Rational Blow-Downs in Heegaard–floer Homology

Abstract

We examine the effect rational blow-downs have on the Ozsváth-Szabó four-manifold invariant, verify that it is identical to that on the Seiberg-Witten invariants, and extend to new classes of blow-downs.

Surgery along star-shaped plumbings and exotic smooth structures on 4–manifolds

“…It suffices to know all the Seiberg-Witten basic classes of X . The effect of such cut and paste operations on Seiberg-Witten invariants was studied by Michalogiorgaki in [30] in a more general framework; see also [39] for an analogous result from the perspective of Heegaard Floer theory.…”

On the Heegaard Floer homology of a surface times a circle

“…Certainly ϕ is injective, and the fact that Im(ϕ) ⊂ Ker(F ) is easy to check. On the other hand, by considering top-degree parts one can verify that any element of Ker(F ) is of the form given on the right-hand side of (17). Namely, let us pick ξ ∈ KerF of the form ξ = ξ ℓ + ξ ℓ−2|k| + ξ ℓ−4|k| + ... where ξ i is the homogeneous part of ξ of degree i and where we assume that ξ ℓ+2n|k| = 0 for all n > 0.…”

“…These examples are typically distinguished via gauge-theoretic invariants using either ad-hoc arguments or an appeal to general product theorems (or a combination). The 4-manifold invariants of Ozsváth and Szabó [10,11] are expected to give the same information as the Seiberg-Witten invariants but are in some cases easier to work with (results in this direction include the vanishing theorems for sums along L-spaces by Ozsváth and Szabó [10] and the rational blowdown formulae by Roberts [17]). An approach to understanding the behavior of Ozsváth-Szabó invariants under fiber sum is to interpret the invariants as the result of a certain pairing of relative invariants of the complements of the surfaces in each 4-manifold, invariants taking values in the Floer homology of the boundary Σ g × S 1 .…”

Projective Naturality in Heegaard Floer Homology

“…If one can establish naturality with respect to cobordisms over Z/±, we would obtain corresponding mixed invariants Φ X,t : Λ * (H 1 (X; Z)/Tors) ⊗ Z Z[U ] → Z/± which we expect would provide fruitful extra information. In fact, before the gap in the literature was noticed, the integral mixed invariants had already been extensively studied in papers including [OS04c], [JM08] and [Rob08], so establishing naturality with respect to cobordisms over Z would immediately prove useful, and would likely also be useful for computations and applications in the future.…”