Rational blow-downs in Heegaard-Floer homology Preprint

In [2], R. Fintushel and R. Stern introduced the rational blow down, a process which could be applied to a smooth four manifold containing one of a family of codimension 0, negative definite sub-manifolds. This family of examples were extended in the work of J. Park, [21]. In each case, the procedure called for removing the negative definite piece and replacing it with a rational homology four ball. Furthermore, if the four manifold had a symplectic structure, and the negative definite piece was appropriately embedded with respect to this structure, M. Symington showed that the resulting manifold would have a symplectic structure. The importance of the procedure derives from the possibility that it preserves some of the Seiberg-Witten1 basic classes of the original manifold. It could thus be used to construct new examples of homotopy equivalent but non-diffeomorphic four manifolds. Additionally, through clever topo-logical manipulation, it allowed the computation of the Seiberg-Witten invariants for the result of p-log transforms on certain four manifolds. In particular, the computation of the Seiberg-Witten invariants for logarithmic transforms in cusp neighborhoods was then achieved. Recently, further extensions of this procedure have been announced, [24], [3]. In [24] it is noted that another family of examples built around Wahl singularities, [25], could also give rise to ways t