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

TL;DR

This paper constructs a compact 7-manifold with a closed G_2-structure, first Betti number 1, which does not admit a torsion-free G_2-structure, and explores associative submanifolds and fibrations within it.

Contribution

It provides the first example of a compact formal 7-manifold with a closed G_2-structure and first Betti number 1 that lacks a torsion-free G_2-structure, along with detailed submanifold analysis.

Findings

Existence of a compact formal 7-manifold with closed G_2-structure and b_1=1

Construction of associative calibrated 3-tori with deformation families

Fibration over S^2×S^1 with singular fibers

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.