Embedding punctured lens spaces in four-manifolds (Q1921343)

It is a well-known fact that no 3-dimensional lens space can embed into 4-space. However, a punctured lens space embeds into 4-space if and only if it has odd fundamental group. In this paper the authors study the question for which numbers \(n\) the (punctured) lens spaces \(L(p,1)\) embed into the connected sum of \(n\) copies of complex projective space \(CP^2\). In the topological category they find the minimal such \(n\) for \(L(p,1)\) and almost find it in the punctured case. The smooth case is shown to differ considerably from the topological one but here the minimal number \(n\) remains an open problem for most \(p\).