Abstract
A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form $S^1 \times N^3$ admits a symplectic structure, then $N$ fibers over $S^1$. In fact we will completely determine the symplectic cone of $S^1\times N$ in terms of the fibered faces of the Thurston norm ball of $N$.
-
[Ag08]
I. Agol, "Criteria for virtual fibering," J. Topol., vol. 1, iss. 2, pp. 269-284, 2008.@article {Ag08, MRKEY = {2399130},
AUTHOR = {Agol, Ian},
TITLE = {Criteria for virtual fibering},
JOURNAL = {J. Topol.},
FJOURNAL = {Journal of Topology},
VOLUME = {1},
YEAR = {2008},
NUMBER = {2},
PAGES = {269--284},
ISSN = {1753-8416},
MRCLASS = {57M50 (57N10)},
MRNUMBER = {2399130},
MRREVIEWER = {Darren D. Long},
DOI = {10.1112/jtopol/jtn003},
ZBLNUMBER = {1148.57023},
} -
[AHKS07] 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.
@article {AHKS07, MRKEY = {2318535},
AUTHOR = {Arzhantseva, Goulnara and de la Harpe, Pierre and Kahrobaei, Delaram and {Š}uni{ć},
Zoran},
TITLE = {The true prosoluble completion of a group: examples and open problems},
JOURNAL = {Geom. Dedicata},
FJOURNAL = {Geometriae Dedicata},
VOLUME = {124},
YEAR = {2007},
PAGES = {5--26},
ISSN = {0046-5755},
CODEN = {GEMDAT},
MRCLASS = {20E18 (20F19 20F38)},
MRNUMBER = {2318535},
MRREVIEWER = {Javier Otal},
DOI = {10.1007/s10711-006-9103-y},
ZBLNUMBER = {1138.20027},
} -
[AF10] M. Aschenbrenner and S. Friedl, 3-manifold groups are virtually residually $p$, 2010.
@misc{AF10,
author={Aschenbrenner, M. and Friedl, S.},
TITLE={3-manifold groups are virtually residually $p$},
NOTE={preprint},
ARXIV={1004.3619},
YEAR={2010},
} -
[BG04] 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.
@article {BG04, MRKEY = {2119723},
AUTHOR = {Bridson, Martin R. and Grunewald, Fritz J.},
TITLE = {Grothendieck's problems concerning profinite completions and representations of groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {160},
YEAR = {2004},
NUMBER = {1},
PAGES = {359--373},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {20E18 (20E26)},
MRNUMBER = {2119723},
MRREVIEWER = {Jan-Christoph Schlage-Puchta},
DOI = {10.4007/annals.2004.160.359},
ZBLNUMBER = {1083.20023},
} -
[Br94] K. S. Brown, Cohomology of Groups, New York: Springer-Verlag, 1994, vol. 87.
@book {Br94, MRKEY = {1324339},
AUTHOR = {Brown, Kenneth S.},
TITLE = {Cohomology of Groups},
SERIES = {Grad. Texts in Math.},
VOLUME = {87},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {x+306},
ISBN = {0-387-90688-6},
MRCLASS = {20J05 (20-02)},
MRNUMBER = {1324339},
ZBLNUMBER = {0823.20049},
} -
[CC03] A. Candel and L. Conlon, Foliations. II, Providence, RI: Amer. Math. Soc., 2003, vol. 60.
@book {CC03, MRKEY = {1994394},
AUTHOR = {Candel, Alberto and Conlon, Lawrence},
TITLE = {Foliations. {{\rm II}}},
SERIES = {Grad. Stud. Math.},
VOLUME = {60},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2003},
PAGES = {xiv+545},
ISBN = {0-8218-0881-8},
MRCLASS = {57R30 (37C85 58J35)},
MRNUMBER = {1994394},
MRREVIEWER = {P. G. Walczak},
ZBLNUMBER = {1035.57001},
} -
[Ch03] J. C. Cha, "Fibred knots and twisted Alexander invariants," Trans. Amer. Math. Soc., vol. 355, iss. 10, pp. 4187-4200, 2003.
@article {Ch03, MRKEY = {1990582},
AUTHOR = {Cha, Jae Choon},
TITLE = {Fibred knots and twisted {A}lexander invariants},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {355},
YEAR = {2003},
NUMBER = {10},
PAGES = {4187--4200},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {57M25},
MRNUMBER = {1990582},
MRREVIEWER = {Peiyi Zhao},
DOI = {10.1090/S0002-9947-03-03348-8},
ZBLNUMBER = {1028.57004},
} -
[CM00] W. Chen and R. Matveyev, "Symplectic Lefschetz fibrations on $S^1\times M^3$," Geom. Topol., vol. 4, pp. 517-535, 2000.
