Geometry & Topology

Surgery obstructions and Heegaard Floer homology

Jennifer Hom, Çağrı Karakurt, and Tye Lidman

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Using Taubes’ periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We use Heegaard Floer homology to give an obstruction to a homology sphere being surgery on a knot, and then use this obstruction to construct infinitely many small Seifert fibered examples.

Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 2219-2251.

Dates
Received: 7 January 2015
Revised: 12 August 2015
Accepted: 10 November 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859024

Digital Object Identifier
doi:10.2140/gt.2016.20.2219

Mathematical Reviews number (MathSciNet)
MR3548466

Zentralblatt MATH identifier
1352.57021

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology 57R65: Surgery and handlebodies

Keywords
Dehn surgery $3$–manifold Floer homology

Citation

Hom, Jennifer; Karakurt, Çağrı; Lidman, Tye. Surgery obstructions and Heegaard Floer homology. Geom. Topol. 20 (2016), no. 4, 2219--2251. doi:10.2140/gt.2016.20.2219. https://projecteuclid.org/euclid.gt/1510859024


Export citation

References

  • D Auckly, Surgery numbers of $3$–manifolds: a hyperbolic example, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 21–34
  • S Boyer, D Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990) 181–220
  • M B Can, Ç Karakurt, Calculating Heegaard–Floer homology by counting lattice points in tetrahedra, Acta Math. Hungar. 144 (2014) 43–75
  • M I Doig, Finite knot surgeries and Heegaard Floer homology, Algebr. Geom. Topol. 15 (2015) 667–690
  • D Eisenbud, W Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies 110, Princeton Univ. Press (1985)
  • F Gainullin, Mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$, preprint (2014)
  • C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371–415
  • N R Hoffman, G S Walsh, The big Dehn surgery graph and the link of $S^3$, preprint (2013)
  • J Hom, T Lidman, N Zufelt, Reducible surgeries and Heegaard Floer homology, Math. Res. Lett. 22 (2015) 763–788
  • Ç Karakurt, T Lidman, Rank inequalities for the Heegaard Floer homology of Seifert homology spheres, Trans. Amer. Math. Soc. 367 (2015) 7291–7322
  • R Kirby, editor, Problems in low-dimensional topology, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 35–473
  • W B R Lickorish, A representation of orientable combinatorial $3$–manifolds, Ann. of Math. 76 (1962) 531–540
  • C Manolescu, P Ozsvath, Heegaard Floer homology and integer surgeries on links, preprint (2010)
  • V D Mazurov, E I Khukhro (editors), The Kourovka notebook: unsolved problems in group theory, 18th edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk (2014) In Russian; translated in arXiv
  • A Némethi, On the Ozsváth–Szabó invariant of negative definite plumbed $3$–manifolds, Geom. Topol. 9 (2005) 991–1042
  • W D Neumann, F Raymond, Seifert manifolds, plumbing, $\mu $–invariant and orientation reversing maps, from: “Algebraic and geometric topology”, (K C Millett, editor), Lecture Notes in Math. 664, Springer, Berlin (1978) 163–196
  • Y Ni, Z Wu, Cosmetic surgeries on knots in $S\sp 3$, J. Reine Angew. Math. 706 (2015) 1–17
  • P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
  • P Ozsváth, Z Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185–224
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P S Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1–68
  • J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305332635 {\unhbox0
  • N Saveliev, Invariants for homology $3$–spheres, Encyclopaedia of Mathematical Sciences 140, Springer, Berlin (2002)
  • H Seifert, W Threlfall, A textbook of topology, Pure and Applied Mathematics 89, Academic Press, New York (1980)
  • C H Taubes, Gauge theory on asymptotically periodic $4$–manifolds, J. Differential Geom. 25 (1987) 363–430
  • A H Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503–528