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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On the cut number of a $3$–manifold

Shelly L Harvey

Geometry & Topology 6 (2002) 409–424

arXiv: math.GT/0112193

Abstract

The question was raised as to whether the cut number of a 3–manifold X is bounded from below by 1 3β1(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.

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
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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