Topology of Asymptotically Conical Calabi-Yau and G2 Manifolds and Desingularization of Nearly Kahler and Nearly G2 Conifolds

A natural approach to the construction of nearly G(2) manifolds lies in resolving nearly G(2) spaces with isolated conical singularities by gluing in asymptotically conical G(2) manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G(2) manifolds, whose endpoint is the original nearly G(2) conifold and whose parameter is the scale of the glued in asymptotically conical G(2) manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G(2) manifolds: if the rate of the metric is below -3, then the G(2) 4-form is exact if and only if the manifold is Euclidean R-7. A similar construction is possible in the nearly K & auml;hler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi-Yau 6-manifolds: if the rate of the metric is below -3, then the square of the K & auml;hler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R-6.