We propose a compact, dimension-independent data structure for manifold, non-manifold and non-regular simplicial complexes, that we call the Generalized Indexed Data Structure with Adjacencies (IA* data structure). It encodes only top simplices, i.e. the ones that are not on the boundary of any other simplex, plus a suitable subset of the adjacency relations. We describe the IA⁎ data structure in arbitrary dimensions, and compare the storage requirements of its 2D and 3D instances with both dimension-specific and dimension-independent representations. We show that the IA* data structure is more cost effective than other dimension-independent representations and is even slightly more compact than the existing dimension-specific ones. We present efficient algorithms for navigating a simplicial complex described as an IA* data structure. This shows that the IA* data structure allows retrieving all topological relations of a given simplex by considering only its local neighborhood and thus it is a more efficient alternative to incidence-based representations when information does not need to be encoded for boundary simplices.