On attaching 3-handles to a 1-connected 4-manifold (Q788329)

The author proves several results about 4-manifolds which are obtained from a simply connected 4-manifold by adding 3-handles \(\{h_ i^ 3\}^ k_{i=1}\). In particular he proves: Theorem 1. Suppose \(M^ 4=W^ 4\cup h_ 1^ 3\cup...\cup h_ k^ 3\) and \(\tilde M^ 4=\) \(\tilde W^ 4\cup \tilde h_ 1^ 3\cup...\cup \tilde h_ k^ 3\) are 1-connected 4-manifolds with connected boundary. Then \(M^ 4\) is diffeomorphic to \(\tilde M^ 4\) if and only if \(W^ 4\) is diffeomorphic to \(\tilde W^ 4\). Theorem 2. Suppose \(M^ 4\) and \(W^ 4\) are as in the statement of Theorem 1. Then, if \(N^ 4\) is an arbitrary 4-manifold, \(M^ 4\) embeds in \(N^ 4\) if and only if \(W^ 4\) embeds in \(N^ 4\). Theorem 3. Suppose \(M^ 4\) and \(W^ 4\) are as in the statement of Theorem 1, and \(W^ 4=W_ 1^ 4\#_{\partial}W_ 2^ 4\). Then \(M^ 4=M_ 1^ 4\#_{\partial}M_ 2^ 4\) where \(M_ i^ 4=W_ i^ 4\cup(3\)-handles) for \(i=1,2\). The key result used is the following: Proposition 1. Suppose \(W^ 4\) is a 1-connected 4-manifold with connected boundary. If K is a knot in \(\partial W^ 4\) which meets the embedded 2-sphere \(\Sigma^ 2\) in \(\partial W^ 4\) transversally in one point then K is slice in \(W^ 4\). This means that any slicing disc for K is a core of a 2-handle which is complementary to any 3-handle attached by \(\Sigma^ 2\).