Computing the cohomological information of a cubical complex based on its surface can be very helpful in practice: feature recognition, faster algorithms, and so on. However, usual cubical complexes present topological issues called "pinches". There is then a real need to find a way to delete these pinches in the cubical complexes while preserving its cohomological information; the proposed method is then to "repair" the initial complex into a simplicial complex which is homotopy equivalent and such that its boundary is a discrete surface; this property is known as "well-composedness in the sense of Alexandrov" or "AWCness". A similar method already exists in 3D, our aim is then to extend this method to nD since modern signals are generally 2D (photographies), 3D (MR images), or even 4D (CT scans). After some recalls in matter of partially ordered set theory, we will then introduce the notion of border of a complex and the notion of discrete surfaces with borders. This way, we will be able to present a brief sketch of the proof that we propose to justify the efficiency of our method.