## 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.]).

### 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

### Citations:

Zbl 0483.57003; Zbl 1168.53024; Zbl 1286.20051
Full Text:

### References:

 [1] Brunnbauer, M.; Hanke, B., Large and small group homology, J. Topol., 3, 463-486, (2010) · Zbl 1196.53028 [2] Carlson, J.A.; Toledo, D., Harmonic mapping of Kähler manifolds to locally symmetric spaces, Publ. Math. I.H.E.S., 69, 173-201, (1989) · Zbl 0695.58010 [3] Casson, A.; Jungreis, D., Convergence groups and Seifert fibered 3-manifolds, Invent. Math., 118, 441-456, (1994) · Zbl 0840.57005 [4] Derbez, P.; Sun, H.; Wang, S., Finiteness of mapping degree sets for 3-manifolds, Acta Math. Sinica Engl. Ser., 27, 807-812, (2011) · Zbl 1221.55005 [5] Epstein, D.B.A., Factorization of 3-manifolds, Comment. Math. Helv., 36, 91-102, (1961) · Zbl 0102.38704 [6] Gabai, D., Convergence groups are Fuchsian groups, Ann. Math., 136, 447-510, (1992) · Zbl 0785.57004 [7] Gromov, M., Volume and bounded cohomology, Publ. Math. I.H.E.S., 56, 5-99, (1982) · Zbl 0516.53046 [8] Gromov, M.; Lawson, H.B., Spin and scalar curvature in the presence of a fundamental group I, Ann. Math., 111, 209-230, (1980) · Zbl 0445.53025 [9] Hanke, B.; Schick, T., Enlargeability and index theory, J. Differ. Geom., 74, 293-320, (2006) · Zbl 1122.58011 [10] de la Harpe P.: Topics in Geometric Group Theory. The University of Chicago Press, Chicago and London (2000) · Zbl 0965.20025 [11] Hempel, J.: Residual finiteness for 3-manifolds. In: Gersten, S.M., Stallings, J.R. (eds.) Combinatorial Groups Theory and Topology. Annals of Mathematics Studies, vol. 111. Princeton University Press, NJ (1987) · Zbl 0772.57002 [12] Jaco, W.: Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics no. 43. American Mathematical Society, USA (1980) · Zbl 1221.55005 [13] Kleiner, B.; Lott, J., Notes on perelman’s papers, Geom. Topol., 12, 2587-2855, (2008) · Zbl 1204.53033 [14] Kotschick, D.; Löh, C., Fundamental classes not representable by products, J. Lond. Math. Soc., 79, 545-561, (2009) · Zbl 1168.53024 [15] Kotschick, D., Löh, C.: Groups not presentable by products. Groups Geom. Dyn. (to appear) · Zbl 1286.20051 [16] Milnor, J.W., A unique decomposition theorem for 3-manifolds, Amer. J. Math., 84, 1-7, (1962) · Zbl 0108.36501 [17] Morgan, J.W., Tian, G.: Ricci flow and the Poincaré conjecture. Am. Math. Soc. Clay Math. Inst. (2007) · Zbl 0840.57005 [18] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint arXiv:math/ 0211159v1 [math.DG] (2002) · Zbl 0695.58010 [19] Perelman, G.: Ricci flow with surgery on three-manifolds. Preprint arXiv:math/0303109v1 [math.DG] 10 Mar (2003) · Zbl 0516.53046 [20] Sakuma, M., Surface bundles over $$S$$\^{}{1} which are 2-fold branched cyclic coverings of $$S$$\^{}{3}, Math. Sem. Notes Kobe Univ., 9, 159-180, (1981) · Zbl 0483.57003 [21] Scott, P., There are no fake Seifert fibre spaces with infinite π_{1}, Ann. Math., 117, 35-70, (1983) · Zbl 0516.57006 [22] Scott, P., The geometries of 3-manifolds, Bull. Lond. Math. Soc., 15, 401-487, (1983) · Zbl 0561.57001 [23] Waldhausen, F., Gruppen mit zentrum und 3-dimensionale mannigfaltigkeiten, Topology, 6, 505-517, (1967) · Zbl 0172.48704 [24] Wang, S., The existence of maps of non-zero degree between aspherical 3-manifolds, Math. Z., 208, 147-160, (1991) · Zbl 0737.57006 [25] Wang, S.: Non-zero degree maps between 3-manifolds. In: Proceedings of the ICM Beijing 2002. vol. II, pp. 457-468, Higher Education Press, Beijing (2002) · Zbl 1009.57025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.