Geometric rank and the Tucker property. (English) Zbl 1369.57035
Let \(V^{3}\) be an open \(3\)-manifold and \(B^{n}\) the ball of dimension \(n.\) Suppose that a manifold \(W^{n+3}\) has a handlebody decomposition with finitely many \(1\)-handles. The author proves that if there exist a positive integer \(n\) and such a manifold \(W^{n+3}\) such that \(V^{3}\subset W^{n+3}\subset V^{3}\times B^{n},\) where the latter inclusion is proper, then \(\pi _{1}(V^{3}-C)\) is finitely generated for any compact and connected submanifold \(C\subset V.\)
Reviewer: Shengkui Ye (Suzhou)
MSC:
57R65 | Surgery and handlebodies |
57Q15 | Triangulating manifolds |
57N35 | Embeddings and immersions in topological manifolds |