@article {CM00, MRKEY = {1800295},
AUTHOR = {Chen, Weimin and Matveyev, Rostislav},
TITLE = {Symplectic {L}efschetz fibrations on {$S\sp 1\times M\sp 3$}},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry and Topology},
VOLUME = {4},
YEAR = {2000},
PAGES = {517--535},
ISSN = {1465-3060},
MRCLASS = {57R17 (57M50 57R57)},
MRNUMBER = {1800295},
MRREVIEWER = {Andrei Yu. Vesnin},
DOI = {10.2140/gt.2000.4.517},
ZBLNUMBER = {0968.57012},
} -
[Do96] S. K. Donaldson, "Symplectic submanifolds and almost-complex geometry," J. Differential Geom., vol. 44, iss. 4, pp. 666-705, 1996.
@article {Do96, MRKEY = {1438190},
AUTHOR = {Donaldson, S. K.},
TITLE = {Symplectic submanifolds and almost-complex geometry},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {44},
YEAR = {1996},
NUMBER = {4},
PAGES = {666--705},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C15 (53C40 53C65 57R15 58F05)},
MRNUMBER = {1438190},
MRREVIEWER = {Daniel Pollack},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214459407},
ZBLNUMBER = {0883.53032},
} -
[EN85] D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Princeton, NJ: Princeton Univ. Press, 1985, vol. 110.
@book {EN85, MRKEY = {0817982},
AUTHOR = {Eisenbud, David and Neumann, Walter},
TITLE = {Three-Dimensional Link Theory and Invariants of Plane Curve Singularities},
SERIES = {Annals of Math. Studies},
VOLUME = {110},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1985},
PAGES = {vii+173},
ISBN = {0-691-08380-0; 0-691-08381-9},
MRCLASS = {57M25 (14H20 57S15)},
MRNUMBER = {0817982},
MRREVIEWER = {Christopher W. Stark},
ZBLNUMBER = {0628.57002},
} -
[Et01] T. Etgü, "Lefschetz fibrations, complex structures and Seifert fibrations on $S^1\times M^3$," Algebr. Geom. Topol., vol. 1, pp. 469-489, 2001.
@article {Et01, MRKEY = {1852768},
AUTHOR = {Etg{ü},
Tolga},
TITLE = {Lefschetz fibrations, complex structures and {S}eifert fibrations on {$S\sp 1\times M\sp 3$}},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {1},
YEAR = {2001},
PAGES = {469--489},
ISSN = {1472-2747},
MRCLASS = {57R57 (57N13 57R17)},
MRNUMBER = {1852768},
MRREVIEWER = {Ivan Smith},
DOI = {10.2140/agt.2001.1.469},
} -
[FK06] S. Friedl and T. Kim, "The Thurston norm, fibered manifolds and twisted Alexander polynomials," Topology, vol. 45, iss. 6, pp. 929-953, 2006.
@article {FK06, MRKEY = {2263219},
AUTHOR = {Friedl, Stefan and Kim, Taehee},
TITLE = {The {T}hurston norm, fibered manifolds and twisted {A}lexander polynomials},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {45},
YEAR = {2006},
NUMBER = {6},
PAGES = {929--953},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {57M27 (57N10)},
MRNUMBER = {2263219},
MRREVIEWER = {Stephan Tillmann},
DOI = {10.1016/j.top.2006.06.003},
ZBLNUMBER = {1105.57009},
} -
[FV11] S. Friedl and S. Vidussi, Construction of symplectic structures on 4-manifolds with a free circle action.
@misc{FV11,
author = {Friedl, Stefan and Vidussi, Stefano},
KEY={FV11},
TITLE={Construction of symplectic structures on 4-manifolds with a free circle action},
NOTE={to appear in {\it {P}roc. {R}. {S}oc. {E}dinb., {S}ect. {{\rm A}},
{M}ath.}},
SORTYEAR={2011},
} -
[FV08a] S. Friedl and S. Vidussi, "Twisted Alexander polynomials and symplectic structures," Amer. J. Math., vol. 130, iss. 2, pp. 455-484, 2008.
