Finiteness of mapping degrees and \(\text{PSL}(2,R\mathbf)\)-volume on graph manifolds.

*(English)*Zbl 1187.57021Let \(M\) and \(N\) be two closed oriented 3-manifolds. Let \({\mathcal D}(M, N)\) be the set of degrees of maps from \(M\) to \(N\), that is \({\mathcal D}(M, N)= \{d\in\mathbb{Z}\mid f: M\to N\), \(\deg(f)= d\}\). By the work of W. P. Thurston [Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982; Zbl 0496.57005)]; R. Brooks and W. Goldman [Trans. Am. Math. Soc. 286, 651–664 (1984; Zbl 0548.57016) and Duke Math. J. 51, 529–545 (1984; Zbl 0546.57003)] and S. Wang [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 457–468 (2002; Zbl 1009.57025)], we know that (i) if \(N\) is closed oriented geometric 3-manifold which supports the hyperbolic or the \(\widetilde{\text{PSL}}(2, \mathbb{R})\) geometry then \({\mathcal D}(M, N)\) is finite for any \(M\); (ii) if \(N\) admits one of the six remainding geometries, \(S^3, S^2\times\mathbb{R}\), \(\text{Nil},\mathbb{R}^3,\mathbb{H}^2\times \mathbb{R}\) or Sol then \({\mathcal D}(N, N)\) is infinite.

In the paper under review the authors prove that for any given closed prime nontrivial graph manifold \(N\), \({\mathcal D}(M, N)\) is finite for any graph manifold \(M\). The proof uses a recently developed standard form of maps between graph manifolds and the estimation of the \(\widetilde{\text{PSL}}(2,\mathbb{R})\)-volume for a certain class of graph manifolds.

In the paper under review the authors prove that for any given closed prime nontrivial graph manifold \(N\), \({\mathcal D}(M, N)\) is finite for any graph manifold \(M\). The proof uses a recently developed standard form of maps between graph manifolds and the estimation of the \(\widetilde{\text{PSL}}(2,\mathbb{R})\)-volume for a certain class of graph manifolds.

Reviewer: Andrzej Szczepański (Gdańsk)

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

55M25 | Degree, winding number |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

54C50 | Topology of special sets defined by functions |

57N65 | Algebraic topology of manifolds |

##### Keywords:

geometric manifolds; graph manifolds; three-dimensional geometries; degree of map; three-dimensional manifolds
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\textit{P. Derbez} and \textit{S. Wang}, Algebr. Geom. Topol. 9, No. 3, 1727--1749 (2009; Zbl 1187.57021)

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