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Twisted Alexander polynomials detect fibered 3-manifolds. (English) Zbl 1231.57020
It is well known that, if a knot \(K \subset S^3\) is fibered, then the Alexander polynomial is monic and the degree equals twice the genus of \(K.\) Various generalizations of this result have been performed, showing that twisted Alexander polynomials give necessary conditions for \((N, \phi)\) (where \(N\) is a compact, connected, oriented 3-manifolds with empty or toroidal boundary and \(\phi \in H^1(N; Z)\)) to fiber: see [C. T. McMullen, Ann. Sci. Ecol. Norm. Super. (4) 35, No. 2, 153–171 (2002; Zbl 1009.57021)], [J. C. Cha, Trans. Am. Math. Soc. 355, No.10, 4187–4200 (2003; Zbl 1028.57004)], [H. Goda, T. Kitano, T. Morifuji, Comment. Math. Helv. 80, No. 1, 51–61 (2005; Zbl 1066.57008)], [S. Friedl, T. Kim, Topology 45, No. 6, 929–953 (2006; Zbl 1105.57009)] and [T. Kitayama, “Normalization of twisted Alexander invariants”, preprint 2007 (arXiv:0705.2371)].
In general, the constraint of monicness and degree for the ordinary Alexander polynomial falls short from characterizing fibered 3-manifolds. The main result of present paper shows that that the collection of all twisted Alexander polynomials does detect fiberedness; equivalently, it proves that twisted Alexander polynomials detect whether \((N, \phi)\) fibers under the assumption that the Thurston norm of \(\phi\) is known.
Moreover, by making use of some of their previous works (see in particular [S. Friedl and S. Vidussi, Am. J. Math. 130, No. 2, 455–484 (2008; Zbl 1154.57021)]), the authors show that, if a manifold of the form \(S^1 \times N^3\) admits a symplectic structure, then \(N\) fibers over \(S^1\).

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:
[1] I. Agol, ”Criteria for virtual fibering,” J. Topol., vol. 1, iss. 2, pp. 269-284, 2008. · Zbl 1148.57023 · doi:10.1112/jtopol/jtn003 · arxiv:0707.4522
[2] G. Arzhantseva, P. de la Harpe, D. Kahrobaei, and Z. vSunić, ”The true prosoluble completion of a group: examples and open problems,” Geom. Dedicata, vol. 124, pp. 5-26, 2007. · Zbl 1138.20027 · doi:10.1007/s10711-006-9103-y · arxiv:math/0512242
[3] M. Aschenbrenner and S. Friedl, 3-manifold groups are virtually residually \(p\), 2010. · Zbl 1328.57002 · doi:10.1090/S0065-9266-2013-00682-X · arxiv:1004.3619
[4] M. R. Bridson and F. J. Grunewald, ”Grothendieck’s problems concerning profinite completions and representations of groups,” Ann. of Math., vol. 160, iss. 1, pp. 359-373, 2004. · Zbl 1083.20023 · doi:10.4007/annals.2004.160.359 · euclid:annm/1105737695
[5] K. S. Brown, Cohomology of Groups, New York: Springer-Verlag, 1994, vol. 87. · Zbl 0823.20049 · doi:10.1016/0022-4049(94)90036-1
[6] A. Candel and L. Conlon, Foliations. II, Providence, RI: Amer. Math. Soc., 2003, vol. 60. · Zbl 1035.57001
[7] J. C. Cha, ”Fibred knots and twisted Alexander invariants,” Trans. Amer. Math. Soc., vol. 355, iss. 10, pp. 4187-4200, 2003. · Zbl 1028.57004 · doi:10.1090/S0002-9947-03-03348-8 · arxiv:math/0109136
[8] W. Chen and R. Matveyev, ”Symplectic Lefschetz fibrations on \(S^1\times M^3\),” Geom. Topol., vol. 4, pp. 517-535, 2000. · Zbl 0968.57012 · doi:10.2140/gt.2000.4.517 · emis:journals/UW/gt/GTVol4/paper18.abs.html · eudml:121212 · arxiv:math/0002022
[9] S. K. Donaldson, ”Symplectic submanifolds and almost-complex geometry,” J. Differential Geom., vol. 44, iss. 4, pp. 666-705, 1996. · Zbl 0883.53032 · projecteuclid.org
[10] D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Princeton, NJ: Princeton Univ. Press, 1985, vol. 110. · Zbl 0628.57002 · doi:10.1515/9781400881925
[11] T. Etgü, ”Lefschetz fibrations, complex structures and Seifert fibrations on \(S^1\times M^3\),” Algebr. Geom. Topol., vol. 1, pp. 469-489, 2001. · Zbl 0977.57010 · doi:10.2140/agt.2001.1.469 · emis:journals/UW/agt/AGTVol1/agt-1-24.abs.html · eudml:121509 · arxiv:math/0109150
[12] S. Friedl and T. Kim, ”The Thurston norm, fibered manifolds and twisted Alexander polynomials,” Topology, vol. 45, iss. 6, pp. 929-953, 2006. · Zbl 1105.57009 · doi:10.1016/j.top.2006.06.003 · arxiv:math/0505594
[13] S. Friedl and S. Vidussi, Construction of symplectic structures on 4-manifolds with a free circle action. · Zbl 1244.57045 · doi:10.1017/S0308210510000727 · arxiv:1102.0821
