Coassociative 4-folds with Conical Singularities

This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds. In each case, the map in question can be considered as a projection from the infinitesimal deformation space onto the obstruction space. Thus, when there are no obstructions the moduli space is a smooth manifold. Furthermore, we calculate a lower bound on the expected dimension of the moduli space. Finally we show that, in weakening the condition on the G_2 structure of the ambient 7-manifold, there is a generic smoothness result for the moduli spaces of deformations corresponding to our second and third problems.