Geometry & Topology 12 (2008) 1557–1651

DOI: 10.2140/gt.2008.12.1557

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.

Keywords

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

Mathematical Subject Classification

Primary: 57R58

Secondary: 57M99

Publication

Received: 5 July 2007
Revised: 4 March 2008
Accepted: 15 April 2008
Published: 19 June 2008
Proposed: Ron Fintushel
Seconded: Ron Stern, Peter Ozsváth

References

Authors