Product formulae for Ozsváth–Szabó 4–manifold invariants

Jabuka, Stanislav¹ ; Mark, Thomas E²

¹ Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557
² Department of Mathematics, University of Virginia, Charlottesville, VA 22904

Geometry & topology, Tome 12 (2008) no. 3, pp. 1557-1651.

Résumé

We give formulae for the Ozsváth–Szabó invariants of $4$–manifolds $X$ obtained by fiber sum of two manifolds $M_1$, $M_2$ along surfaces ${\Sigma }_1$, ${\Sigma }_2$ having trivial normal bundle and genus $g\ge 1$. The formulae follow from a general theorem on the Ozsváth–Szabó invariants of the result of gluing two $4$–manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this “perturbed” version of Heegaard Floer theory recovers the usual Ozsváth–Szabó invariants, when the $4$–manifold in question has $b^{+}\ge 2$. The construction allows an extension of the definition of Ozsváth–Szabó invariants to $4$–manifolds having $b^{+}=1$ depending on certain choices, in close analogy with Seiberg–Witten theory. The product formulae lead quickly to calculations of the Ozsváth–Szabó invariants of various $4$–manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsváth–Szabó and Seiberg–Witten invariants.

Détail

DOI : 10.2140/gt.2008.12.1557

Keywords: four manifolds, product formula, Ozsváth–Szabó invariant, Heegaard Floer homology

Bibliographie

[1] D Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer (1995)

[2] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc. (1999)

[3] S Jabuka, T Mark, Heegaard Floer homology of a surfaces times a circle, Adv. Math. 218 (2008) 728

[4] P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press (2007)

[5] T J Li, A Liu, General wall crossing formula, Math. Res. Lett. 2 (1995) 797

[6] T Mark, Knotted surfaces in $4$–manifolds

[7] J W Morgan, T S Mrowka, Z Szabó, Product formulas along $T^3$ for Seiberg–Witten invariants, Math. Res. Lett. 4 (1997) 915

[8] J W Morgan, Z Szabó, C H Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706

[9] P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179

[10] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[11] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $(2)$ 159 (2004) 1159

[12] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[13] P Ozsváth, Z Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004) 1

[14] P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39

[15] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326

[16] P Ozsváth, Z Szabó, An introduction to Heegaard Floer homology, from: "Floer homology, gauge theory, and low-dimensional topology", Clay Math. Proc. 5, Amer. Math. Soc. (2006) 3

[17] B D Park, A gluing formula for the Seiberg–Witten invariant along $T^3$, Michigan Math. J. 50 (2002) 593

[18] J A Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757

[19] R J Stern, Will we ever classify simply-connected smooth $4$–manifolds?, from: "Floer homology, gauge theory, and low-dimensional topology", Clay Math. Proc. 5, Amer. Math. Soc. (2006) 225

[20] C H Taubes, $\mathrm{GR}=\mathrm{SW}$: counting curves and connections, J. Differential Geom. 52 (1999) 453

[21] C H Taubes, The Seiberg–Witten invariants and $4$–manifolds with essential tori, Geom. Topol. 5 (2001) 441

[22] E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769