Introduction to Heegaard Floer homology. (English. Russian original) Zbl 1433.57003
Russ. Math. Surv. 74, No. 1, 1-35 (2019); translation from Usp. Mat. Nauk 74, No. 1, 3-40 (2019).
Summary: Heegaard Floer homology is an invariant of knots, links, and 3-manifolds introduced by Ozsváth and Szabó about 15 years ago. This survey defines Heegaard Floer homology and describes its basic properties. Also discussed is the relation between Heegaard Floer homology and invariants of singularities of curves and surfaces.
MSC:
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
| 57K10 | Knot theory |
| 57R58 | Floer homology |
| 32S25 | Complex surface and hypersurface singularities |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |
Keywords:
Heegaard Floer homology; survey; d-invariants; \(L\)-spaces; singularities of curves and surfaces; plane curve singularitiesReferences:
| [1] | В. И. Арнольд 1967 О характеристическом классе, входящем в условия квантования Функц. анализ и его прил.1 1 1-14 |
| [2] | English transl. V. I. Arnol’d 1967 Characteristic class entering in quantization conditions Funct. Anal. Appl.1 1 1-13 · Zbl 0175.20303 · doi:10.1007/BF01075861 |
| [3] | M. Artin 1966 On isolated rational singularities of surfaces Amer. J. Math.88 129-136 · Zbl 0142.18602 · doi:10.2307/2373050 |
| [4] | M. F. Atiyah, R. Bott, and A. Shapiro 1964 Clifford modules Topology3 3-38 suppl. 1 · Zbl 0146.19001 · doi:10.1016/0040-9383(64)90003-5 |
| [5] | M. Borodzik, M. Hedden, and C. Livingston 2017 Plane algebraic curves of arbitrary genus via Heegaard Floer homology Comment. Math. Helv.92 2 215-256 · Zbl 1372.14026 · doi:10.4171/CMH/411 |
| [6] | M. Borodzik and C. Livingston 2014 Heegaard Floer homology and rational cuspidal curves Forum Math. Sigma2 e28 23 pp. · Zbl 1325.14047 · doi:10.1017/fms.2014.28 |
| [7] | M. Borodzik and C. Livingston 2016 Semigroups, \(d\)-invariants and deformations of cuspidal singular points of plane curves J. Lond. Math. Soc. (2)93 2 439-463 · Zbl 1339.14004 · doi:10.1112/jlms/jdv068 |
| [8] | S. Boyer and A. Clay 2017 Foliations, orders, representations, L-spaces and graph manifolds Adv. Math.310 159-234 · Zbl 1381.57003 · doi:10.1016/j.aim.2017.01.026 |
| [9] | S. Boyer, C. McA. Gordon, and L. Watson 2013 On L-spaces and left-orderable fundamental groups Math. Ann.356 4 1213-1245 · Zbl 1279.57008 · doi:10.1007/s00208-012-0852-7 |
| [10] | S. Boyer, D. Rolfsen, and B. Wiest 2005 Orderable 3-manifold groups Ann. Inst. Fourier (Grenoble)55 1 243-288 · Zbl 1068.57001 · doi:10.5802/aif.2098 |
| [11] | A. Campillo, F. Delgado, and S. M. Gusein-Zade 2003 The Alexander polynomial of a plane curve singularity via the ring of functions on it Duke Math. J.117 1 125-156 · Zbl 1028.32013 · doi:10.1215/S0012-7094-03-11712-3 |
| [12] | P. Dehornoy 1994 Braid groups and left distributive operations Trans. Amer. Math. Soc.345 1 115-150 · Zbl 0837.20048 · doi:10.1090/S0002-9947-1994-1214782-4 |
| [13] | P. Dehornoy, I. Dynnikov, D. Rolfsen, and B. Wiest 2002 Why are braids orderable? Panor. Synthèses 14 Soc. Math. France, Paris xiv+190 pp. · Zbl 1048.20021 |
| [14] | S. K. Donaldson 1983 An application of gauge theory to four-dimensional topology J. Differential Geom.18 2 279-315 · Zbl 0507.57010 · doi:10.4310/jdg/1214437665 |