[14] S. Friedl and S. Vidussi, ”Twisted Alexander polynomials and symplectic structures,” Amer. J. Math., vol. 130, iss. 2, pp. 455-484, 2008. · Zbl 1154.57021 · doi:10.1353/ajm.2008.0014 · arxiv:math/0604398
[15] S. Friedl and S. Vidussi, ”Symplectic \(S^1\times N^3\), subgroup separability, and vanishing Thurston norm,” J. Amer. Math. Soc., vol. 21, iss. 2, pp. 597-610, 2008. · Zbl 1142.57014 · doi:10.1090/S0894-0347-07-00577-2 · arxiv:math/0701717
[16] S. Friedl and S. Vidussi, Twisted Alexander polynomials and fibered 3-manifolds. · Zbl 1244.57024
[17] D. Gabai, ”Foliations and the topology of \(3\)-manifolds,” J. Differential Geom., vol. 18, iss. 3, pp. 445-503, 1983. · Zbl 0533.57013 · projecteuclid.org
[18] P. Ghiggini, ”Knot Floer homology detects genus-one fibred knots,” Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008. · Zbl 1149.57019 · doi:10.1353/ajm.0.0016 · arxiv:math/0603445
[19] H. Goda, T. Kitano, and T. Morifuji, ”Reidemeister torsion, twisted Alexander polynomial and fibered knots,” Comment. Math. Helv., vol. 80, iss. 1, pp. 51-61, 2005. · Zbl 1066.57008 · doi:10.4171/CMH/3 · arxiv:math/0311155
[20] A. Grothendieck, ”Représentations linéaires et compactification profinie des groupes discrets,” Manuscripta Math., vol. 2, pp. 375-396, 1970. · Zbl 0239.20065 · doi:10.1007/BF01719593 · eudml:154006
[21] P. Hall, ”On the finiteness of certain soluble groups,” Proc. London Math. Soc., vol. 9, pp. 595-622, 1959. · Zbl 0091.02501 · doi:10.1112/plms/s3-9.4.595
[22] E. Hamilton, ”Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic \(n\)-orbifolds,” Proc. London Math. Soc., vol. 83, iss. 3, pp. 626-646, 2001. · Zbl 1025.57007 · doi:10.1112/plms/83.3.626
[23] J. Hempel, \(3\)-Manifolds, Princeton, NJ: Princeton Univ. Press, 1976, vol. 86. · Zbl 0345.57001
[24] J. Hempel, ”Residual finiteness for \(3\)-manifolds,” in Combinatorial Group Theory and Topology, Princeton, NJ: Princeton Univ. Press, 1987, vol. 111, pp. 379-396. · Zbl 0772.57002
[25] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Second ed., New York: Springer-Verlag, 1997, vol. 4. · Zbl 0863.18001
[26] R. Kirby, ”Problems in low-dimensional topology,” in Geometric Topology, Providence, RI, 1997, pp. 35-473. · Zbl 0888.57014
[27] P. Kirk and C. Livingston, ”Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants,” Topology, vol. 38, iss. 3, pp. 635-661, 1999. · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1
[28] T. Kitayama, Normalization of twisted Alexander invariants, 2007. · Zbl 1339.57012 · doi:10.1142/S0129167X15500779 · arxiv:0705.2371
[29] P. B. Kronheimer, ”Embedded surfaces and gauge theory in three and four dimensions,” in Surveys in Differential Geometry, Vol. III, Internat. Press, Boston, 1998, pp. 243-298. · Zbl 0965.57030
[30] P. B. Kronheimer, ”Minimal genus in \(S^1\times M^3\),” Invent. Math., vol. 135, iss. 1, pp. 45-61, 1999. · Zbl 0917.57017 · doi:10.1007/s002220050279
[31] P. B. Kronheimer and T. Mrowka, ”Knots, sutures and excision,” J. Differential Geom., vol. 84, pp. 301-364, 2010. · Zbl 1208.57008 · euclid:jdg/1274707316 · arxiv:0807.4891
[32] &. Kutluhan and C. H. Taubes, ”Seiberg-Witten Floer homology and symplectic forms on \(S^1\times M^3\),” Geom. Topol., vol. 13, iss. 1, pp. 493-525, 2009. · Zbl 1182.57020 · doi:10.2140/gt.2009.13.493 · arxiv:0804.1371
[33] X. S. Lin, ”Representations of knot groups and twisted Alexander polynomials,” Acta Math. Sin. \((\)Engl. Ser.\()\), vol. 17, iss. 3, pp. 361-380, 2001. · Zbl 0986.57003 · doi:10.1007/s101140100122
[34] D. D. Long and G. A. Niblo, ”Subgroup separability and \(3\)-manifold groups,” Math. Z., vol. 207, iss. 2, pp. 209-215, 1991. · Zbl 0711.57002 · doi:10.1007/BF02571384 · eudml:174265
[35] A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212. · Zbl 1071.20033
[36] J. D. McCarthy, ”On the asphericity of a symplectic \(M^3\times S^1\),” Proc. Amer. Math. Soc., vol. 129, iss. 1, pp. 257-264, 2001. · Zbl 0972.53045 · doi:10.1090/S0002-9939-00-05571-4
[37] C. T. McMullen, ”The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology,” Ann. Sci. École Norm. Sup., vol. 35, iss. 2, pp. 153-171, 2002. · Zbl 1009.57021 · doi:10.1016/S0012-9593(02)01086-8 · numdam:ASENS_2002_4_35_2_153_0 · eudml:82567
[38] J. Morgan and G. Tian, Ricci Flow and the Poincaré Conjecture, Providence, RI: Amer. Math. Soc., 2007, vol. 3. · Zbl 1179.57045
[39] Y. Ni, Addendum to: “Knots, sutures and excision”, 2008.
