Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams

TL;DR

This paper extends the concept of modified Heegaard complexity to non-orientable 3-manifolds and proves its equivalence with Gem-Matveev complexity for closed 3-manifolds, providing tools to estimate Matveev complexity.

Contribution

It generalizes the modified Heegaard complexity to non-orientable cases and establishes its equivalence with Gem-Matveev complexity for closed 3-manifolds.

Findings

Gem-Matveev and modified Heegaard complexities coincide for closed 3-manifolds.

Both complexities provide upper bounds for Matveev complexity.

Extension of modified Heegaard complexity to non-orientable 3-manifolds.

Abstract

The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.