Abstract. Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper we give a sufficiently condition and a necessary condition for selfamalgamated Heegaard surfaces to be critical.
We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m, then surgery on some (m − 1) components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.
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.
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