We construct a compact, formal 7-manifold with a closed ${\mathrm{G}}_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free ${\mathrm{G}}_2$-structure, that is, it does not admit any ${\mathrm{G}}_2$-structure such that the holonomy group of the associated metric is a subgroup of ${\mathrm{G}}_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed ${\mathrm{G}}_2$-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.

中文翻译：

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紧凑的G ₂校准歧管，其第一贝蒂数b ₁  = 1

我们构造一个封闭的，紧凑的，正规的7流形 ${\mathrm{G}}_2$结构并带有第一个贝蒂数 $b_{\mathrm{1个}}=\mathrm{1个}$，它不允许无扭转 ${\mathrm{G}}_2$-结构，即它不允许任何 ${\mathrm{G}}_2$-结构，使得相关度量的完整组是 ${\mathrm{G}}_2$。我们还针对此封闭构造了关联校准（因此使体积最小）的3-tori${\mathrm{G}}_2$-结构，对于这些3-tori中的每一个，我们都显示了3维非平凡的关联变形族。我们还构造了一个7流形的纤维${\mathrm{小号}}^2\times {\mathrm{小号}}^{\mathrm{1个}}$ 带有通用纤维a（未校准）共聚4托和一些奇异纤维。