Quantum mechanics

Submanifold

Engineering

Pure mathematics

Surgery

Gravitational singularity

Normal subgroup

Subgroup

Torsion (gastropod)

Mechanical engineering

Physics

Organic chemistry

Holonomy

Mathematics

Mathematical analysis

Orbifold

Medicine

Chemistry

Manifold (fluid mechanics)

Group (periodic table)

Journal of Differential Geometry

We give a new construction of compact Riemannian $7$-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the $G_2$-structure. Then $M / {\langle \iota \rangle}$ is a $G_2$- orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb{R}^3 \times (\mathbb{R}^4 / {\lbrace \pm 1 \rbrace})$. We resolve this orbifold by gluing in a family of Eguchi–Hanson spaces, parametrized by a nonvanishing closed and coclosed 1-form $\lambda$ on $L$. Much of the analytic difficulty lies in constructing appropriate closed $G_2$-structures with sufficiently small torsion to be able to apply the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi–Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces