Classification of orientable PL 4-manifolds beyond regular genus five

Classify orientable closed PL 4-manifolds by regular genus six and higher, extending the classification currently known through regular genus five.

Background

The paper situates regular-genus classification as a central problem in combinatorial topology. For orientable PL 4-manifolds, existing results classify the cases through regular genus five, while examples with regular genus six are known but do not yield a complete classification. The unresolved scope therefore concerns the classification at regular genus six and all subsequent values.

References

The classification beyond regular genus five remains open.

— Regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$, $4$-torus, and small covers over $Δ^2 \times Δ^2$  (2506.01315 - Agarwal et al., 2 Jun 2025) in Section 1, Introduction