## Duke Mathematical Journal

### Involutive Heegaard Floer homology

#### 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 $\mathbb{Z}_{4}$-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$, and two invariants of smooth knot concordance, $\underline{V}_{0}$ and $\overline{V}_{0}$. 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 $\underline{V}_{0}$ 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
Revised: 12 July 2016
First available in Project Euclid: 11 January 2017

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

#### 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

#### 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].