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2008 Product formulae for Ozsváth–Szabó $4$–manifold invariants
Stanislav Jabuka, Thomas E Mark
Geom. Topol. 12(3): 1557-1651 (2008). 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 g1. 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.

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Stanislav Jabuka. Thomas E Mark. "Product formulae for Ozsváth–Szabó $4$–manifold invariants." Geom. Topol. 12 (3) 1557 - 1651, 2008. https://doi.org/10.2140/gt.2008.12.1557

Information

Received: 5 July 2007; Revised: 4 March 2008; Accepted: 15 April 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1156.57026
MathSciNet: MR2421135
Digital Object Identifier: 10.2140/gt.2008.12.1557

Subjects:
Primary: 57R58
Secondary: 57M99

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

Rights: Copyright © 2008 Mathematical Sciences Publishers

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