#### Geometry & Topology, Vol. 6 (2002)
Paper no. 15, pages 409--424.

## On the Cut Number of a 3-manifold

### Shelly L Harvey

**Abstract**.
The question was raised as to whether the cut number of a 3-manifold X
is bounded from below by 1/3 beta_1(X). We show that the answer to
this question is `no.' For each m>0, we construct explicit examples of
closed 3-manifolds X with beta_1(X)=m and cut number 1. That is,
pi_1(X) 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

**AMS subject classification**.
Primary: 57M27, 57N10.
Secondary: 57M05, 57M50, 20F34, 20F67.

**DOI:** 10.2140/gt.2002.6.409

**E-print:** `arXiv:math.GT/0112193`

Submitted to GT on 27 February 2002.
Paper accepted 22 August 2002.
Paper published 15 September 2002.

Notes on file formats
Shelly L Harvey

Department of Mathematics

University of California at San Diego

La Jolla, CA 92093-0112, USA

Email: sharvey@math.ucsd.edu

URL: http://math.ucsd.edu/~sharvey

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