Duke Mathematical Journal

Involutive Heegaard Floer homology

Kristen Hendricks and Ciprian Manolescu

Full-text: Access denied (no subscription detected)

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 the conjugation symmetry on Heegaard Floer complexes, we define a 3-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d̲ and d¯, and two invariants of smooth knot concordance, V̲0 and V¯0. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that V̲0 detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.

Article information

Source
Duke Math. J., Volume 166, Number 7 (2017), 1211-1299.

Dates
Received: 24 November 2015
Revised: 12 July 2016
First available in Project Euclid: 11 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1484103841

Digital Object Identifier
doi:10.1215/00127094-3793141

Mathematical Reviews number (MathSciNet)
MR3649355

Zentralblatt MATH identifier
1383.57036

Subjects
Primary: 57R58: Floer homology
Secondary: 57M27: Invariants of knots and 3-manifolds

Keywords
Heegaard Floer homology cobordism 3-manifold knot

Citation

Hendricks, Kristen; Manolescu, Ciprian. Involutive Heegaard Floer homology. Duke Math. J. 166 (2017), no. 7, 1211--1299. doi:10.1215/00127094-3793141. https://projecteuclid.org/euclid.dmj/1484103841


Export citation

References

  • [1] C. Bohr and R. Lee, Homology cobordism and classical knot invariants, Comment. Math. Helv. 77 (2002), 363–382.
  • [2] V. Colin, P. Ghiggini, and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011), 8100–8105.
  • [3] J. Eells, Jr. and N. H. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110.
  • [4] R. Fintushel and R. J. Stern, “A $\mu$-invariant one homology $3$-sphere that bounds an orientable rational ball” in Four-manifold Theory (Durham, N.H., 1982), Contemp. Math. 35, Amer. Math. Soc., Providence, 1984, 265–268.
  • [5] R. H. Fox and J. W. Milnor, Singularities of $2$-spheres in $4$-space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267.
  • [6] K. A. Frøyshov, Monopole Floer homology for rational homology 3-spheres, Duke Math. J. 155 (2010), 519–576.
  • [7] D. E. Galewski and R. J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. (2) 111 (1980), 1–34.
  • [8] J. Hom, The knot Floer complex and the smooth concordance group, Comment. Math. Helv. 89 (2014), 537–570.
  • [9] J. Hom and Z. Wu, Four-ball genus bounds and a refinement of the Ozsváth-Szabó tau invariant, J. Symplectic Geom. 14 (2016), 305–323.
  • [10] A. Juhász, Cobordisms of sutured manifolds and the functoriality of link Floer homology, Adv. Math. 299 (2016), 940–1038.
  • [11] A. Juhász and D. P. Thurston, Naturality and mapping class groups in Heegaard Floer homology, preprint, arXiv:1210.4996v3 [math.GT].
  • [12] P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-manifolds, New Math. Monogr. 10, Cambridge Univ. Press, Cambridge, 2007.
  • [13] P. B. Kronheimer, T. S. Mrowka, P. S. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007), 457–546.
  • [14] C. Kutluhan, Y.-J. Lee, and C. H. Taubes, HF=HM I: Heegaard Floer homology and Seiberg–Witten Floer homology, preprint, arXiv:1007.1979v5 [math.GT].
  • [15] F. Laudenbach, A proof of Reidemeister-Singer’s theorem by Cerf’s methods, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 197–221.
  • [16] T. Lidman and C. Manolescu, The equivalence of two Seiberg-Witten Floer homologies, preprint, arXiv:1603.00582v1 [math.GT].
  • [17] F. Lin, A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture, preprint, arXiv:1404.4561v2 [math.GT].
  • [18] F. Lin, The surgery exact triangle in $\operatorname{Pin}(2)$-monopole Floer homology, preprint, arXiv:1504.01993v2 [math.GT].
  • [19] R. Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006), 955–1097.
  • [20] P. Lisca and B. Owens, Signatures, Heegaard Floer correction terms and quasi-alternating links, Proc. Amer. Math. Soc. 143 (2015), 907–914.
  • [21] P. Lisca and A. I. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds, II, J. Differential Geom. 75 (2007), 109–141.
  • [22] C. Manolescu, Seiberg-Witten-Floer stable homotopy type of three-manifolds with $b_{1}=0$, Geom. Topol. 7 (2003), 889–932.
