Classification of PL 4-manifolds by balanced genus

Classify all PL 4-manifolds according to their balanced genus.

Background

The authors establish lower bounds for the balanced genus of non-spherical PL 4-manifolds and characterize the 4-sphere using an upper threshold. A broader unresolved problem is to determine whether balanced genus can provide a systematic classification of all PL 4-manifolds.

References

Question 6.2. In dimension 4, we have provided a lower bound for the balanced genus of PL 4-manifolds. An interesting research direction is to classify all PL 4-manifolds based on their balanced genus.

— Balanced genus and a lower bound theorem for balanced 3- and 4-manifolds  (2503.06133 - Basak et al., 8 Mar 2025) in Question 6.2, Section 6, p. 15