On the cut number of a 3–manifold

The question was raised as to whether the cut number of a 3–manifold $X$ is bounded from below by $\frac{1}{3} β_{1} (X)$ . We show that the answer to this question is “no.” For each $m \geq 1$ , we construct explicit examples of closed 3–manifolds $X$ with $β_{1} (X) = m$ and cut number 1. That is, $π_{1} (X)$ 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

Mathematical Subject Classification 2000

Primary: 57M27, 57N10

Secondary: 57M05, 57M50, 20F34, 20F67

References

Forward citations

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

Volume 6, issue 1 (2002)

Shelly L Harvey