We give formulae for the Ozsváth–Szabó invariants of
–manifolds
obtained by fiber sum
of two manifolds ,
along
surfaces ,
having trivial normal
bundle and genus .
The formulae follow from a general theorem on the Ozsváth–Szabó invariants of the result of gluing
two –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
–manifold in
question has .
The construction allows an extension of the definition of Ozsváth–Szabó invariants to
–manifolds
having
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
–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