| [15] | D. Eisenbud and W. Neumann 1985 Three-dimensional link theory and invariants of plane curve singularities Ann. of Math. Stud. 110 Princeton Univ. Press, Princeton, NJ vii+173 pp. · Zbl 0628.57002 |
| [16] | Ya. Eliashberg and L. Traynor (eds.) 1999 Symplectic geometry and topology, Lecture notes from the graduate summer school programPark City, UT, USA 1997 IAS/Park City Math. Ser. 7 Amer. Math. Soc., Providence, RI, Institute for Advanced Study (IAS), Princeton, NJ xiv+430 pp. |
| [17] | N. D. Elkies 1995 A characterization of the \(\mathbb{Z}^n\) lattice Math. Res. Lett.2 3 321-326 · Zbl 0855.11032 · doi:10.4310/MRL.1995.v2.n3.a9 |
| [18] | J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi 2006 On rational cuspidal projective plane curves Proc. London Math. Soc. (3)92 1 99-138 · Zbl 1115.14021 · doi:10.1017/S0024611505015467 |
| [19] | P. Ghiggini 2008 Knot Floer homology detects genus-one fibred knots Amer. J. Math.130 5 1151-1169 · Zbl 1149.57019 · doi:10.1353/ajm.0.0016 |
| [20] | E. Gorsky and A. Némethi 2015 Lattice and Heegaard Floer homologies of algebraic links Int. Math. Res. Not. IMRN2015 23 12737-12780 · Zbl 1342.57005 · doi:10.1093/imrn/rnv075 |
| [21] | E. Gorsky and A. Némethi 2016 Links of plane curve singularities are \(L\)-space links Algebr. Geom. Topol.16 4 1905-1912 · Zbl 1364.14022 · doi:10.2140/agt.2016.16.1905 |
| [22] | H. Grauert 1962 Über Modifikationen und exzeptionelle analytische Mengen Math. Ann.146 4 331-368 · Zbl 0173.33004 · doi:10.1007/BF01441136 |
| [23] | D. V. Gugnin 2017 (v1 - 2015) On integral cohomology ring of symmetric products 1502.01862 17 pp. |
| [24] | A. Haefliger 1956 Sur l’extension du groupe structural d’un espace fibré C. R. Acad. Sci. Paris243 558-560 · Zbl 0070.40001 |
| [25] | J. Hanselman 2016 Bordered Heegaard Floer homology and graph manifolds Algebr. Geom. Topol.16 6 3103-3166 · Zbl 1364.57012 · doi:10.2140/agt.2016.16.3103 |
| [26] | J. Hanselman, J. Rasmussen, S. D. Rasmussen, and L. Watson 2015 Taut foliations on graph manifolds 1508.05911 9 pp. |
| [27] | J. Hom 2016 A note on the concordance invariants epsilon and upsilon Proc. Amer. Math. Soc.144 2 897-902 · Zbl 1337.57020 · doi:10.1090/proc/12706 |
| [28] | J. Hom, Ç. Karakurt, and T. Lidman 2016 Surgery obstructions and Heegaard Floer homology Geom. Topol.20 4 2219-2251 · Zbl 1352.57021 · doi:10.2140/gt.2016.20.2219 |
| [29] | H. B. Laufer 1972 On rational singularities Amer. J. Math.94 2 597-608 · Zbl 0251.32002 · doi:10.2307/2374639 |
| [30] | H. B. Lawson, Jr. and M.-L. Michelsohn 1989 Spin geometry Princeton Math. Ser. 38 Princeton Univ. Press, Princeton, NJ xii+427 pp. · Zbl 0688.57001 |
| [31] | W. B. R. Lickorish 1962 A representation of orientable combinatorial 3-manifolds Ann. of Math. (2)76 3 531-540 · Zbl 0106.37102 · doi:10.2307/1970373 |
| [32] | P. A. Linnell 1999 Left ordered amenable and locally indicable groups J. Lond. Math. Soc. (2)60 1 133-142 · Zbl 0940.20047 · doi:10.1112/S0024610799007462 |
| [33] | P. Lisca and A. I. Stipsicz 2007 Ozsváth-Szabó invariants and tight contact 3-manifolds. III J. Symplectic Geom.5 4 357-384 · Zbl 1149.57037 · doi:10.4310/JSG.2007.v5.n4.a1 |
| [34] | B. Liu 2017 Heegaard Floer homology of \(L\)-space links with two components 1704.02538 23 pp. |
