A construction of G2-manifolds from K3 surfaces with a [formula omitted]-action

Abstract

A product of a K3 surface S and a flat 3-dimensional torus T3 is a manifold with holonomy SU(2). Since SU(2) is a subgroup of G2, S×T3 carries a torsion-free G2-structure. We assume that S admits an action of Z22 with certain properties. There are several possibilities to extend this action to S×T3. A recent result of Joyce and Karigiannis allows us to resolve the singularities of (S×T3)/Z22 such that we obtain smooth G2-manifolds. We classify the quotients (S×T3)/Z22 under certain restrictions and compute the Betti numbers of the corresponding G2-manifolds. Moreover, we study a class of quotients by a non-abelian group. Several of our examples have new values of (b2,b3).