Geometry & Topology

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

Stanislav Jabuka and Thomas E Mark

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 12, Number 3 (2008), 1557-1651.

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800104

Digital Object Identifier
doi:10.2140/gt.2008.12.1557

Mathematical Reviews number (MathSciNet)
MR2421135

Zentralblatt MATH identifier
1156.57026

Subjects
Primary: 57R58: Floer homology
Secondary: 57M99: None of the above, but in this section

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

Citation

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


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