A topologically minimal canonical Heegaard splitting of a surface bundle is critical

Abstract

Every surface bundle with genus [Formula: see text] fiber has a canonical Heegaard splitting of genus [Formula: see text]. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2.