Volume 12, issue 3 (2008)

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Product formulae for Ozsváth–Szabó $4$–manifold invariants

Stanislav Jabuka and Thomas E Mark

Geometry & Topology 12 (2008) 1557–1651
Abstract

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.

Keywords
four manifolds, product formula, Ozsváth–Szabó invariant, Heegaard Floer homology
Primary: 57R58
Secondary: 57M99