Critical Heegaard surfaces obtained by amalgamation

TL;DR

This paper provides a condition for constructing critical Heegaard surfaces through amalgamation, including boundary stabilization, resulting in non-minimal genus surfaces in certain 3-manifolds.

Contribution

It introduces a new criterion for obtaining critical Heegaard surfaces via amalgamation, expanding the understanding of their existence in specific 3-manifolds.

Findings

Critical Heegaard surfaces can be obtained by amalgamation under certain conditions.

Boundary stabilization can produce critical Heegaard surfaces.

Examples include non-minimal genus surfaces in manifolds with unique Heegaard splittings.

Abstract

Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It gives critical Heegaard surfaces of non-minimal genus, for 3-manifolds which do not admit distinct Heegaard splittings (up to isotopy).