Isotopy Uniqueness of Self-diffeomorphism of Handlebodies

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The mapping class group MCG(Σg) of a surface of genus g has a long-history in topology and group theory. More recently, the mapping class group MCG(Vg) of a handlebody Vg of genus g has become an interesting topic in the study of 3 manifolds, largely thanks to Heegaard splitting. While MCG(Vg) can be regarded naturally as a sub group of MCG(Σg), we could not find any complete proof of this fundamental theorem. It is the purpose of this paper that we give a rigorous proof of embedding of MCG(Vg) into MCG(Vg). The key step is: Any self-homeomorphism f of handlebody Vg of genus g is ambient isotopic to identity if the restriction f|∂ ᵥg is isotopic to identity.