[40] Y. Ni, ”Heegaard Floer homology and fibred 3-manifolds,” Amer. J. Math., vol. 131, iss. 4, pp. 1047-1063, 2009. · Zbl 1184.57026 · doi:10.1353/ajm.0.0064 · muse.jhu.edu · arxiv:0706.2032
[41] L. Ribes and P. Zalesskii, Profinite Groups, New York: Springer-Verlag, 2000, vol. 40. · Zbl 0949.20017
[42] R. Roussarie, ”Plongements dans les variétés feuilletées et classification de feuilletages sans holonomie,” Inst. Hautes Études Sci. Publ. Math., iss. 43, pp. 101-141, 1974. · Zbl 0356.57017 · doi:10.1007/BF02684367 · numdam:PMIHES_1974__43__101_0 · eudml:103924
[43] M. Scharlemann, ”\(3\)-manifolds with \(H_2(A,\partial A)=0\) and a conjecture of Stallings,” in Knot Theory and Manifolds, New York: Springer-Verlag, 1985, vol. 1144, pp. 138-145. · Zbl 0585.57011 · doi:10.1007/BFb0075017
[44] M. Scharlemann, ”Sutured manifolds and generalized Thurston norms,” J. Differential Geom., vol. 29, iss. 3, pp. 557-614, 1989. · Zbl 0673.57015 · projecteuclid.org
[45] J. Stallings, ”On fibering certain \(3\)-manifolds,” in Topology of 3-manifolds and Related Topics, Englewood Cliffs, N.J.: Prentice-Hall, 1962, pp. 95-100. · Zbl 0132.20102
[46] J. R. Stallings, ”Surfaces in three-manifolds and nonsingular equations in groups,” Math. Z., vol. 184, iss. 1, pp. 1-17, 1983. · Zbl 0496.57006 · doi:10.1007/BF01162003 · eudml:173342
[47] C. H. Taubes, ”The Seiberg-Witten invariants and symplectic forms,” Math. Res. Lett., vol. 1, iss. 6, pp. 809-822, 1994. · Zbl 0853.57019 · doi:10.4310/MRL.1994.v1.n6.a15
[48] C. H. Taubes, ”More constraints on symplectic forms from Seiberg-Witten invariants,” Math. Res. Lett., vol. 2, iss. 1, pp. 9-13, 1995. · Zbl 0854.57019 · doi:10.4310/MRL.1995.v2.n1.a2
[49] W. P. Thurston, ”Some simple examples of symplectic manifolds,” Proc. Amer. Math. Soc., vol. 55, iss. 2, pp. 467-468, 1976. · Zbl 0324.53031 · doi:10.2307/2041749
[50] W. P. Thurston, ”Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bull. Amer. Math. Soc., vol. 6, iss. 3, pp. 357-381, 1982. · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0
[51] W. P. Thurston, ”A norm for the homology of \(3\)-manifolds,” Mem. Amer. Math. Soc., vol. 59, iss. 339, p. i-vi and 99, 1986. · Zbl 0585.57006
[52] V. Turaev, Introduction to Combinatorial Torsions, Basel: Birkhäuser, 2001. · Zbl 0970.57001
[53] S. Vidussi, The Alexander norm is smaller than the Thurston norm; a Seiberg-Witten proof, 1999.
[54] S. Vidussi, ”Norms on the cohomology of a 3-manifold and SW theory,” Pacific J. Math., vol. 208, iss. 1, pp. 169-186, 2003. · Zbl 1051.57014 · doi:10.2140/pjm.2003.208.169 · pjm.math.berkeley.edu · arxiv:math/0204211
[55] M. Wada, ”Twisted Alexander polynomial for finitely presentable groups,” Topology, vol. 33, iss. 2, pp. 241-256, 1994. · Zbl 0822.57006 · doi:10.1016/0040-9383(94)90013-2
[56] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, New York: Springer-Verlag, 1973, vol. 76. · Zbl 0261.20038
[57] J. S. Wilson, Profinite Groups, New York: The Clarendon Press Oxford University Press, 1998, vol. 19. · Zbl 0909.20001
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