Deformations of glued G_2-manifolds

TL;DR

This paper investigates how the process of gluing asymptotically cylindrical G_2-manifolds affects their deformation space, establishing a smooth correspondence and a partial compactification of the moduli space of torsion-free G_2-structures.

Contribution

It demonstrates that the gluing construction induces a local diffeomorphism between moduli spaces and introduces a partial compactification including boundary points as equivalence classes of asymptotic structures.

Findings

Gluing defines a smooth map between moduli spaces

The map is a local diffeomorphism

Partial compactification includes boundary points as matching pairs

Abstract

We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map from a moduli space of gluing data to the moduli space of torsion-free G_2-structures on the glued manifold, and that this is a local diffeomorphism. We use this to partially compactify the moduli space of torsion-free G_2-structures, including it as the interior of a topological manifold with boundary. The boundary points are equivalence classes of matching pairs of torsion-free asymptotically cylindrical G_2-structures.