Geometry & Topology
- Geom. Topol.
- Volume 12, Number 3 (2008), 1557-1651.
Product formulae for Ozsváth–Szabó $4$–manifold invariants
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.
Geom. Topol., Volume 12, Number 3 (2008), 1557-1651.
Received: 5 July 2007
Revised: 4 March 2008
Accepted: 15 April 2008
First available in Project Euclid: 20 December 2017
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Jabuka, Stanislav; Mark, Thomas E. Product formulae for Ozsváth–Szabó $4$–manifold invariants. Geom. Topol. 12 (2008), no. 3, 1557--1651. doi:10.2140/gt.2008.12.1557. https://projecteuclid.org/euclid.gt/1513800104