@article {FV08a, MRKEY = {2405164},
AUTHOR = {Friedl, Stefan and Vidussi, Stefano},
TITLE = {Twisted {A}lexander polynomials and symplectic structures},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {130},
YEAR = {2008},
NUMBER = {2},
PAGES = {455--484},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {57R17 (57M25)},
MRNUMBER = {2405164},
MRREVIEWER = {Vladimir Krouglov},
DOI = {10.1353/ajm.2008.0014},
ZBLNUMBER = {1154.57021},
} -
[FV08b] 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.
@article {FV08b, MRKEY = {2373361},
AUTHOR = {Friedl, Stefan and Vidussi, Stefano},
TITLE = {Symplectic {$S\sp 1\times N\sp 3$},
subgroup separability, and vanishing {T}hurston norm},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {21},
YEAR = {2008},
NUMBER = {2},
PAGES = {597--610},
ISSN = {0894-0347},
MRCLASS = {57R17 (57M27)},
MRNUMBER = {2373361},
MRREVIEWER = {Tolga Etg{ü}},
DOI = {10.1090/S0894-0347-07-00577-2},
ZBLNUMBER = {1142.57014},
} -
[FV10] S. Friedl and S. Vidussi, Twisted Alexander polynomials and fibered 3-manifolds.
@misc{FV10, SORTYEAR={2009},
KEY={FV09},
AUTHOR= {Friedl, Stefan and Vidussi, Stefano},
TITLE = {Twisted {A}lexander polynomials and fibered 3-manifolds},
NOTE={to be published by the {P}roceedings of the {G}eorgia {I}nternational {T}opology {C}onference ({M}ay 18--29, 2009, {A}thens, {GA}), to appear},
} -
[Ga83] D. Gabai, "Foliations and the topology of $3$-manifolds," J. Differential Geom., vol. 18, iss. 3, pp. 445-503, 1983.
@article {Ga83, MRKEY = {0723813},
AUTHOR = {Gabai, David},
TITLE = {Foliations and the topology of {$3$}-manifolds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {18},
YEAR = {1983},
NUMBER = {3},
PAGES = {445--503},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {57N10 (57R30)},
MRNUMBER = {0723813},
MRREVIEWER = {Jean-Pierre Otal},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214437784},
ZBLNUMBER = {0533.57013},
} -
[Gh08] P. Ghiggini, "Knot Floer homology detects genus-one fibred knots," Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008.
@article {Gh08, MRKEY = {2450204},
AUTHOR = {Ghiggini, Paolo},
TITLE = {Knot {F}loer homology detects genus-one fibred knots},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {130},
YEAR = {2008},
NUMBER = {5},
PAGES = {1151--1169},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {57M25 (57R58)},
MRNUMBER = {2450204},
DOI = {10.1353/ajm.0.0016},
ZBLNUMBER = {1149.57019},
} -
[GKM05] 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.
@article {GKM05, MRKEY = {2130565},
AUTHOR = {Goda, Hiroshi and Kitano, Teruaki and Morifuji, Takayuki},
TITLE = {Reidemeister torsion, twisted {A}lexander polynomial and fibered knots},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
VOLUME = {80},
YEAR = {2005},
NUMBER = {1},
PAGES = {51--61},
ISSN = {0010-2571},
MRCLASS = {57M25 (57M05 57Q10)},
MRNUMBER = {2130565},
MRREVIEWER = {Charles Livingston},
DOI = {10.4171/CMH/3},
ZBLNUMBER = {1066.57008},
} -
[Gr70] A. Grothendieck, "Représentations linéaires et compactification profinie des groupes discrets," Manuscripta Math., vol. 2, pp. 375-396, 1970.
@article {Gr70, MRKEY = {0262386},
AUTHOR = {Grothendieck, Alexander},
TITLE = {Représentations linéaires et compactification profinie des groupes discrets},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {2},
YEAR = {1970},
PAGES = {375--396},
ISSN = {0025-2611},
MRCLASS = {20.80 (14.00)},
MRNUMBER = {0262386},
MRREVIEWER = {F. Oort},
DOI = {10.1007/BF01719593},
} -
[Ha59] P. Hall, "On the finiteness of certain soluble groups," Proc. London Math. Soc., vol. 9, pp. 595-622, 1959.
@article {Ha59, MRKEY = {0110750},
AUTHOR = {Hall, P.},
TITLE = {On the finiteness of certain soluble groups},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {9},
YEAR = {1959},
PAGES = {595--622},
ISSN = {0024-6115},
MRCLASS = {20.00},
MRNUMBER = {0110750},
MRREVIEWER = {K. Gruenberg},
DOI = {10.1112/plms/s3-9.4.595},
ZBLNUMBER = {0091.02501},
} -
[Ha01] 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.
