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

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.