An algorithm for computing some Heegaard Floer homologies

Pages 1213-1236 from Volume 171 (2010), Issue 2 by Sucharit Sarkar, Jiajun Wang

Abstract

In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtration coming from a null-homologous link in a three-manifold.

  • [Ghiggini] Go to document P. Ghiggini, "Knot Floer homology detects genus-one fibred knots," Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008.
    @article {Ghiggini, 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},
      }
  • [Juhasz] Go to document A. Juhász, "Floer homology and surface decompositions," Geom. Topol., vol. 12, iss. 1, pp. 299-350, 2008.
    @article {Juhasz, MRKEY = {2390347},
      AUTHOR = {Juh{á}sz, Andr{á}s},
      TITLE = {Floer homology and surface decompositions},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {12},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {299--350},
      MRCLASS = {57M27 (57R58)},
      MRNUMBER = {2009a:57021},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.2140/gt.2008.12.299},
      ZBLNUMBER = {05256859},
      }
  • [Lipshitz] Go to document R. Lipshitz, "A cylindrical reformulation of Heegaard Floer homology," Geom. Topol., vol. 10, pp. 955-1097, 2006.
    @article {Lipshitz, MRKEY = {2240908},
      AUTHOR = {Lipshitz, Robert},
      TITLE = {A cylindrical reformulation of {H}eegaard {F}loer homology},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {10},
      YEAR = {2006},
      PAGES = {955--1097},
      ISSN = {1465-3060},
      MRCLASS = {57R58 (57M27)},
      MRNUMBER = {2007h:57040},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.2140/gt.2006.10.955},
      ZBLNUMBER = {1130.57035},
      }
  • [LipManWang] Go to document R. Lipshitz, C. Manolescu, and J. Wang, "Combinatorial cobordism maps in hat Heegaard Floer theory," Duke Math. J., vol. 145, iss. 2, pp. 207-247, 2008.
    @article {LipManWang, MRKEY = {2449946},
      AUTHOR = {Lipshitz, Robert and Manolescu, Ciprian and Wang, Jiajun},
      TITLE = {Combinatorial cobordism maps in hat {H}eegaard {F}loer theory},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {145},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {207--247},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {57R58 (57R56)},
      MRNUMBER = {2009f:57050},
      MRREVIEWER = {Burak Ozbagci},
      DOI = {10.1215/00127094-2008-050},
      ZBLNUMBER = {1153.57027},
      }
  • [ManOzsSar] C. Manolescu, P. S. Ozsváth, and S. Sarkar, "A combinatorial description of knot Floer homology," Ann. of Math., vol. 169, pp. 633-660, 2009.
    @article{ManOzsSar,
      author={Manolescu, C. and Ozsváth, P. S. and Sarkar, S.},
      TITLE={A combinatorial description of knot Floer homology},
      JOURNAL={Ann. of Math.},
      VOLUME={169},
      YEAR={2009},
      PAGES={633--660},
      MRNUMBER={2009k:57047},
      ZBLNUMBER={pre05578753},
      }
  • [NiFibred] Go to document Y. Ni, "Knot Floer homology detects fibred knots," Invent. Math., vol. 170, iss. 3, pp. 577-608, 2007.
    @article {NiFibred, MRKEY = {2357503},
      AUTHOR = {Ni, Yi},
      TITLE = {Knot {F}loer homology detects fibred knots},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {170},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {577--608},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57R58 (57M27 57R30)},
      MRNUMBER = {2008j:57053},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.1007/s00222-007-0075-9},
      ZBLNUMBER = {1138.57031},
      }
  • [N522] Y. Ni, Link Floer homology detects the Thurston norm, 2006.
    @misc{N522,
      author={Ni, Yi},
      TITLE={Link Floer homology detects the Thurston norm},
      ARXIV = {math.GT/0604360},
      YEAR={2006},
      }
  • [OSzClosed3M] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and topological invariants for closed three-manifolds," Ann. of Math., vol. 159, iss. 3, pp. 1027-1158, 2004.
    @article {OSzClosed3M, MRKEY = {2113019},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Holomorphic disks and topological invariants for closed three-manifolds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {159},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {1027--1158},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {57M27 (32Q65 57R58)},
      MRNUMBER = {2006b:57016},
      MRREVIEWER = {Thomas E. Mark},
      DOI = {10.4007/annals.2004.159.1027},
      ZBLNUMBER = {1073.57009},
      }
  • [OSzClosed3MApp] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and three-manifold invariants: properties and applications," Ann. of Math., vol. 159, iss. 3, pp. 1159-1245, 2004.
    @article {OSzClosed3MApp, MRKEY = {2113020},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Holomorphic disks and three-manifold invariants: properties and applications},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {159},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {1159--1245},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {57M27 (32Q65 57R58)},
      MRNUMBER = {2006b:57017},
      MRREVIEWER = {Thomas E. Mark},
      DOI = {10.4007/annals.2004.159.1159},
      ZBLNUMBER = {1081.57013},
      }
  • [OSzAbsoluteGrade] Go to document P. Ozsváth and Z. Szabó, "Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary," Adv. Math., vol. 173, iss. 2, pp. 179-261, 2003.
    @article {OSzAbsoluteGrade, MRKEY = {1957829},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Absolutely graded {F}loer homologies and intersection forms for four-manifolds with boundary},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {173},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {179--261},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {57R58 (57M27)},
      MRNUMBER = {2003m:57066},