@article {Ha01, MRKEY = {1851085},
AUTHOR = {Hamilton, Emily},
TITLE = {Abelian subgroup separability of {H}aken 3-manifolds and closed hyperbolic {$n$}-orbifolds},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {83},
YEAR = {2001},
NUMBER = {3},
PAGES = {626--646},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {57M50 (20E26 57M05 57N10)},
MRNUMBER = {1851085},
MRREVIEWER = {Darryl McCullough},
DOI = {10.1112/plms/83.3.626},
ZBLNUMBER = {1025.57007},
} -
[He76] J. Hempel, $3$-Manifolds, Princeton, NJ: Princeton Univ. Press, 1976, vol. 86.
@book {He76, MRKEY = {0415619},
AUTHOR = {Hempel, John},
TITLE = {{$3$}-{M}anifolds},
SERIES = {Ann. of Math. Stud.},
VOLUME={86},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1976},
PAGES = {xii+195},
MRCLASS = {57A10},
MRNUMBER = {0415619},
MRREVIEWER = {J. S. Birman},
ZBLNUMBER = {0345.57001},
} -
[He87] J. Hempel, "Residual finiteness for $3$-manifolds," in Combinatorial Group Theory and Topology, Princeton, NJ: Princeton Univ. Press, 1987, vol. 111, pp. 379-396.
@incollection {He87, MRKEY = {0895623},
AUTHOR = {Hempel, John},
TITLE = {Residual finiteness for {$3$}-manifolds},
BOOKTITLE = {Combinatorial Group Theory and Topology},
VENUE={{A}lta, {U}tah, 1984},
SERIES = {Ann. of Math. Stud.},
VOLUME = {111},
PAGES = {379--396},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1987},
MRCLASS = {57M05 (20E26 20F34 57N10)},
MRNUMBER = {0895623},
ZBLNUMBER = {0772.57002},
} -
[HS97] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Second ed., New York: Springer-Verlag, 1997, vol. 4.
@book {HS97, MRKEY = {1438546},
AUTHOR = {Hilton, P. J. and Stammbach, U.},
TITLE = {A Course in Homological Algebra},
SERIES = {Grad. Texts Math.},
VOLUME = {4},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {xii+364},
ISBN = {0-387-94823-6},
MRCLASS = {18-02 (17B56 18Gxx 20J05)},
MRNUMBER = {1438546},
ZBLNUMBER = {0863.18001},
} -
[Ki] R. Kirby, "Problems in low-dimensional topology," in Geometric Topology, Providence, RI, 1997, pp. 35-473.
@inproceedings {Ki,
author={Kirby, R.},
MRKEY = {1470751},
TITLE = {Problems in low-dimensional topology},
BOOKTITLE = {Geometric Topology},
VENUE={1993 {G}eorgia {I}nternat. {T}opology {C}onf., {A}ug. 2--13, 1993, {A}thens, {GA}},
SERIES = {AMS/IP Stud. Adv. Math.},
VOLUME = {2},
PAGES = {35--473},
NOTE = {edited by Rob Kirby},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1997},
MRCLASS = {57-02},
MRNUMBER = {1470751},
ZBLNUMBER = {0888.57014},
} -
[KL99] P. Kirk and C. Livingston, "Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants," Topology, vol. 38, iss. 3, pp. 635-661, 1999.
@article {KL99, MRKEY = {1670420},
AUTHOR = {Kirk, Paul and Livingston, Charles},
TITLE = {Twisted {A}lexander invariants, {R}eidemeister torsion, and {C}asson-{G}ordon invariants},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {38},
YEAR = {1999},
NUMBER = {3},
PAGES = {635--661},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {57M25 (57Q45)},
MRNUMBER = {1670420},
MRREVIEWER = {C. Kearton},
DOI = {10.1016/S0040-9383(98)00039-1},
ZBLNUMBER = {0928.57005},
} -
[Ki07] T. Kitayama, Normalization of twisted Alexander invariants, 2007.
@misc{Ki07,
author={Kitayama, T.},
TITLE={Normalization of twisted {A}lexander invariants},
NOTE={preprint},
YEAR={2007},
ARXIV={0705.2371},
} -
[Kr98] 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.
