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

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.

arXiv comment31 pages. v2 corrects a mistake on the definition of \hatω_1. To appear in Math Nach

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