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

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) satisfies the inequalities k(M)\geq 3\chi(M)+7m+7h-10 and k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, and its regular genus \mathcal{G}(M) satisfies the inequalities \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and \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.

Handle decomposition for a class of compact orientable PL 4-manifolds

“…The definition of semi simple crystallizations is for the compact 4-manifolds with connected boundary. In [3], there is a similar concept for the manifolds with any number of boundary components.…”