Duke Mathematical Journal

Combinatorial cobordism maps in hat Heegaard Floer theory

Robert Lipshitz, Ciprian Manolescu, and Jiajun Wang

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In a previous article, Sarkar and Wang [15] gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version

Article information

Duke Math. J., Volume 145, Number 2 (2008), 207-247.

First available in Project Euclid: 20 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology
Secondary: 57R56: Topological quantum field theories


Lipshitz, Robert; Manolescu, Ciprian; Wang, Jiajun. Combinatorial cobordism maps in hat Heegaard Floer theory. Duke Math. J. 145 (2008), no. 2, 207--247. doi:10.1215/00127094-2008-050. https://projecteuclid.org/euclid.dmj/1224508836

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  • R. Fintushel and R. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), 181–235.
  • R. E. Gompf and A. I. Stipsicz, $4$-manifolds and Kirby Calculus, Grad. Stud. Math. 20, Amer. Math. Soc., Providence, 1999.
  • R. Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006), 955–1097.
  • P. Lisca, On the Donaldson polynomials of elliptic surfaces, Math. Ann. 299 (1994), 629–639.
  • C. Manolescu, P. OzsváTh, and S. Sarkar, A combinatorial description of knot Floer homology, preprint,\arxivmath/0607691v2[math.GT]
  • J. W. Morgan and T. S. Mrowka, On the diffeomorphism classification of regular elliptic surfaces, Internat. Math. Res. Notices 1993, no. 6, 183–184.
  • P. S. OzsváTh and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), 179–261.
  • —, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), 58–116.
  • —, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027–1158.
  • —, Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. (2) 159 (2004), 1159–1245.
  • —, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), 326–400.
  • —, Holomorphic disks and link invariants, preprint,\arxivmath/0512286v2[math.GT]
  • L. Roberts, Rational blow-downs in Heegaard-Floer homology, preprint,\arxivmath/0607675v1[math.GT]
  • S. Sarkar, Maslov index of holomorphic triangles, preprint,\arxivmath/0609673v2[math.GT]
  • S. Sarkar and J. Wang, An algorithm for computing some Heegaard Floer homologies, preprint,\arxivmath/0607777v3[math.GT]
  • A. I. Stipsicz and Z. Szabó, Gluing $4$-manifolds along $\Sigma(2,3,11)$, Topology Appl. 106 (2000), 293–304.