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Finiteness of mapping degrees and $$\text{PSL}(2,R\mathbf)$$-volume on graph manifolds. (English) Zbl 1187.57021
Let $$M$$ 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.

##### 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
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