Cut numbers of $3$-manifolds
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- by Adam S. Sikora PDF
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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.References
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Additional Information
- Adam S. Sikora
- Affiliation: Department of Mathematics, 244 Mathematics Building, SUNY at Buffalo, Buffalo, New York 14260
- MR Author ID: 364939
- Email: asikora@buffalo.edu
- Received by editor(s): October 28, 2002
- Received by editor(s) in revised form: December 2, 2003
- Published electronically: October 7, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2007-2020
- MSC (2000): Primary 57M05, 57M27, 20F34, 11E76
- DOI: https://doi.org/10.1090/S0002-9947-04-03581-0
- MathSciNet review: 2115088