Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy

Abstract

In Berger's classification of Riemannian holonomy groups there are several infinite families and two exceptional cases: the groups Spin(7) and G_2. This thesis is mainly concerned with 7-dimensional manifolds with holonomy G_2. A metric with holonomy contained in G_2 can be defined in terms of a torsion-free G_2-structure, and a G_2-manifold is a 7-dimensional manifold equipped with such a structure.

There are two known constructions of compact manifolds with holonomy exactly G_2. Joyce found examples by resolving singularities of quotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentially asymptotically cylindrical (EAC) G_2-manifolds (not necessarily with holonomy exactly G_2) whose cylinders match. The result of this gluing construction can be regarded as a generalised connected sum of the EAC components, and has a long approximately cylindrical neck region. We consider the deformation theory of EAC G_2-manifolds and show, generalising from the compact case, that there is a smooth moduli space of torsion-free EAC G_2-structures.

As an application we study the deformations of the gluing construction for compact G_2-manifolds, and find that the glued torsion-free G_2-structures form an open subset of the moduli space on the compact connected sum. For a fixed pair of matching EAC G_2-manifolds the gluing construction provides a path of torsion-free G_2-structures on the connected sum with increasing neck length. Intuitively this defines a boundary point for the moduli space on the connected sum, representing a way to `pull apart' the compact G_2-manifold into a pair of EAC components. We use the deformation theory to make this more precise.

We then consider the problem whether compact G_2-manifolds constructed by Joyce's method can be deformed to the result of a gluing construction. By proving a result for resolving singularities of EAC G_2-manifolds we show that some of Joyce's examples can be pulled apart in the above sense. Some of the EAC G_2-manifolds that arise this way satisfy a necessary and sufficient topological condition for having holonomy exactly G_2.

We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8 manifolds with holonomy contained in Spin(7). On such manifolds there is a smooth moduli space of torsion-free EAC Spin(7)-structures. Generalising a result of Wang for compact manifolds we show that for EAC G_2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.