Product formulae for Ozsváth–Szabó 4–manifold invariants

Abstract

We give formulae for the Ozsváth-Szabó invariants of 4-manifolds X obtained by fiber sum of two manifolds M 1 , M 2 along surfaces † 1 , † 2 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 57R58; 57M99

Rational Blow-Downs in Heegaard–floer Homology

“…In particular, we will verify that, as conjectured, all the previously known effects of rational blow-downs on Seiberg-Witten or monopole invariants have direct analogs for the Ozsváth and Szabó four-manifold invariants. Pending the results of Jabuka and Mark, [11], these will extend the calculations of the four-manifold invariants to log transforms, just as the original blow-downs did. However, more can be said about when the invariants will be preserved, and, to a topologist, a more hands on account of the structure can be given.…”

“…We follow [11] in saying Definition 6.1. A Spin c structure u on a closed, oriented, smooth four-manifold,…”

Heegaard Floer invariants in codimension one

“…See Ozsváth-Szabó [27] for the original definition, and Jabuka-Mark [12] for an excellent exposition. Here we emphasize two aspects of the theory that will be needed later: passing from HF + to HF ∞ via the U-completed version HF ∞ , and the behavior of the coefficient modules under cobordism maps.…”

“…. , A ζn are null-homotopic (see [12,Remark 5.2]), but they are still defined on the chain level. Indeed, we extend each A ζ i to a map on C * by tensoring with the identity map on F * .…”

The twisted Floer homology of torus bundles

“…We will describe this group explicitly in Section 2.1. It is worth noting that Heegaard Floer homology with twisted coefficients in a certain Novikov ring has already been studied extensively by Jabuka and Mark [7]. The main theorem we prove in this paper is the following: Theorem 1.3 Suppose Y is a closed oriented 3-manifold which fibers over the circle with torus fiber F , and OE!…”