Product Formulae for Ozsvath-Szabo 4-manifold Invariants

TL;DR

This paper derives formulae for Ozsvath-Szabo 4-manifold invariants under fiber sum operations, extending their definition to certain cases and confirming their consistency with Seiberg-Witten invariants.

Contribution

It provides a general theorem for computing Ozsvath-Szabo invariants of fiber-summed 4-manifolds using relative invariants and extends the invariants' definition to cases with b^+ = 1.

Findings

Derived explicit fiber sum formulae for Ozsvath-Szabo invariants.

Extended the invariants' applicability to 4-manifolds with b^+ = 1.

Confirmed consistency with Seiberg-Witten invariants.

Abstract

We give formulae for the Ozsvath-Szabo invariants of 4-manifolds X obtained by fiber sum of two manifolds M_1, M_2 along surfaces S_1, S_2 having trivial normal bundle and genus g>0. The formulae follow from a general theorem on the Ozsvath-Szabo 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 Ozsvath-Szabo invariants, when the 4-manifold in question has b^+>1. The construction allows an extension of the definition of the Ozsvath-Szabo 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 Ozsvath-Szabo invariants of various 4-manifolds; in all cases the results are in accord with the conjectured equivalence between the Ozsvath-Szabo and Seiberg-Witten invariants.