Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

Hedden, Matthew; Kim, Min Hoon; Mark, Thomas; Park, Kyungbae

doi:10.1090/tran/7786

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

Authors: Matthew Hedden, Min Hoon Kim, Thomas E. Mark and Kyungbae Park
Journal: Trans. Amer. Math. Soc. 372 (2019), 7619-7638
MSC (2010): Primary 57M25, 57M27, 57R58, 57R65
DOI: https://doi.org/10.1090/tran/7786
Published electronically: May 20, 2019
MathSciNet review: 4029676
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give two infinite families of examples of closed, orientable, irreducible 3-manifolds such that and $\pi _1(M)$ has weight 1, but is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl, and Wilton and provides the first examples of irreducible manifolds with that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.

[AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton, 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015. MR 3444187
[Ale24] J. W. Alexander, On the subdivision of -space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 6-8.
[Auc97] David Auckly, Surgery numbers of 3-manifolds: a hyperbolic example, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 21–34. MR 1470719
[Bal08] John A. Baldwin, Heegaard Floer homology and genus one, one-boundary component open books, J. Topol. 1 (2008), no. 4, 963–992. MR 2461862, https://doi.org/10.1112/jtopol/jtn029
[BL90] Steven Boyer and Daniel Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990), 181–220. MR 1041002
[BN13] Maciej Borodzik and András Némethi, Heegaard-Floer homologies of (+1) surgeries on torus knots, Acta Math. Hungar. 139 (2013), no. 4, 303–319. MR 3061478, https://doi.org/10.1007/s10474-012-0280-x
[Elk95] Noam D. Elkies, A characterization of the 𝑍ⁿ lattice, Math. Res. Lett. 2 (1995), no. 3, 321–326. MR 1338791, https://doi.org/10.4310/MRL.1995.v2.n3.a9
[Gab87] David Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479–536. MR 910018
[GAnS86] Francisco González-Acuña and Hamish Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102. MR 809502, https://doi.org/10.1017/S0305004100063969
[Ger87] S. M. Gersten, Selected problems, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 545–551. MR 895631
[GL89] C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, https://doi.org/10.1090/S0894-0347-1989-0965210-7
[GS99] Robert E. Gompf and András I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327
[Hed07] Matthew Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338. MR 2372849, https://doi.org/10.2140/gt.2007.11.2277
[HHL18] K. Hendricks, J. Hom, and T. Lidman, Applications of involutive Heegaard Floer homology, arXiv:1802.02008, 2018.
[HKL16a] Matthew Hedden, Se-Goo Kim, and Charles Livingston, Topologically slice knots of smooth concordance order two, J. Differential Geom. 102 (2016), no. 3, 353–393. MR 3466802
[HKL16b] Jennifer Hom, Çağrı Karakurt, and Tye Lidman, Surgery obstructions and Heegaard Floer homology, Geom. Topol. 20 (2016), no. 4, 2219–2251. MR 3548466, https://doi.org/10.2140/gt.2016.20.2219
[HL18] Jennifer Hom and Tye Lidman, A note on surgery obstructions and hyperbolic integer homology spheres, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1363–1365. MR 3750247, https://doi.org/10.1090/proc/13925
[Hom17] Jennifer Hom, A survey on Heegaard Floer homology and concordance, J. Knot Theory Ramifications 26 (2017), no. 2, 1740015, 24. MR 3604497, https://doi.org/10.1142/S0218216517400156
[HW15] Neil R. Hoffman and Genevieve S. Walsh, The big Dehn surgery graph and the link of 𝑆³, Proc. Amer. Math. Soc. Ser. B 2 (2015), 17–34. MR 3422666, https://doi.org/10.1090/S2330-1511-2015-00020-7
[HW18] Matthew Hedden and Liam Watson, On the geography and botany of knot Floer homology, Selecta Math. (N.S.) 24 (2018), no. 2, 997–1037. MR 3782416, https://doi.org/10.1007/s00029-017-0351-5
[Jac80] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
[Lic62] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, https://doi.org/10.2307/1970373
[LR14] Adam Simon Levine and Daniel Ruberman, Generalized Heegaard Floer correction terms, Proceedings of the Gökova Geometry-Topology Conference 2013, Gökova Geometry/Topology Conference (GGT), Gökova, 2014, pp. 76–96. MR 3287799
[LS07] Paolo Lisca and András I. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds. II, J. Differential Geom. 75 (2007), no. 1, 109–141. MR 2282726
[MK14] V. D. Mazurov and E. I. Khukhro (eds.), The Kourovka notebook, Eighteenth edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2014. Unsolved problems in group theory. MR 3408705
[Mye83] Robert Myers, Homology cobordisms, link concordances, and hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 1, 271–288. MR 697074, https://doi.org/10.1090/S0002-9947-1983-0697074-4
[Neu80] Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125–144. MR 585657
[NR78] Walter D. Neumann and Frank Raymond, Seifert manifolds, plumbing, 𝜇-invariant and orientation reversing maps, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 163–196. MR 518415
[NW15] Yi Ni and Zhongtao Wu, Cosmetic surgeries on knots in 𝑆³, J. Reine Angew. Math. 706 (2015), 1–17. MR 3393360, https://doi.org/10.1515/crelle-2013-0067
[OS03] Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, https://doi.org/10.1016/S0001-8708(02)00030-0
[OS04a] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, https://doi.org/10.1016/j.aim.2003.05.001
[OS04b] Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, https://doi.org/10.4007/annals.2004.159.1159
[OS05] Peter Ozsváth and Zoltán Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005), no. 1, 39–61. MR 2153455, https://doi.org/10.1215/S0012-7094-04-12912-4
[OS08] Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101–153. MR 2377279, https://doi.org/10.2140/agt.2008.8.101
[Ras03] Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683
[Sav02] Nikolai Saveliev, Invariants for homology 3-spheres, Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002. Low-Dimensional Topology, I. MR 1941324
[Sei33] H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, https://doi.org/10.1007/BF02398271
[Sto15] M. Stoffregen, $\operatorname {Pin}(2)$ -equivariant Seiberg-Witten Floer homology of Seifert fibrations, arXiv:1505.03234, https://arxiv.org/pdf/1505.03234.pdf, 2015.
[Sto15] Matthew Henry Stoffregen, Applications of Pin(2)-equivariant Seiberg-Witten Floer Homology, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–University of California, Los Angeles. MR 3697767
[Tau87] Clifford Henry Taubes, Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geom. 25 (1987), no. 3, 363–430. MR 882829
[Wal60] Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, https://doi.org/10.4153/CJM-1960-045-7