A compact G2-calibrated manifold with first Betti number b1 = 1

Abstract

We construct a compact, formal 7-manifold with a closed G2-structure and with first Betti number b1=1, which does not admit any torsion-free G2-structure, that is, it does not admit any G2-structure such that the holonomy group of the associated metric is a subgroup of G2. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed G2-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over S2×S1 with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.