Compact Riemannian 7-manifolds with holonomy $G\sb 2$. II

Abstract

This is the second of two papers about metrics of holonomy G 2 on compact 7-manifolds.In our first paper [15] we established the existence of a family of metrics of holonomy G 2 on a single, compact, simply-connected 7-manifold M, using three general results, Theorems A, B and C. Our purpose in this paper is to explore the theory of compact riemannian 7-manifolds with holonomy G 2 in greater detail.By relying on Theorems A-C we will be able to avoid the emphasis on analysis that characterized [15], so that this sequel will have a more topological flavour.The paper has four chapters.The first chapter consists of introductory material.Section 1.1 gives some elementary geometric and topological material on compact 7-manifolds with torsion-free G 2 -structures.Then §1.2 describes the holonomy groups SU(2) and 5C/(3), and §1.3 explains the concept of asymptotically locally Euclidean riemannian manifolds (shortened to ALE spaces) with special holonomy.Recall that in [15], a compact 7-manifold M was defined by desingularizing a quotient T 7 /Γ of the 7-torus by a finite group of isometries Γ = ΊJ\.The subject of Chapters 2 and 3 is a generalization of this idea.Chapter 2 defines a general construction for compact 7-manifolds with torsion-free G 2 -structures, which works by desingularizing quotients T 7 /Γ for finite groups Γ.The ALE spaces with holonomy SU(2) and SU(3) discussed in §1.3 are an essential ingredient in performing this desingularization.The central result of Chapter 2 is Theorem 2.2.3, which states that given a suitable finite group Γ and certain other data, one may con-