Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

Abstract

Abstract Let 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k ⁢ ( M ) k(M) satisfies the inequalities k ⁢ ( M ) ≥ 3 ⁢ χ ⁢ ( M ) + 7 ⁢ m + 7 ⁢ h - 10 k(M)\geq 3\chi(M)+7m+7h-10 and k ⁢ ( M ) ≥ k ⁢ ( ∂ ⁡ M ) + 3 ⁢ χ ⁢ ( M ) + 4 ⁢ m + 6 ⁢ h - 9 k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9 , and its regular genus G ⁢ ( M ) \mathcal{G}(M) satisfies the inequalities G ⁢ ( M ) ≥ 2 ⁢ χ ⁢ ( M ) + 3 ⁢ m + 2 ⁢ h - 4 \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and G ⁢ ( M ) ≥ G ⁢ ( ∂ ⁡ M ) + 2 ⁢ χ ⁢ ( M ) + 2 ⁢ m + 2 ⁢ h - 4 \mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4 , where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.