@incollection {Kr98, MRKEY = {1677890},
AUTHOR = {Kronheimer, P. B.},
TITLE = {Embedded surfaces and gauge theory in three and four dimensions},
BOOKTITLE = {Surveys in Differential Geometry, {V}ol. {{\rm III}}},
VENUE={{C}ambridge, {MA},
1996},
PAGES = {243--298},
PUBLISHER = {Internat. Press, Boston},
YEAR = {1998},
MRCLASS = {57R57 (53C27 57R58 57R95)},
MRNUMBER = {1677890},
MRREVIEWER = {Paolo Lisca},
ZBLNUMBER = {0965.57030},
} -
[Kr99] P. B. Kronheimer, "Minimal genus in $S^1\times M^3$," Invent. Math., vol. 135, iss. 1, pp. 45-61, 1999.
@article {Kr99, MRKEY = {1664695},
AUTHOR = {Kronheimer, P. B.},
TITLE = {Minimal genus in {$S\sp 1\times M\sp 3$}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {135},
YEAR = {1999},
NUMBER = {1},
PAGES = {45--61},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {57R95 (57M25 57N10 57R17 57R30 57R57)},
MRNUMBER = {1664695},
MRREVIEWER = {Liviu I. Nicolaescu},
DOI = {10.1007/s002220050279},
ZBLNUMBER = {0917.57017},
} -
[KM08] P. B. Kronheimer and T. Mrowka, "Knots, sutures and excision," J. Differential Geom., vol. 84, pp. 301-364, 2010.
@article{KM08,
author = {Kronheimer, P. B. and Mrowka, T.},
TITLE={Knots, sutures and excision},
JOURNAL={J. Differential Geom.},
VOLUME={84},
YEAR={2010},
PAGES={301--364},
MRNUMBER = {2652464},
ZBLNUMBER = {1208.57008},
} -
[KT09] &. 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.
@article {KT09, MRKEY = {2469523},
AUTHOR = {Kutluhan, {Ç}a{\u{g}}atay and Taubes, Clifford Henry},
TITLE = {Seiberg-{W}itten {F}loer homology and symplectic forms on {$S\sp 1\times M\sp 3$}},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {13},
YEAR = {2009},
NUMBER = {1},
PAGES = {493--525},
ISSN = {1465-3060},
MRCLASS = {57R58 (53D40 57R57)},
MRNUMBER = {2469523},
MRREVIEWER = {Stanislav Jabuka},
DOI = {10.2140/gt.2009.13.493},
ZBLNUMBER = {1182.57020},
} -
[L01] X. S. Lin, "Representations of knot groups and twisted Alexander polynomials," Acta Math. Sin. $($Engl. Ser.$)$, vol. 17, iss. 3, pp. 361-380, 2001.
@article {L01, MRKEY = {1852950},
AUTHOR = {Lin, Xiao Song},
TITLE = {Representations of knot groups and twisted {A}lexander polynomials},
JOURNAL = {Acta Math. Sin. $($Engl. Ser.$)$},
FJOURNAL = {Acta Mathematica Sinica. English Series},
VOLUME = {17},
YEAR = {2001},
NUMBER = {3},
PAGES = {361--380},
ISSN = {1000-9574},
CODEN = {AMSSFV},
MRCLASS = {57M25 (57M27)},
MRNUMBER = {1852950},
MRREVIEWER = {Michael Heusener},
DOI = {10.1007/PL00011612},
ZBLNUMBER = {0986.57003},
} -
[LN91] D. D. Long and G. A. Niblo, "Subgroup separability and $3$-manifold groups," Math. Z., vol. 207, iss. 2, pp. 209-215, 1991.
@article {LN91, MRKEY = {1109662},
AUTHOR = {Long, D. D. and Niblo, G. A.},
TITLE = {Subgroup separability and {$3$}-manifold groups},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {207},
YEAR = {1991},
NUMBER = {2},
PAGES = {209--215},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {20E26 (57M05)},
MRNUMBER = {1109662},
MRREVIEWER = {G. Peter Scott},
DOI = {10.1007/BF02571384},
ZBLNUMBER = {0711.57002},
} -
[LS03] A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212.
@book {LS03, MRKEY = {1978431},
AUTHOR = {Lubotzky, Alexander and Segal, Dan},
TITLE = {Subgroup Growth},
SERIES = {Progr. Math.},
VOLUME = {212},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2003},
PAGES = {xxii+453},
ISBN = {3-7643-6989-2},
MRCLASS = {20E07 (20E18 20E26 20F69)},
MRNUMBER = {1978431},
MRREVIEWER = {Avinoam Mann},
ZBLNUMBER = {1071.20033},
} -
[McC01] 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.
