In this paper we consider the problem of decomposing a nonmanifold n-dimensional object described by an abstract simplicial complex into an assembly of ‘more-regular’ components. Manifolds, which would be natural candidates for components, cannot be used to this aim in high dimensions because they are not decidable sets. Therefore, we define d-quasi-manifolds, a decidable superset of the class of combinatorial d-manifolds that coincides with d-manifolds in dimension less or equal than two. We first introduce the notion of d-quasi-manifold complexes, then we sketch an algorithm to decompose an arbitrary complex into an assembly of quasi-manifold components abutting at non-manifold joints. This result provides a rigorous starting point for our future work, which includes designing efficient data structures for non-manifold modeling, as well as defining a notion of measure of shape complexity of such models.