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 M1, M2 along surfaces Σ1, Σ2 having trivial normal bundle and genus g≥ 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+≥ 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. © 2008 Geometry & Topology.