Deformations and Gluing of Asymptotically Cylindrical Manifolds with Exceptional Holonomy

Download or read book Deformations and Gluing of Asymptotically Cylindrical Manifolds with Exceptional Holonomy written by Johannes Nordström and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In Berger's classification of Riemannian holonomy groupsthere are several infinite families and two exceptional cases:the groups Spin(7) and G-2. This thesis is mainly concerned with 7-dimensional manifoldswith holonomy G-2.A metric with holonomy contained in G-2 can be defined in terms ofa torsion-free G-2-structure, and a G-2-manifold is a 7-dimensional manifoldequipped with such a structure. There are two known constructions of compact manifolds with holonomyexactly G-2. Joyce found examples by resolving singularities ofquotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentiallyasymptotically cylindrical (EAC) G-2-manifolds (not necessarily with holonomyexactly G-2) whose cylinders match. The result of this gluing constructioncan be regarded as a generalised connected sum of the EAC components, and hasa long approximately cylindrical neck region. We consider the deformation theory of EAC G-2-manifolds and show, generalising fromthe compact case, that there is a smooth moduli space of torsion-free EACG-2-structures. As an application we study the deformations of the gluing construction forcompact G-2-manifolds, and find that the glued torsion-free G-2-structures form an opensubset of the moduli space on the compact connected sum. For a fixed pair ofmatching EAC G-2-manifolds the gluing construction provides a path of torsion-freeG-2-structures on the connected sum with increasing neck length. Intuitively this defines a boundary point for the moduli space on the connectedsum, representing a way to `pull apart' the compact G-2-manifold into a pair of EACcomponents. We use the deformation theory to make this more precise. We then consider the problem whether compact G-2-manifolds constructed by Joyce'smethod can be deformed to the result of a gluing construction. By proving a result for resolving singularities of EAC G-2-manifolds we showthat 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 andsufficient topological condition for having holonomy exactly G-2. We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8manifolds with holonomy contained in Spin(7). On such manifolds there isa smooth moduli space of torsion-free EAC Spin(7)-structures. Generalising a result of Wang for compact manifolds we show that forEAC G-2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.