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.