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.

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Shelly L Harvey
Department of Mathematics
University of California at San Diego
La Jolla, CA 92093-0112, USA
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