@article {McC01, MRKEY = {1707526},
AUTHOR = {McCarthy, John D.},
TITLE = {On the asphericity of a symplectic {$M\sp 3\times S\sp 1$}},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {129},
YEAR = {2001},
NUMBER = {1},
PAGES = {257--264},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {57N10 (57M50 57N13 57R17)},
MRNUMBER = {1707526},
MRREVIEWER = {Martin Scharlemann},
DOI = {10.1090/S0002-9939-00-05571-4},
ZBLNUMBER = {0972.53045},
} -
[McM02] 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.
@article {McM02, MRKEY = {1914929},
AUTHOR = {McMullen, Curtis T.},
TITLE = {The {A}lexander polynomial of a 3-manifold and the {T}hurston norm on cohomology},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {35},
YEAR = {2002},
NUMBER = {2},
PAGES = {153--171},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {57N10 (57M25)},
MRNUMBER = {1914929},
MRREVIEWER = {Athanase Papadopoulos},
DOI = {10.1016/S0012-9593(02)01086-8},
ZBLNUMBER = {1009.57021},
} -
[MT07] J. Morgan and G. Tian, Ricci Flow and the Poincaré Conjecture, Providence, RI: Amer. Math. Soc., 2007, vol. 3.
@book {MT07, MRKEY = {2334563},
AUTHOR = {Morgan, John and Tian, Gang},
TITLE = {Ricci Flow and the {P}oincaré Conjecture},
SERIES = {Clay Math. Monographs},
VOLUME = {3},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2007},
PAGES = {xlii+521},
ISBN = {978-0-8218-4328-4},
MRCLASS = {57M40 (53C44)},
MRNUMBER = {2334563},
MRREVIEWER = {Terence C. Tao},
ZBLNUMBER = {1179.57045},
} -
[Ni08] Y. Ni, Addendum to: “Knots, sutures and excision", 2008.
@misc{Ni08,
author={Ni, Yi},
TITLE={Addendum to: ``{K}nots, sutures and excision"},
NOTE={preprint},
ARXIV={0808.1327},
YEAR={2008},
} -
[Ni09] Y. Ni, "Heegaard Floer homology and fibred 3-manifolds," Amer. J. Math., vol. 131, iss. 4, pp. 1047-1063, 2009.
@article {Ni09, MRKEY = {2543922},
AUTHOR = {Ni, Yi},
TITLE = {Heegaard {F}loer homology and fibred 3-manifolds},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {131},
YEAR = {2009},
NUMBER = {4},
PAGES = {1047--1063},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {57M27 (57N10)},
MRNUMBER = {2543922},
MRREVIEWER = {Martin Scharlemann},
DOI = {10.1353/ajm.0.0064},
ZBLNUMBER = {1184.57026},
} -
[RZ00] L. Ribes and P. Zalesskii, Profinite Groups, New York: Springer-Verlag, 2000, vol. 40.
@book {RZ00, MRKEY = {1775104},
AUTHOR = {Ribes, Luis and Zalesskii, Pavel},
TITLE = {Profinite Groups},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {40},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2000},
PAGES = {xiv+435},
ISBN = {3-540-66986-8},
MRCLASS = {20E18},
MRNUMBER = {1775104},
MRREVIEWER = {Marcus du Sautoy},
ZBLNUMBER = {0949.20017},
} -
[Ro74] 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.
@article {Ro74, MRKEY = {0358809},
AUTHOR = {Roussarie, R.},
TITLE = {Plongements dans les variétés feuilletées et classification de feuilletages sans holonomie},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {43},
YEAR = {1974},
PAGES = {101--141},
ISSN = {0073-8301},
MRCLASS = {57D30 (58F99)},
MRNUMBER = {0358809},
MRREVIEWER = {G. Chatelet},
URL = {http://www.numdam.org/item?id=PMIHES_1974__43__101_0},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0356.57017},
} -
[Sc85] 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.
@incollection {Sc85, MRKEY = {0823287},
AUTHOR = {Scharlemann, Martin},
TITLE = {{$3$}-manifolds with {$H\sb 2(A,\partial A)=0$} and a conjecture of {S}tallings},
BOOKTITLE = {Knot Theory and Manifolds},
VENUE={{V}ancouver, {B}.{C}., 1983},
SERIES = {Lecture Notes in Math.},
VOLUME = {1144},
PAGES = {138--145},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1985},
MRCLASS = {57N10 (20F10)},
MRNUMBER = {0823287},
MRREVIEWER = {John G. Ratcliffe},
DOI = {10.1007/BFb0075017},
ZBLNUMBER = {0585.57011},
} -
[Sc89] M. Scharlemann, "Sutured manifolds and generalized Thurston norms," J. Differential Geom., vol. 29, iss. 3, pp. 557-614, 1989.