  • [23] C. Manolescu, $\operatorname{Pin}(2)$-equivariant Seiberg-Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016), 147–176.
  • [24] C. Manolescu, An introduction to knot Floer homology, preprint, arXiv:1401.7107v2 [math.GT].
  • [25] C. Manolescu and B. Owens, A concordance invariant from the Floer homology of double branched covers, Int. Math. Res. Not. IMRN 2007, no. 20, art. ID rnm077.
  • [26] C. Manolescu and P. S. Ozsváth, “On the Khovanov and knot Floer homologies of quasi-alternating links” in Proceedings of Gökova Geometry-Topology Conference 2007 (Gökova, 2007), Gökova Geometry/Topology Conference (GGT), Gökova, 2008, 60–81.
  • [27] C. Manolescu and P. S. Ozsváth, Heegaard Floer homology and integer surgeries on links, preprint, arXiv:1011.1317v3 [math.GT].
  • [28] C. Manolescu, P. S. Ozsváth, and S. Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. (2) 169 (2009), 633–660.
  • [29] C. Manolescu, P. S. Ozsváth, Z. Szabó, and D. P. Thurston, On combinatorial link Floer homology, Geom. Topol. 11 (2007), 2339–2412.
  • [30] C. Manolescu, P. S. Ozsváth, and D. P. Thurston, Grid diagrams and Heegaard Floer invariants, preprint, arXiv:0910.0078v2 [math.GT].
  • [31] T. Matumoto, “Triangulation of manifolds” in Algebraic and Geometric Topology (Stanford, 1976), Proc. Sympos. Pure Math. XXXII, Part 2, Amer. Math. Soc., Providence, 1978, 3–6.
  • [32] Y. Ni and Z. Wu, Cosmetic surgeries on knots in $S^{3}$, J. Reine Angew. Math. 706 (2015), 1–17.
  • [33] P. S. Ozsváth and A. Stipsicz, Contact surgeries and the transverse invariant in knot Floer homology, J. Inst. Math. Jussieu 9 (2010), 601–632.
  • [34] P. S. Ozsváth, A. Stipsicz, and Z. Szabó, Concordance homomorphisms from knot Floer homology, preprint, arXiv:1407.1795v2 [math.GT].
  • [35] P. S. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), 179–261.
  • [36] P. S. Ozsváth and Z. Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254.
  • [37] P. S. Ozsváth and Z. Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639.
  • [38] P. S. Ozsváth and Z. Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), 58–116.
  • [39] P. S. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. (2) 159 (2004), 1159–1245.
  • [40] P. S. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027–1158.
  • [41] P. S. Ozsváth and Z. Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), 1281–1300.
  • [42] P. S. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), 1–33.
  • [43] P. S. Ozsváth and Z. Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), 326–400.
  • [44] P. S. Ozsváth and Z. Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008), 101–153.
  • [45] P. S. Ozsváth and Z. Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), 1–68.
  • [46] T. Peters, A concordance invariant from the Floer homology of $\pm1$ surgeries, preprint, arXiv:1003.3038v4 [math.GT].
  • [47] I. Petkova, Cables of thin knots and bordered Heegaard Floer homology, Quantum Topol. 4 (2013), 377–409.
  • [48] J. A. Rasmussen, Lens space surgeries and a conjecture of Goda and Teragaito, Geom. Topol. 8 (2004), 1013–1031.
  • [49] J. A. Rasmussen, “Knot polynomials and knot homologies” in Geometry and Topology of Manifolds, Fields Inst. Commun. 47, Amer. Math. Soc., Providence, 2005, 261–280.
  • [50] J. A. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419–447.
  • [51] J. A. Rasmussen, Floer homology and knot complements, preprint, arXiv:math/0306378v1 [math.GT].
  • [52] S. Sarkar, Moving basepoints and the induced automorphisms of link Floer homology, Algebr. Geom. Topol. 15 (2015), 2479–2515.
  • [53] P. Seidel and I. Smith, Localization for involutions in Floer cohomology, Geom. Funct. Anal. 20 (2010), 1464–1501.
  • [54] M. Stoffregen, $\operatorname{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert fibrations, preprint, arXiv:1505.03234v2 [math.GT].
  • [55] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology, I, Geom. Topol. 14 (2010), 2497–2581.
  • [56] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994.
  • [57] I. Zemke, A graph TQFT for hat Heegaard Floer homology, preprint, arXiv:1503.05846v2 [math.GT].
  • [58] I. Zemke, Quasi-stabilization and basepoint moving maps in link Floer homology, preprint, arXiv:1604.04316v2 [math.GT].