The toric cobordisms

TL;DR

This paper characterizes 3-manifolds fibered by tori as boundaries of 4-manifolds with torus fibrations, introduces cobordism notions, and computes the associated cobordism groups, revealing their algebraic structures.

Contribution

It establishes a criterion for when a torus-fibered 3-manifold bounds a torus-fibered 4-manifold and describes the algebraic structure of the cobordism groups.

Findings

Cobordism groups are isomorphic to  in the oriented case and  in the unoriented case.

A criterion for boundary characterization involves the commutator subgroup of GL(2,).

Minimal genus of the base surface can be computed using Culler's algorithm.

Abstract

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.