@article {Sc89, MRKEY = {0992331},
AUTHOR = {Scharlemann, Martin},
TITLE = {Sutured manifolds and generalized {T}hurston norms},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {29},
YEAR = {1989},
NUMBER = {3},
PAGES = {557--614},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {57M99},
MRNUMBER = {0992331},
MRREVIEWER = {N. V. Ivanov},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214443063},
ZBLNUMBER = {0673.57015},
} -
[St62] 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.
@incollection {St62, MRKEY = {0158375},
AUTHOR = {Stallings, John},
TITLE = {On fibering certain {$3$}-manifolds},
BOOKTITLE = {Topology of 3-manifolds and Related Topics},
VENUE={{P}roc. {T}he {U}niv. of {G}eorgia {I}nstitute, 1961},
PAGES = {95--100},
PUBLISHER = {Prentice-Hall},
ADDRESS = {Englewood Cliffs, N.J.},
YEAR = {1962},
MRCLASS = {54.78},
MRNUMBER = {0158375},
MRREVIEWER = {O. G. Harrold},
ZBLNUMBER = {0132.20102},
} -
[St83] J. R. Stallings, "Surfaces in three-manifolds and nonsingular equations in groups," Math. Z., vol. 184, iss. 1, pp. 1-17, 1983.
@article {St83, MRKEY = {0711725},
AUTHOR = {Stallings, John R.},
TITLE = {Surfaces in three-manifolds and nonsingular equations in groups},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {184},
YEAR = {1983},
NUMBER = {1},
PAGES = {1--17},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {20E05 (57M99)},
MRNUMBER = {0711725},
MRREVIEWER = {Johannes Huebschmann},
DOI = {10.1007/BF01162003},
} -
[Ta94] C. H. Taubes, "The Seiberg-Witten invariants and symplectic forms," Math. Res. Lett., vol. 1, iss. 6, pp. 809-822, 1994.
@article {Ta94, MRKEY = {1306023},
AUTHOR = {Taubes, Clifford Henry},
TITLE = {The {S}eiberg-{W}itten invariants and symplectic forms},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {1},
YEAR = {1994},
NUMBER = {6},
PAGES = {809--822},
ISSN = {1073-2780},
MRCLASS = {57R57 (57N13 57R15)},
MRNUMBER = {1306023},
MRREVIEWER = {Jim A. Bryan},
ZBLNUMBER = {0853.57019},
} -
[Ta95] C. H. Taubes, "More constraints on symplectic forms from Seiberg-Witten invariants," Math. Res. Lett., vol. 2, iss. 1, pp. 9-13, 1995.
@article {Ta95, MRKEY = {1312973},
AUTHOR = {Taubes, Clifford Henry},
TITLE = {More constraints on symplectic forms from {S}eiberg-{W}itten invariants},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {2},
YEAR = {1995},
NUMBER = {1},
PAGES = {9--13},
ISSN = {1073-2780},
MRCLASS = {57R57 (57N13 57R15 58D29)},
MRNUMBER = {1312973},
MRREVIEWER = {Jim A. Bryan},
ZBLNUMBER = {0854.57019},
} -
[Th76] W. P. Thurston, "Some simple examples of symplectic manifolds," Proc. Amer. Math. Soc., vol. 55, iss. 2, pp. 467-468, 1976.
@article {Th76, MRKEY = {0402764},
AUTHOR = {Thurston, William P.},
TITLE = {Some simple examples of symplectic manifolds},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {55},
YEAR = {1976},
NUMBER = {2},
PAGES = {467--468},
ISSN = {0002-9939},
MRCLASS = {57D15},
MRNUMBER = {0402764},
MRREVIEWER = {Edgar H. Brown, Jr.},
DOI = {10.2307/2041749},
ZBLNUMBER = {0324.53031},
} -
[Th82] W. P. Thurston, "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry," Bull. Amer. Math. Soc., vol. 6, iss. 3, pp. 357-381, 1982.
@article {Th82, MRKEY = {0648524},
AUTHOR = {Thurston, William P.},
TITLE = {Three-dimensional manifolds, {K}leinian groups and hyperbolic geometry},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {6},
YEAR = {1982},
NUMBER = {3},
PAGES = {357--381},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {57N10 (20H15 30F40 57M35 57M40 57S17)},
MRNUMBER = {0648524},
MRREVIEWER = {Klaus Johannson},
DOI = {10.1090/S0273-0979-1982-15003-0},
ZBLNUMBER = {0496.57005},
} -
[Th86] 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.
