Mathematics – Algebraic Topology
Scientific paper
2000-02-05
Proc. Amer. Math. Soc. 132 (2004), no. 1, 299--303
Mathematics
Algebraic Topology
3 pages, final version
Scientific paper
A smooth closed 3-manifold $M$ fibered by tori $T^2$ is characterized by an element $\phi \in GL(2,\mathbb{Z})$. We show that $M$ is the boundary of a 4-manifold fibered by tori over a surface such that the bundle structure on $M$ is the restriction of the bundle structure on the 4-manifold if and only if $\phi$ is from the commutator subgroup $(GL(2,\mathbb{Z}))'$. The notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori are introduced. It turns out that in this case the cobordisms form a group, namely $\mathbb{Z}_{12}$ in the oriented case and $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ in the unoriented one. When the surface on the base of oriented cobordism is orientable, it is shown that its minimal genus can be calculated by Culler's algorithm.
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