The work investigates data structures for representing and manipulating \(d\)-dimensional geometric objects for arbitrary \(d\geq 1\). A class of geometric objects is defined, the `subdivided \(d\)-manifolds', which is large enough to encompass many applications. A new representation is given for such objects, the `cell-tuple structure', which provides direct access to topological structure, ordering information among cells, the topological dual, and boundaries. The cell-tuple structure gives a simple, uniform representation of subdivided manifolds which unifies the existing work in the field and provides intuitive clarity in all dimensions. The dual subdivision, and boundaries, are represented consistently. The work has direct applications in solid modeling, computer graphics, and computational geometry.