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• W747226871 abstract "In Berger’s classification of Riemannian holonomy groups there are several infinite families and two exceptional cases: the groups Spin(7) and G2. This thesis is mainly concerned with 7-dimensional manifolds with holonomy G2. A metric with holonomy contained in G2 can be defined in terms of a torsion-free G2-structure, and a G2-manifold is a 7-dimensional manifold equipped with such a structure. There are two known constructions of compact manifolds with holonomy exactly G2. Joyce found examples by resolving singularities of quotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentially asymptotically cylindrical (EAC) G2-manifolds (not necessarily with holonomy exactlyG2) 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 G2-manifolds and show, generalising from the compact case, that there is a smooth moduli space of torsion-free EACG2-structures. As an application we study the deformations of the gluing construction for compact G2-manifolds, and find that the glued torsion-free G2-structures form an open subset of the moduli space on the compact connected sum. For a fixed pair of matching EAC G2-manifolds the gluing construction provides a path of torsion-free G2-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 G2-manifold into a pair of EAC components. We use the deformation theory to make this more precise. We then consider the problem whether compact G2-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 G2-manifolds we show that some of Joyce’s examples can be pulled apart in the above sense. Some of the EAC G2-manifolds that arise this way satisfy a necessary and sufficient topological condition for having holonomy exactly G2. 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 G2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics." @default.
• W747226871 created "2016-06-24" @default.
• W747226871 creator A5052382587 @default.
• W747226871 date "2008-10-14" @default.
• W747226871 modified "2023-09-25" @default.
• W747226871 title "Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy" @default.
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• W747226871 doi "https://doi.org/10.17863/cam.16204" @default.
• W747226871 hasPublicationYear "2008" @default.
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