Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of $2$-torsion elements in the class groups and narrow class groups of these cubic fields when they are ordered by their absolute discriminants. For an order $\mathcal{O}$ in a cubic field, we study three groups: ${\mathrm{Cl}}_2\left(\mathcal{O}\right)$, the group of ideal classes of $\mathcal{O}$ of order $2$; ${\mathrm{Cl}}_2^{+}\left(\mathcal{O}\right)$, the group of narrow ideal classes of $\mathcal{O}$ of order $2$; and ${\mathcal{I}}_2\left(\mathcal{O}\right)$, the group of ideals of $\mathcal{O}$ of order $2$. We prove that the mean value of the difference $\left|{\mathrm{Cl}}_2\right(\mathcal{O}\left)\right|-\frac{1}{4}\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is always equal to $1$, regardless of whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the average value of the difference $\left|{\mathrm{Cl}}_2^{+}\right(\mathcal{O}\left)\right|-\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is equal to $1$ for any such family. Also, for any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $\left|{\mathrm{Cl}}_2\right(\mathcal{O}\left)\right|-\frac{1}{2}\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is always equal to $1$, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove—in stark contrast to the case of quadratic fields—that (1) a positive proportion of cubic fields have odd class number, (2) a positive proportion of real cubic fields have isomorphic $2$-torsion in the class group and the narrow class group, and (3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in ${\mathcal{P}}^c$ we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length $x$ with primes including some in $(\sqrt{x},x]$, using methods motivated by additive combinatorics.

We construct many new topological types of compact ${\mathrm{G}}_2$-manifolds, that is, Riemannian $7$-manifolds with holonomy group ${\mathrm{G}}_2$. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau $3$-folds built from semi-Fano $3$-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth $7$-manifolds completely; we find that many $2$-connected $7$-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of ${\mathrm{G}}_2$-metrics. Many of the ${\mathrm{G}}_2$-manifolds we construct contain compact rigid associative $3$-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of ${\mathrm{G}}_2$-metrics. By varying the semi-Fanos used to build different ${\mathrm{G}}_2$-metrics on the same $7$-manifold we can change the number of rigid associative $3$-folds we produce.

We correct a mistake in Section 2.4 of the said paper.