Volume 12, issue 3 (2008)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Product formulae for Ozsváth–Szabó $4$–manifold invariants

Stanislav Jabuka and Thomas E Mark

Geometry & Topology 12 (2008) 1557–1651
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 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.

Keywords
four manifolds, product formula, Ozsváth–Szabó invariant, Heegaard Floer homology
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M99
References
Publication
Received: 5 July 2007
Revised: 4 March 2008
Accepted: 15 April 2008
Published: 19 June 2008
Proposed: Ron Fintushel
Seconded: Ron Stern, Peter Ozsváth
Authors
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