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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cut numbers of $3$-manifolds


Author: Adam S. Sikora
Journal: Trans. Amer. Math. Soc. 357 (2005), 2007-2020
MSC (2000): Primary 57M05, 57M27, 20F34, 11E76
Published electronically: October 7, 2004
MathSciNet review: 2115088
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Abstract: We investigate the relations between the cut number, $c(M),$ and the first Betti number, $b_1(M),$ of $3$-manifolds $M.$ We prove that the cut number of a ``generic'' $3$-manifold $M$ is at most $2.$ This is a rather unexpected result since specific examples of $3$-manifolds with large $b_1(M)$ and $c(M)\leq 2$ are hard to construct. We also prove that for any complex semisimple Lie algebra $\mathfrak g$ there exists a $3$-manifold $M$ with $b_1(M)=dim\, \mathfrak g$ and $c(M)\leq rank\, \mathfrak g.$ Such manifolds can be explicitly constructed.


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Additional Information

Adam S. Sikora
Affiliation: Department of Mathematics, 244 Mathematics Building, SUNY at Buffalo, Buffalo, New York 14260
Email: asikora@buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03581-0
PII: S 0002-9947(04)03581-0
Keywords: Cut number, 3-manifold, corank, skew-symmetric form, cohomology ring
Received by editor(s): October 28, 2002
Received by editor(s) in revised form: December 2, 2003
Published electronically: October 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society