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

TL;DR

This paper constructs a compact, non-formal manifold with a closed $G_2$ structure, first Betti number 1, and no torsion-free $G_2$ structure, using a novel orbifold resolution method involving a nowhere-vanishing closed 1-form.

Contribution

It introduces a new method for resolving orbifolds with $G_2$ structures, producing examples with specific topological and geometric properties.

Findings

Constructed a non-formal closed $G_2$ manifold with $b_1=1$.

Demonstrated the existence of manifolds with closed but non-torsion-free $G_2$ structures.

Developed a resolution technique for orbifolds with singular loci carrying closed 1-forms.

Abstract

We construct a compact manifold with a closed $G_2$ structure not admitting any torsion-free $G_2$ structure, which is non-formal and has first Betti number $b_1=1$. We develop a method of resolution for orbifolds that arise as a quotient $M/Z_2$ with $M$ a closed $G_2$ manifold under the assumption that the singular locus carries a nowhere-vanishing closed 1-form.