In precedent works, we have proved that our nD topological reparation method is able to make weak well-composed any given cubical image in a homotopy-preserving manner. However, in 4D and beyond, weak well-composedness is weaker that homology manifoldness, and then weak well-composedness is not sufficient anymore for cohomological computation based on the boundary, which is useful for example in image recognition. To overcome this problem, we propose a new method, corresponding to the old simplicial method with only one modification, the detection of the critical points, which is now based on local homology. We also expose some possible tracks that can be followed to prove that the final repaired complex provided with the new method is a homology manifold.
