Computing Matveev's complexity via crystallization theory: the boundary case (Q2848021)

\textit{S. V. Matveev}'s complexity is a well-known invariant for 3-manifolds defined in [Acta Appl. Math. 19, No. 2, 101--130 (1990; Zbl 0724.57012)]. The Gem-Matveev complexity (GM-complexity) has been introduced within crystallization theory, \textit{M. R. Casali} [Topology Appl. 144, No. 1--3, 201--209 (2004; Zbl 1059.57010)]. The modified Heegaard complexity (HM-complexity) has been introduced by \textit{A. Cattabriga} et al. [J. Korean Math. Soc. 47, No. 3, 585--599 (2010; Zbl 1198.57014)]. Both GM-complexity and HM-complexity are useful as upper bounds of Matveev complexity. The main result of the paper says that for every compact irreducible and boundary-irreducible 3-manifold the GM-complexity and HM-complexity coincide.