A 4D counter-example showing that DWCness does not imply CWCness in n-D

Abstract

In this paper, we prove that the two avours of well-compo- sedness called Continuous Well-Composedness (shortly CWCness), stat- ing that the boundary of the continuous analog of a discrete set is a manifold, and Digital Well-Composedness (shortly DWCness), stating that a discrete set does not contain any critical con guration, are not equivalent in dimension 4. To prove this, we exhibit the example of a con- guration of 8 tesseracts (4D cubes) sharing a common corner (vertex), which is DWC but not CWC. This result is surprising since we know that CWCness and DWCness are equivalent in 2D and 3D. To reach our goal, we use local homology.