Higher-dimensional bounds and classification

Determine general bounds for the balanced genus of PL \(d\)-manifolds in dimensions \(d>4\), and develop corresponding classification results in higher dimensions.

Background

The main lower-bound and sphere-characterization results are proved specifically in dimensions three and four, while a dimension-independent elementary lower bound is also given. The authors leave unresolved the development of stronger bounds and classification theories for PL manifolds in dimensions greater than four.

References

Question 6.3. One may investigate general bounds for the balanced genus of PL d-manifolds for d > 4 and explore classification problems in higher dimensions.

— Balanced genus and a lower bound theorem for balanced 3- and 4-manifolds  (2503.06133 - Basak et al., 8 Mar 2025) in Question 6.3, Section 6, p. 15