| [35] | Y. Liu \(2017 L\)-space surgeries on links Quantum Topol.8 3 505-570 · Zbl 1381.57019 · doi:10.4171/QT/96 |
| [36] | I. G. MacDonald 1962 Symmetric products of an algebraic curve Topology1 319-343 · Zbl 0121.38003 · doi:10.1016/0040-9383(62)90019-8 |
| [37] | C. Manolescu \(2016 \operatorname{Pin}(2)\)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture J. Amer. Math. Soc.29 1 147-176 · Zbl 1343.57015 · doi:10.1090/jams829 |
| [38] | C. Manolescu and P. Ozsváth 2017 (v1 - 2010) Heegaard Floer homology and integer surgeries on links 1011.1317 231 pp. |
| [39] | С. В. Матвеев, А. Т. Фоменко 1998 Алгоритмические и компьютерные методы в трехмерной топологии Наука, М. 2-е изд., перераб. и доп.304 pp. |
| [40] | English transl. of 1st ed. A. T. Fomenko and S. V. Matveev 1997 Algorithmic and computer methods for three-manifolds Math. Appl. 425 Kluwer Acad. Publ., Dordrecht xii+334 pp. · Zbl 0885.57009 · doi:10.1007/978-94-017-0699-5 |
| [41] | D. McDuff and D. Salamon 1998 Introduction to symplectic topology Oxford Math. Monogr. The Clarendon Press, Oxford Univ. Press, New York 2nd ed., x+486 pp. · Zbl 1066.53137 |
| [42] | A. Némethi 2005 On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds Geom. Topol.9 991-1042 · Zbl 1138.57301 · doi:10.2140/gt.2005.9.991 |
| [43] | A. Némethi 2017 Links of rational singularities, L-spaces and LO fundamental groups Invent. Math.210 1 69-83 · Zbl 1380.32027 · doi:10.1007/s00222-017-0724-6 |
| [44] | Y. Ni 2007 Knot Floer homology detects fibred knots Invent. Math.170 3 577-608 · Zbl 1138.57031 · doi:10.1007/s00222-007-0075-9 |
| [45] | Y. Ni and Z. Wu 2014 Heegaard Floer correction terms and rational genus bounds Adv. Math.267 360-380 · Zbl 1312.57017 · doi:10.1016/j.aim.2014.09.006 |
| [46] | P. Ozsváth, A. Stipsicz, and Z. Szabó 2017 Concordance homomorphisms from knot Floer homology Adv. Math.315 366-426 · Zbl 1383.57020 · doi:10.1016/j.aim.2017.05.017 |
| [47] | P. Ozsváth and Z. Szabó 2003 Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary Adv. Math.173 2 179-261 · Zbl 1025.57016 · doi:10.1016/S0001-8708(02)00030-0 |
| [48] | P. Ozsváth and Z. Szabó 2003 On the Floer homology of plumbed three-manifolds Geom. Topol.7 185-224 · Zbl 1130.57302 · doi:10.2140/gt.2003.7.185 |
| [49] | P. Ozsváth and Z. Szabó 2003 Heegaard Floer homology and alternating knots Geom. Topol.7 225-254 · Zbl 1130.57303 · doi:10.2140/gt.2003.7.225 |
| [50] | P. Ozsváth and Z. Szabó 2003 Knot Floer homology and the four-ball genus Geom. Topol.7 615-639 · Zbl 1037.57027 · doi:10.2140/gt.2003.7.615 |
| [51] | P. Ozsváth and Z. Szabó 2004 Holomorphic disks and genus bounds Geom. Topol.8 311-334 · Zbl 1056.57020 · doi:10.2140/gt.2004.8.311 |
| [52] | P. Ozsváth and Z. Szabó 2004 Holomorphic disks and knot invariants Adv. Math.186 1 58-116 · Zbl 1062.57019 · doi:10.1016/j.aim.2003.05.001 |
| [53] | P. Ozsváth and Z. Szabó 2004 Holomorphic disks and topological invariants for closed three-manifolds Ann. of Math. (2)159 3 1027-1158 · Zbl 1073.57009 · doi:10.4007/annals.2004.159.1027 |
| [54] | P. Ozsváth and Z. Szabó 2004 Holomorphic disks and three-manifold invariants: properties and applications Ann. of Math. (2)159 3 1159-1245 · Zbl 1081.57013 · doi:10.4007/annals.2004.159.1159 |
