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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.

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
Full Text: DOI arXiv
[1] R Brooks, W Goldman, The Godbillon-Vey invariant of a transversely homogeneous foliation, Trans. Amer. Math. Soc. 286 (1984) 651 · Zbl 0548.57016
[2] R Brooks, W Goldman, Volumes in Seifert space, Duke Math. J. 51 (1984) 529 · Zbl 0546.57003
[3] J A Carlson, D Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. (1989) 173 · Zbl 0695.58010
[4] C Connell, B Farb, The degree theorem in higher rank, J. Differential Geom. 65 (2003) 19 · Zbl 1067.53032
[5] P Derbez, Topological rigidity and Gromov simplicial volume, to appear in Comment. Math. Helv. · Zbl 1204.57012
[6] P Derbez, Nonzero degree maps between closed orientable three-manifolds, Trans. Amer. Math. Soc. 359 (2007) 3887 · Zbl 1120.57009
[7] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) · Zbl 0516.53046
[8] W H Jaco, P B Shalen, Seifert fibered spaces in \(3\)-manifolds, Mem. Amer. Math. Soc. 21 (1979) · Zbl 0415.57005
[9] J F Lafont, B Schmidt, Simplicial volume of closed locally symmetric spaces of non-compact type, Acta Math. 197 (2006) 129 · Zbl 1111.57020
[10] J Luecke, Y Q Wu, Relative Euler number and finite covers of graph manifolds (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 80 · Zbl 0889.57024
[11] J Milnor, W Thurston, Characteristic numbers of \(3\)-manifolds, Enseignement Math. \((2)\) 23 (1977) 249 · Zbl 0393.57002
[12] A Reznikov, Volumes of discrete groups and topological complexity of homology spheres, Math. Ann. 306 (1996) 547 · Zbl 0859.20027
[13] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. \((\)N.S.\()\) 6 (1982) 357 · Zbl 0496.57005
[14] S Wang, The \(\pi_1\)-injectivity of self-maps of nonzero degree on \(3\)-manifolds, Math. Ann. 297 (1993) 171 · Zbl 0793.57008
[15] S Wang, Non-zero degree maps between 3-manifolds, Higher Ed. Press (2002) 457 · Zbl 1009.57025
[16] S Wang, Q Zhou, Any \(3\)-manifold \(1\)-dominates at most finitely many geometric \(3\)-manifolds, Math. Ann. 322 (2002) 525 · Zbl 0992.57017
[17] F Yu, S Wang, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv. 74 (1999) 238 · Zbl 0930.57011
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