Volume 6, issue 1 (2002)

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On the cut number of a $3$–manifold

Shelly L Harvey

Geometry & Topology 6 (2002) 409–424
 arXiv: math.GT/0112193
Abstract

The question was raised as to whether the cut number of a 3–manifold $X$ is bounded from below by $\frac{1}{3}{\beta }_{1}\left(X\right)$. We show that the answer to this question is “no.” For each $m\ge 1$, we construct explicit examples of closed 3–manifolds $X$ with ${\beta }_{1}\left(X\right)=m$ and cut number 1. That is, ${\pi }_{1}\left(X\right)$ cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.

Keywords
3–manifold, fundamental group, corank, Alexander module, virtual betti number, free group
Mathematical Subject Classification 2000
Primary: 57M27, 57N10
Secondary: 57M05, 57M50, 20F34, 20F67