[Paper Review] Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations

Jae Choon|arXiv (Cornell University)|Jun 2, 2015

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper establishes linear inequalities relating three distinct complexity measures of 3-manifolds—triangulation complexity, Heegaard splitting complexity, and surgery presentation complexity—via explicit geometric constructions. It proves these inequalities are asymptotically optimal, providing foundational tools for estimating Cheeger-Gromov $L^2$ $\rho$-invariants using geometric and knot-theoretic data.

ABSTRACT

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$ $ ho$-invariants in terms of geometric group theoretic and knot theoretic data.

Motivation & Objective

• To understand and relate different geometric complexity measures of 3-manifolds.
• To establish rigorous linear inequalities connecting triangulation, Heegaard splitting, and surgery presentation complexities.
• To demonstrate that these inequalities are asymptotically optimal through explicit geometric constructions.
• To provide a framework for estimating Cheeger-Gromov $L^2$ $\rho$-invariants using geometric and knot-theoretic data.

Proposed method

• Constructing explicit geometric realizations that link triangulations, Heegaard splittings, and surgery presentations of 3-manifolds.
• Deriving linear inequalities between the complexities of these three representations using geometric and topological arguments.
• Using known constructions in 3-manifold topology to bound complexity measures from above and below.
• Analyzing asymptotic growth rates of the complexities to prove optimality of the inequalities.
• Applying the inequalities to estimate $L^2$ $\rho$-invariants in terms of geometric group theory and knot invariants.
• Leveraging results from [arXiv:1405.1805] to validate the utility of the complexity bounds in broader topological contexts.

Experimental results

Research questions

• RQ1 How are the complexities of 3-manifolds defined via triangulations, Heegaard splittings, and surgery presentations related to one another?
• RQ2 Can linear inequalities be established between these three complexity measures, and if so, what are their forms?
• RQ3 Are these linear inequalities asymptotically optimal, meaning do they capture the best possible growth rate relationships?
• RQ4 To what extent can these complexity bounds be used to estimate Cheeger-Gromov $L^2$ $\rho$-invariants?
• RQ5 What geometric constructions underlie the relationships between different 3-manifold complexity representations?

Key findings

• The paper establishes linear inequalities that relate the complexity of a 3-manifold as measured by triangulations, Heegaard splittings, and surgery presentations.
• These inequalities are shown to be asymptotically optimal, meaning no tighter linear bounds can be achieved in the large complexity limit.
• Explicit geometric constructions are provided that realize the extremal cases of the inequalities, confirming their tightness.
• The results are applied to estimate Cheeger-Gromov $L^2$ $\rho$-invariants using data from geometric group theory and knot theory.
• The framework enables quantitative control over $L^2$ $\rho$-invariants through combinatorial and geometric invariants of 3-manifolds.
• The findings provide a bridge between topological complexity and $L^2$-invariants, enhancing their computability and theoretical analysis.