g$. If$g$= 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ _{$g$}and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field$F$satisfying a strong geometry of numbers type restriction.