Geometry & Topology

On the cut number of a $3$–manifold

Shelly L Harvey

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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 m1, 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

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

Received: 27 February 2002
Accepted: 22 August 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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