(untitled) (Communications in Mathematical Research, 2021)
Let M be a compact connected oriented 3-manifold with boundary, Q_1, Q_2 ⊂ ∂M be two disjoint homeomorphic subsurfaces of ∂M, and h : Q_1 → Q_2 be an orientation-reversing homeomorphism. Denote by M_h or M_Q_1=Q_2 the 3-manifold obtained from M by gluing Q_1 and Q_2 together via h. M_h is called a self-amalgamation of M along Q_1 and Q_2. Suppose Q_1 and Q_2 lie on the same component F' of ∂M', and F' − Q_1 ∪ Q_2 is connected. We give a lower bound to the Heegaard genus of M when M' has a Heegaard splitting with sufficiently high distance.