## Geometry & Topology

### 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 $13β1(X)$. We show that the answer to this question is “no.” For each $m≥1$, we construct explicit examples of closed 3–manifolds $X$ with $β1(X)=m$ and cut number 1. That is, $π1(X)$ cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.

#### Article information

Source
Geom. Topol., Volume 6, Number 1 (2002), 409-424.

Dates
Accepted: 22 August 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882801

Digital Object Identifier
doi:10.2140/gt.2002.6.409

Mathematical Reviews number (MathSciNet)
MR1928841

Zentralblatt MATH identifier
1021.57006

#### Citation

Harvey, Shelly L. On the cut number of a $3$–manifold. Geom. Topol. 6 (2002), no. 1, 409--424. doi:10.2140/gt.2002.6.409. https://projecteuclid.org/euclid.gt/1513882801

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