| [55] | P. Ozsváth and Z. Szabó 2005 Heegaard Floer homology and contact structures Duke Math. J.129 1 39-61 · Zbl 1083.57042 · doi:10.1215/S0012-7094-04-12912-4 |
| [56] | P. Ozsváth and Z. Szabó 2005 On knot Floer homology and lens space surgeries Topology44 6 1281-1300 · Zbl 1077.57012 · doi:10.1016/j.top.2005.05.001 |
| [57] | P. Ozsváth and Z. Szabó 2006 Holomorphic triangles and invariants for smooth four-manifolds Adv. Math.202 2 326-400 · Zbl 1099.53058 · doi:10.1016/j.aim.2005.03.014 |
| [58] | P. Ozsváth and Z. Szabó 2006 Heegaard diagrams and Floer homology International congress of mathematicians II Eur. Math. Soc., Zürich 1083-1099 · Zbl 1116.53062 |
| [59] | P. Ozsváth, Z. Szabó 2008 Knot Floer homology and integer surgeries Algebr. Geom. Topol.8 1 101-153 · Zbl 1181.57018 · doi:10.2140/agt.2008.8.101 |
| [60] | P. Ozsváth and Z. Szabó 2008 Link Floer homology and the Thurston norm J. Amer. Math. Soc.21 3 671-709 · Zbl 1235.53090 · doi:10.1090/S0894-0347-08-00586-9 |
| [61] | P. Ozsváth and Z. Szabó 2008 Holomorphic disks, link invariants and the multi-variable Alexander polynomial Algebr. Geom. Topol.8 2 615-692 · Zbl 1144.57011 · doi:10.2140/agt.2008.8.615 |
| [62] | V. V. Prasolov and A. B. Sossinsky 1997 Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology Transl. Math. Monogr. 154 Amer. Math. Soc., Providence, RI viii+239 pp. · Zbl 0864.57002 · doi:10.1090/mmono/170 |
| [63] | S. D. Rasmussen 2017 L-space intervals for graph manifolds and cables Compos. Math.153 5 1008-1049 · Zbl 1420.57042 · doi:10.1112/S0010437X16008319 |
| [64] | J. Rasmussen and S. D. Rasmussen 2017 Floer simple manifolds and L-space intervals Adv. Math.322 738-805 · Zbl 1379.57024 · doi:10.1016/j.aim.2017.10.014 |
| [65] | J. Rasmussen and L. Watson 2014 Floer Homology and low-dimensional topology 64 pp. http://www.tcd.ie/Hamilton/events/gt/gt2014/notes.pdf |
| [66] | K. Reidemeister 1933 Zur dreidimensionalen Topologie Abh. Math. Sem. Univ. Hamburg9 1 189-194 · Zbl 0007.08005 · doi:10.1007/BF02940644 |
| [67] | M. Scharlemann 2002 Heegaard splittings of compact 3-manifolds Handbook of geometric topology North-Holland, Amsterdam 921-953 · Zbl 0985.57005 |
| [68] | J. Schultens 2014 Introduction to 3-manifolds Grad. Stud. Math. 151 Amer. Math. Soc., Providence, RI x+286 pp. · Zbl 1295.57001 · doi:10.1090/gsm/151 |
| [69] | R. Seroul 1972 Anneau de cohomologie entière et \(KU^*\)-théorie d’un produit symétrique d’une surface de Riemann Publ. Dep. Math. (Lyon)9 4 27-66 · Zbl 0292.57007 |
| [70] | J. Singer 1933 Three-dimensional manifolds and their Heegaard diagrams Trans. Amer. Math. Soc.35 1 88-111 · Zbl 0006.18501 · doi:10.1090/S0002-9947-1933-1501673-5 |
| [71] | P. A. Smith 1936 Manifolds with abelian fundamental groups Ann. of Math. (2)37 3 526-533 · Zbl 0015.08403 · doi:10.2307/1968475 |
| [72] | W. P. Thurston 1986 A norm for the homology of 3-manifolds Mem. Amer. Math. Soc. 59 339 Amer. Math. Soc., Providence, RI i-vi, 99-130 · Zbl 0585.57006 |
| [73] | V. Turaev 1997 Torsion invariants of \(\text{Spin}^c\)-structures on 3-manifolds Math. Res. Lett.4 5 679-695 · Zbl 0891.57019 · doi:10.4310/MRL.1997.v4.n5.a6 |
| [74] | A. H. Wallace 1960 Modifications and cobounding manifolds Canad. J. Math.12 503-528 · Zbl 0108.36101 · doi:10.4153/CJM-1960-045-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
