Constructions of compact G2-holonomy manifolds

TL;DR

This survey reviews methods for constructing compact G2-holonomy manifolds, focusing on resolution of orbifold singularities and gluing asymptotically cylindrical pieces, highlighting the geometric techniques involved.

Contribution

It systematically summarizes known geometric constructions of compact G2-manifolds, clarifying the roles of singularity resolution and gluing techniques.

Findings

Two main construction methods are detailed: resolution of orbifold singularities and gluing of cylindrical pieces.

The paper clarifies the geometric conditions needed for successful G2-holonomy constructions.

It consolidates existing knowledge, serving as a comprehensive reference for researchers in the field.

Abstract

This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen 7-dimensional orbifolds, with the help of asymptotically locally Euclidean spaces. Another method uses the gluing of two asymptotically cylindrical pieces and requires a certain matching condition for their cross-sections `at infinity'.