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Kotschick, D.; Neofytidis, C.

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

Math. Z. 274, No. 1-2, 21-32 (2013).
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.]).
Reviewer: Maria Rita Casali (Modena)

Cited in 1 Review
Cited in 13 Documents

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

Keywords:

3-manifold; non-zero degree map; rational essentialness; coverings; Thurston geometries

Citations:

Zbl 0483.57003; Zbl 1168.53024; Zbl 1286.20051
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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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