Balanced genus of well-known PL manifolds

Determine the balanced genus of other well-known PL \(d\)-manifolds beyond the sphere \(S^d\), extending the formula established for spheres when \(d\geq 3\).

Background

The paper proves that for every d≥3d\geq 3, the balanced genus of the sphere satisfies GSd=1+(d−3)⋅2d−2G_{S^d}=1+(d-3)\cdot 2^{d-2}. Question 6.1 asks for analogous computations for other standard PL manifolds, which would broaden the known examples and help assess the distinguishing power of the balanced genus.

References

Question 6.1. We have shown that for d ≥ 3, the balanced genus of Sd is given by 1 + (d − 3) * 2d−2. A natural extension of this result is to determine the balanced genus of other well-known PL d-manifolds.

— Balanced genus and a lower bound theorem for balanced 3- and 4-manifolds  (2503.06133 - Basak et al., 8 Mar 2025) in Question 6.1, Section 6, p. 15