@article {Th86, MRKEY = {0823443},
AUTHOR = {Thurston, William P.},
TITLE = {A norm for the homology of {$3$}-manifolds},
JOURNAL = {Mem. Amer. Math. Soc.},
FJOURNAL = {Memoirs of the American Mathematical Society},
VOLUME = {59},
YEAR = {1986},
NUMBER = {339},
PAGES = {i--vi and 99--130},
ISSN = {0065-9266},
CODEN = {MAMCAU},
MRCLASS = {57N10 (57M25 57R20 57R30)},
MRNUMBER = {0823443},
MRREVIEWER = {Martin Scharlemann},
ZBLNUMBER = {0585.57006},
} -
[Tu01] V. Turaev, Introduction to Combinatorial Torsions, Basel: Birkhäuser, 2001.
@book {Tu01, MRKEY = {1809561},
AUTHOR = {Turaev, Vladimir},
TITLE = {Introduction to Combinatorial Torsions},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
PAGES = {viii+123},
ISBN = {3-7643-6403-3},
MRCLASS = {57Q10 (19J10 57M27)},
MRNUMBER = {1809561},
MRREVIEWER = {Joan Porti},
ZBLNUMBER = {0970.57001},
} -
[Vi99] S. Vidussi, The Alexander norm is smaller than the Thurston norm; a Seiberg-Witten proof, 1999.
@misc{Vi99,
author = {Vidussi, Stefano},
TITLE = {The {A}lexander norm is smaller than the {T}hurston norm; a {S}eiberg-{W}itten proof},
NOTE={prepublication {\'{E}}cole {P}olytechnique 6},
YEAR={1999},
} -
[Vi03] S. Vidussi, "Norms on the cohomology of a 3-manifold and SW theory," Pacific J. Math., vol. 208, iss. 1, pp. 169-186, 2003.
@article {Vi03, MRKEY = {1979378},
AUTHOR = {Vidussi, Stefano},
TITLE = {Norms on the cohomology of a 3-manifold and {SW} theory},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {208},
YEAR = {2003},
NUMBER = {1},
PAGES = {169--186},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {57R57 (57N10 57R17 57R95)},
MRNUMBER = {1979378},
MRREVIEWER = {Dave Auckly},
DOI = {10.2140/pjm.2003.208.169},
ZBLNUMBER = {1051.57014},
} -
[Wa94] M. Wada, "Twisted Alexander polynomial for finitely presentable groups," Topology, vol. 33, iss. 2, pp. 241-256, 1994.
@article {Wa94, MRKEY = {1273784},
AUTHOR = {Wada, Masaaki},
TITLE = {Twisted {A}lexander polynomial for finitely presentable groups},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {33},
YEAR = {1994},
NUMBER = {2},
PAGES = {241--256},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {57M25},
MRNUMBER = {1273784},
MRREVIEWER = {Charles Livingston},
DOI = {10.1016/0040-9383(94)90013-2},
ZBLNUMBER = {0822.57006},
} -
[We73] 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.
@book {We73, MRKEY = {0335656},
AUTHOR = {Wehrfritz, B. A. F.},
TITLE = {Infinite Linear Groups. {A}n Account of the Group-Theoretic Properties of Infinite Groups of Matrices},
NOTE = {Ergeb. Math. Grenzgeb.},
VOLUME={76},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {xiv+229},
MRCLASS = {20HXX},
MRNUMBER = {0335656},
MRREVIEWER = {I. Stewart},
ZBLNUMBER = {0261.20038},
} -
[Wi98] J. S. Wilson, Profinite Groups, New York: The Clarendon Press Oxford University Press, 1998, vol. 19.
@book {Wi98, MRKEY = {1691054},
AUTHOR = {Wilson, John S.},
TITLE = {Profinite Groups},
SERIES = {London Math. Soc. Monogr. New Series},
VOLUME = {19},
PUBLISHER = {The Clarendon Press Oxford University Press},
ADDRESS = {New York},
YEAR = {1998},
PAGES = {xii+284},
ISBN = {0-19-850082-3},
MRCLASS = {20E18},
MRNUMBER = {1691054},
MRREVIEWER = {Alexander Lubotzky},
ZBLNUMBER = {0909.20001},
}
Full Article