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Product formulae for Ozsváth–Szabó $4$–manifold invariants

Stanislav Jabuka and Thomas E Mark

Geometry & Topology 12 (2008) 1557–1651

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.

four manifolds, product formula, Ozsváth–Szabó invariant, Heegaard Floer homology
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M99
Received: 5 July 2007
Revised: 4 March 2008
Accepted: 15 April 2008
Published: 19 June 2008
Proposed: Ron Fintushel
Seconded: Ron Stern, Peter Ozsváth
Stanislav Jabuka
Department of Mathematics and Statistics
University of Nevada
Reno, NV 89557
Thomas E Mark
Department of Mathematics
University of Virginia
Charlottesville, VA 22904