A lower bound for triangulation complexity for compact $3$-manifolds with boundary

The triangulation complexity of a $3$-manifold with boundary is the minimal number of tetrahedra in any its ideal triangulation. Upper complexity bounds usually arise from the explicit construction of triangulations, while finding lower bounds is a hard problem in general. We will discuss the new lower complexity bound obtained via $\mathbb{Z}_2$-homology. It turns out that this complexity bound is stronger than the one from Frigerio, Martelli and Petronio.