A note on Heegaard genus of self-amalgamated 3-manifold

Abstract

Let M be a connected orientable compact irreducible 3-manifold. Suppose that ∂M consists of two homeomorphic surfaces F ₁ and F ₂, and both F ₁ and F ₂ are compressible in M. Suppose furthermore that g(M,F ₁) = g(M) + g(F ₁), where g(M,F ₁) is the Heegaard genus of M relative to F ₁. Let M _f be the closed orientable 3-manifold obtained by identifying F ₁ and F ₂ using a homeomorphism f: F ₁ → F ₂. The authors show that if f is sufficiently complicated, then g(M _f) = g(M,∂M) + 1.

Keywords

Heegaard splitting / Self-amalgamated / Sufficiently complicated

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