A compact $G_2$-calibrated manifold with first Betti number $b_1=1$

Abstract

We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-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 $S^2\times S^1$ with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.

32 pages, v2 new results on the associative 3-folds, new section on the coassociative 4-folds. v3 corrected a mistake in the proof of Thm.17. v4 corrected the proof of the formality of the resolution (Prop.24). v6: changes made after publication: corrected the calculation of the fixed point sets in eqn.(10) and on p.23, minor simplification before Prop.17, Remark 18 rewritten

References in corpus (3)

• Symplectic resolutions, Lefschetz property and formality
• A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces
• On the automorphism group of a closed G${}_2$-structure

Cited by in corpus (1)

• A compact non-formal closed $G_2$ manifold with $b_1=1$