Systolic volume and complexity of 3-manifolds

TL;DR

This paper establishes a lower bound for the systolic volume of closed aspherical 3-manifolds based on their complexity, linking geometric and topological measures of manifold complexity.

Contribution

It proves that systolic volume is bounded below by the complexity of 3-manifolds, connecting geometric invariants with combinatorial topological complexity.

Findings

Systolic volume is bounded below by manifold complexity.

Systolic volume is a homotopy invariant.

Relation between topological complexity and geometric invariants.

Abstract

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depends on topology of the manifold. More precisely, Gromov proved that the systolic volume is related to some topological invariants measuring complicatedness. In this paper, we work along Gromov's spirit to show that systolic volume of 3-manifolds is related to complexity. The complexity of 3-manifolds is the minimum number of tetrahedra in a triangulation, which is a natural tool to evaluate the combinatorial complicatedness.