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**On three-manifolds dominated by circle bundles.**
*(English)*
Zbl 1277.57003

Given two closed oriented \(n\)-manifolds \(M\) and \(N\), \(M\) is said to dominate \(N\) if a non-zero degree map from \(M\) to \(N\) exists. From dimension \(n=3\) on, the domination relation fails to be an ordering.

By a result of M. Sakuma [Math. Semin. Notes, Kobe Univ. 9, 159–180 (1981; Zbl 0483.57003)], every \(3\)-manifold turns out to be dominated by a surface bundle over the circle; on the other hand, in [D. Kotschick and C. Löh, J. Lond. Math. Soc., II. Ser. 79, No. 3, 545–561 (2009; Zbl 1168.53024), Groups Geom. Dyn. 7, No. 1, 181–204 (2013; Zbl 1286.20051)] it is shown that \(3\)-manifolds dominated by products cannot have hyperbolic or Sol\(^3\)-geometry, and must often be prime.

In the present paper, the authors give a complete characterization of \(3\)-manifolds dominated by products, by proving that a closed oriented \(3\)-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of \(\mathbb S^2 \times \mathbb S^1\). It is worthwhile to note that the same characterization may also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group.

Moreover, the authors determine which \(3\)-manifolds are dominated by non-trivial circle bundles, and which \(3\)-manifold groups are presentable by products (according to [D. Kotschick and C. Löh, loc. cit.]).

By a result of M. Sakuma [Math. Semin. Notes, Kobe Univ. 9, 159–180 (1981; Zbl 0483.57003)], every \(3\)-manifold turns out to be dominated by a surface bundle over the circle; on the other hand, in [D. Kotschick and C. Löh, J. Lond. Math. Soc., II. Ser. 79, No. 3, 545–561 (2009; Zbl 1168.53024), Groups Geom. Dyn. 7, No. 1, 181–204 (2013; Zbl 1286.20051)] it is shown that \(3\)-manifolds dominated by products cannot have hyperbolic or Sol\(^3\)-geometry, and must often be prime.

In the present paper, the authors give a complete characterization of \(3\)-manifolds dominated by products, by proving that a closed oriented \(3\)-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of \(\mathbb S^2 \times \mathbb S^1\). It is worthwhile to note that the same characterization may also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group.

Moreover, the authors determine which \(3\)-manifolds are dominated by non-trivial circle bundles, and which \(3\)-manifold groups are presentable by products (according to [D. Kotschick and C. Löh, loc. cit.]).

Reviewer: Maria Rita Casali (Modena)

### MSC:

57M05 | Fundamental group, presentations, free differential calculus |

57M12 | Low-dimensional topology of special (e.g., branched) coverings |

57M50 | General geometric structures on low-dimensional manifolds |

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\textit{D. Kotschick} and \textit{C. Neofytidis}, Math. Z. 274, No. 1--2, 21--32 (2013; Zbl 1277.57003)

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