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.
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