Applications Of Heegaard Splitting On Haken Manifolds

Let M be a3-manifold,F={F1,F2,…Fn} be a collection of essential closed surfaces in M(Fi is not parallel to Fj if i≠j)and (?)0M be a collection of components of (?)M.Suppose M-U E∈F Fi-×(-1,1) consists of k components M1,M2,…Mk.If each Mi has a Heegaard splitting ViUsiWi with d(Si)>4(g(M1)+…+g(Mk)),then any minimal Heegaard splitting of M relative to (?)M is obtained by doing amalgamations or self-amalgamation from minimal Heegaard splittings of (?)-stabilization of minimal Heegaard splittings of M1,M2,…Mk.There ard many results on the super additivity of tunnel number.In chapter5,we proved the following result.If d(K1)≥2(t(K1)t+(K2)+2)while d(K2)≥3,then t(K1#K2) t(K1)+t(K2)+1.