      MRREVIEWER = {Jacob Andrew Rasmussen},
      DOI = {10.1016/S0001-8708(02)00030-0},
      ZBLNUMBER = {1025.57016},
      }
  • [OSzSmooth4M] Go to document P. Ozsváth and Z. Szabó, "Holomorphic triangles and invariants for smooth four-manifolds," Adv. Math., vol. 202, iss. 2, pp. 326-400, 2006.
    @article {OSzSmooth4M, MRKEY = {2222356},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Holomorphic triangles and invariants for smooth four-manifolds},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {202},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {326--400},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {57R58 (57M27)},
      MRNUMBER = {2007i:57029},
      DOI = {10.1016/j.aim.2005.03.014},
      ZBLNUMBER = {1099.53058},
      }
  • [OSzKnotInv] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and knot invariants," Adv. Math., vol. 186, iss. 1, pp. 58-116, 2004.
    @article {OSzKnotInv, MRKEY = {2065507},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Holomorphic disks and knot invariants},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {186},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {58--116},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {57M27 (57R58)},
      MRNUMBER = {2005e:57044},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.1016/j.aim.2003.05.001},
      ZBLNUMBER = {1062.57019},
      }
  • [OSzGenusBound] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and genus bounds," Geom. Topol., vol. 8, pp. 311-334, 2004.
    @article {OSzGenusBound, MRKEY = {2023281},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Holomorphic disks and genus bounds},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {8},
      YEAR = {2004},
      PAGES = {311--334},
      ISSN = {1465-3060},
      MRCLASS = {57M27 (53D35 57N10 57R58)},
      MRNUMBER = {2004m:57024},
      MRREVIEWER = {Jacob Andrew Rasmussen},
      DOI = {10.2140/gt.2004.8.311},
      ZBLNUMBER = {1056.57020},
      }
  • [OSz4BallGenus] Go to document P. Ozsváth and Z. Szabó, "Knot Floer homology and the four-ball genus," Geom. Topol., vol. 7, pp. 615-639, 2003.
    @article {OSz4BallGenus, MRKEY = {2026543},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Knot {F}loer homology and the four-ball genus},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {7},
      YEAR = {2003},
      PAGES = {615--639},
      ISSN = {1465-3060},
      MRCLASS = {57R58 (57M25 57M27)},
      MRNUMBER = {2004i:57036},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.2140/gt.2003.7.615},
      ZBLNUMBER = {1037.57027},
      }
  • [OSzLinkInv] P. Ozsváth and Z. Szabó, "Holomorphic disks, link invariants and the multi-variable Alexander polynomial," Algebr. Geom. Topol., vol. 8, pp. 615-692, 2008.
    @article{OSzLinkInv,
      author={Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE={Holomorphic disks, link invariants and the multi-variable Alexander polynomial},
      JOURNAL={Algebr. Geom. Topol.},
      VOLUME={8},
      YEAR={2008},
      PAGES={615--692},
      MRNUMBER={2443092},
      ZBLNUMBER={1144.57011},
      }
  • [OSzThurstonNorm] Go to document P. Ozsváth and Z. Szabó, "Link Floer homology and the Thurston norm," J. Amer. Math. Soc., vol. 21, iss. 3, pp. 671-709, 2008.
    @article {OSzThurstonNorm, MRKEY = {2393424},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Link {F}loer homology and the {T}hurston norm},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {21},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {671--709},
      ISSN = {0894-0347},
      MRCLASS = {57R58 (57M27)},
      MRNUMBER = {2009a:57050},
      MRREVIEWER = {Stanislav Jabuka},
      DOI = {10.1090/S0894-0347-08-00586-9},
      }
  • [OSzICMTalk] P. Ozsváth and Z. Szabó, "Heegaard diagrams and Floer homology," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1083-1099.
    @incollection {OSzICMTalk, MRKEY = {2275636},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Heegaard diagrams and {F}loer homology},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm II}},
      PAGES = {1083--1099},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {57R58 (57M27)},
      MRNUMBER = {2008h:57048},
      MRREVIEWER = {Thomas E. Mark},
      }
  • [Differentfaces] P. Ozsváth and Z. Szabó, "Heegaard diagrams and holomorphic disks," in Different Faces of Geometry, New York: Kluwer/Plenum, 2004, pp. 301-348.
    @incollection {Differentfaces, MRKEY = {2102999},
      AUTHOR = {Ozsv{á}th, Peter and Szab{ó},
      Zolt{á}n},
      TITLE = {Heegaard diagrams and holomorphic disks},
      BOOKTITLE = {Different Faces of Geometry},
      SERIES = {Int. Math. Ser.},
      NUMBER = {3},
      PAGES = {301--348},
      PUBLISHER = {Kluwer/Plenum},
      ADDRESS={New York},
      YEAR = {2004},
      MRCLASS = {57R58 (32Q65 57M27 57R17)},
      MRNUMBER = {2005g:57057},
      MRREVIEWER = {Thomas E. Mark},
      ZBLNUMBER = {1091.57010},
      }
  • [Rasmussen] J. Rasmussen, "Floer homology and knot complements," PhD Thesis , Harvard University, 2003.
    @phdthesis{Rasmussen,
      author={Rasmussen, J.},
      TITLE={Floer homology and knot complements},
      SCHOOl={Harvard University},
      YEAR={2003},
      }

Keywords
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Mathematical Subject Classification
Primary:
DOI
http://doi.org/10.4007/annals.2010.171.1213
MR
2630063
zbMATH
05712751
Milestones
Received: 4 August 2006
Revised: 1 May 2008
Accepted: 13 August 2008
Published online: 25 March 2010

Authors

Sucharit Sarkar

Princeton University
Department of Mathematics
Fine Hall - Washington Road
Princeton NJ 08544
United States

Jiajun Wang

California Institute of Technology
Mathematics 253-37
1200 East California Boulevard
Pasadena CA 91125
United States