Cut numbers of 3-manifolds

Sikora, Adam

doi:10.1090/S0002-9947-04-03581-0

Cut numbers of $3$-manifolds
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by Adam S. Sikora PDF

Trans. Amer. Math. Soc. 357 (2005), 2007-2020 Request permission

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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