Unstabilized self-amalgamation of a Heegaard splitting

Let M be a compact orientable 3-manifold, M = V ∪ S W be a Heegaard splitting of M, and F 1 , F 2 be two homeomorphic components of ∂M lying in the minus boundary of W. Let M ⁎ be the manifold obtained from M by gluing F 1 and F 2 together. Then M ⁎ has a natural Heegaard splitting called the self-amalgamation of V ∪ S W . In this paper, we prove that the self-amalgamation of a distance at least 3 Heegaard splitting is unstabilized. There are some examples to show that the lower bound 3 is the best.