Selecta Mathematica

, Volume 24, Issue 2, pp 1183–1245 | Cite as

A connected sum formula for involutive Heegaard Floer homology

  • Kristen Hendricks
  • Ciprian Manolescu
  • Ian Zemke


We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with \(\underline{d}(Y) \ne d(Y) \ne \bar{d}(Y)\). We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence.

Mathematics Subject Classification

57R58 (Primary) 57M27 (Secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken, NJ (2004)zbMATHGoogle Scholar
  2. 2.
    Frøyshov, K.A.: Equivariant aspects of Yang–Mills Floer theory. Topology 41(3), 525–552 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Frøyshov, K.A.: Monopole Floer homology for rational homology 3-spheres. Duke Math. J. 155(3), 519–576 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fukaya, K.: Floer homology of connected sum of homology 3-spheres. Topology 35(1), 89–136 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Galewski, D.E., Stern, R.J.: Classification of simplicial triangulations of topological manifolds. Ann. Math. (2) 111(1), 1–34 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hendricks, K., Manolescu, C.: Involutive Heegaard Floer homology. Duke Math. J. 166(7), 1211–1299 (2017)Google Scholar
  7. 7.
    Hom, J.: The knot Floer complex and the smooth concordance group. Comment. Math. Helv. 89(3), 537–570 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hom, J.: An infinite-rank summand of topologically slice knots. Geom. Topol. 19(2), 1063–1110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Juhász, A.: Cobordisms of sutured manifolds. arXiv:0910.4382v3
  10. 10.
    Kadeishvili, T. V.: On the theory of homology of fiber spaces, Uspekhi Mat. Nauk, 35(1980), no. 3(213), 183–188, International Topology Conference (Moscow State University, Moscow 1979)Google Scholar
  11. 11.
    Kutluhan, C., Lee, Y.-J., Taubes, C.H.: \(HF=HM\) V : Seiberg–Witten Floer homology and handle additions. arXiv:1204.0115
  12. 12.
    Lin, F.: Pin(2)-monopole Floer homology, higher compositions and connected sums. arXiv:1605.03137
  13. 13.
    Lipshitz, R.: A cylindrical reformulation of Heegaard Floer homology. Geom. Topol. 10, 955–1097 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Manolescu, C.: Seiberg–Witten–Floer stable homotopy type of three-manifolds with \(b_1=0\). Geom. Topol. 7, 889–932 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Manolescu, C.: Pin(2)-equivariant Seiberg–Witten Floer homology and the triangulation conjecture. J. Am. Math. Soc. 29(1), 147–176 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Manolescu, C., Ozsváth, P.S., Sarkar, S.: A combinatorial description of knot Floer homology. Ann. Math. (2) 169(2), 633–660 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Matumoto, T.: Triangulation of manifolds. In: Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Amer. Math. Soc., Providence, R.I., Proc. Sympos. Pure Math., XXXII, pp. 3–6 (1978)Google Scholar
  18. 18.
    McCleary, J.: A User’s Guide to Spectral Sequences, Volume 58 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2001)Google Scholar
  19. 19.
    Ni, Y., Wu, Z.: Cosmetic surgeries on knots in \(S^3\). J. Reine Angew. Math. 706, 1–17 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ozsváth, P.S., Szabó, Z.: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173(2), 179–261 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ozsváth, P.S., Szabó, Z.: Holomorphic disks and three-manifold invariants: properties and applications. Ann. Math. (2) 159(3), 1159–1245 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ozsváth, P.S., Szabó, Z.: Holomorphic disks and topological invariants for closed three-manifolds. Ann. Math. (2) 159(3), 1027–1158 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ozsváth, P.S., Szabó, Z.: Holomorphic triangles and invariants for smooth four-manifolds. Adv. Math. 202(2), 326–400 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ozsváth, P.S., Szabó, Z.: Holomorphic disks, link invariants and the multi-variable Alexander polynomial. Algebr. Geom. Topol. 8(2), 615–692 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ozsváth, P.S., Szabó, Z.: Knot Floer homology and integer surgeries. Algebr. Geom. Topol. 8(1), 101–153 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stoffregen, M.: Manolescu invariants of connected sums. arXiv:1510.01286
  27. 27.
    Stoffregen, M.: Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations. arXiv:1505.03234
  28. 28.
    Zemke, I.: Graph cobordisms and Heegaard Floer homology. arXiv:1512.01184

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Kristen Hendricks
    • 1
  • Ciprian Manolescu
    • 2
  • Ian Zemke
    • 2
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations