## Geometry & Topology

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

### On the cut number of a $3$–manifold

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

Received: 27 February 2002

Accepted: 22 August 2002

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

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

**Subjects**

Primary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]

Secondary: 57M05: Fundamental group, presentations, free differential calculus 57M50: Geometric structures on low-dimensional manifolds 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20F67: Hyperbolic groups and nonpositively curved groups

**Keywords**

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

#### 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