#### 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
##### Publication
Received: 27 February 2002
Accepted: 22 August 2002
Published: 15 September 2002
Proposed: Cameron Gordon
Seconded: Joan Birman, Walter Neumann
##### Authors
 Shelly L Harvey Department of Mathematics University of California at San Diego La Jolla California 92093-0112 USA