On the cut number of a 3–manifold

Harvey, Shelly L¹

¹ Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112, USA

Geometry & topology, Tome 6 (2002) no. 1, pp. 409-424.

Résumé

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.

Détail

DOI : 10.2140/gt.2002.6.409

Keywords: 3–manifold, fundamental group, corank, Alexander module, virtual betti number, free group

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