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

Abstract

This is the second of two papers about metrics of holonomy G2 on compact 7manifolds. In our first paper [15] we established the existence of a family of metrics of holonomy G2 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 G2 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 G2structures. Then §1.2 describes the holonomy groups SU(2) and SU(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 /Γ of the 7-torus by a finite group of isometries Γ ∼= Z2. 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 G2structures, which works by desingularizing quotients T /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 construct a compact 7manifold M from T /Γ that admits torsion-free G2structures. This result is proved using Theorems A-C of [15]. Chapter 3 is devoted entirely to examples of this construction. We give many examples of compact 7-manifolds with holonomy G2, and determine their basic topological invariants — the Betti numbers and fundamental group. Finally, in Chapter 4 we discuss some areas of interest, and give a number of open problems. This paper is not written to be read independently of [15]. The language and results of [15] will be used freely, in particular the introductory material in [15, §1.1]. For reference we reproduce here the model 3and 4forms φ, ∗φ defining the flat G2structure on R, as given in [15, §1.1]: