GEOMETRIC RANK AND THE TUCKER PROPERTY
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An open smooth manifold is said of \textit{finite geometric rank} if it admits a handlebody decomposition with a finite number of 1-handles. We prove that, if there exists a proper submanifold $W^{n+3}$ of finite geometric rank between an open 3-manifold $V^3$ and its stabilization $V^3 \times B^n$ (where $B^n$ denotes the standard $n$-ball), then the manifold $V^3$ has the \textit{Tucker property}. This means that for any compact submanifold $C\subset V^3$, the fundamental group $\pi_1 (V^3 - C)$ is finitely generated. In the irreducible case this implies that $V^3$ has a well-behaved compactification.
Abstract
An open smooth manifold is said of \textit{finite geometric rank} if it admits a handlebody decomposition with a finite number of 1-handles. We prove that, if there exists a proper submanifold $W^{n+3}$ of finite geometric rank between an open 3-manifold $V^3$ and its stabilization $V^3 \times B^n$ (where $B^n$ denotes the standard $n$-ball), then the manifold $V^3$ has the \textit{Tucker property}. This means that for any compact submanifold $C\subset V^3$, the fundamental group $\pi_1 (V^3 - C)$ is finitely generated. In the irreducible case this implies that $V^3$ has a well-behaved compactification.
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