A Generalization of the Handle Addition Theorem

We will generalize Jaco’s Handle Addition Theorem to the ncompressibility of surfaces on the boundary of 3-manifolds. Several corollaries are given, which show how the theorem can be applied to different situations. 1 The Handle Addition Theorem was first proved by Przytycki [6] in the case when M is a handlebody. In [4] Jaco proved the general version below. Handle Addition Theorem [4] Suppose M is a 3-manifold with compressible boundary, and J is a simple closed curve on ∂M such that ∂M − J is incompressible. Then the manifold obtained by adding a 2-handle to M along J has incompressible boundary. Note that in the theorem M can be noncompact. So the theorem is still true when ∂M is replaced by a surface S on ∂M . Several alternative proofs have been published ([1, 5, 7]). And it has been applied very successfully in dealing with incompressible surfaces, surgeries and other related topics. (See for example [2, 3, 4, 5, 8]). In this paper we will discuss the ncompressibility of surfaces with respect to a specified 1-manifold, and prove a generalized handle addition theorem for this situation. While 0-compressibility is the usual notion of compressibility of surfaces, 2-compressibility includes ∂-compressibility. Our original motivation is to prove Corollary 3 below, which says that under certain conditions an essential surface in a 3-manifold will remain essential after handle addition. Some other corollaries are given, which illustrate how the theorem is applied in different situations.