Homological reconstruction and simplification in R3R3

We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L)(K,L) can be realized as the homology H⁎(X)H⁎(X) of some complex X   with L⊂X⊂KL⊂X⊂K. We show that this problem is NP-complete even if K   is embedded in R3R3.As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.