We refer to the four groups of results as listed in the summary in the same order and then describe other results related to different lines of investigation.
1) There are two main results which describe the topological analogue of having a unique smoothing.
First Theorem: Let Y be a plumbed three-manifold according to a plumbing tree with no bad vertices and such that each weight is at most -5.
Then, up to stabilization, Y bounds a unique negative-definite intersection form (i.e. the intersection form of the corresponding plumbed 4-manifold)
Second Theorem: Let Y be a connected sum of Seifert Fibered spaces bounding negative definite plumbings with no bad vertices and such that every vertex has weight at most -5 or is a leaf with weight at most -3. Then, up to stabilizations, Y bounds a unique negative-definite intersection form.
Each of these theorems relies on a graph-theoretical statement linked to topology via Donaldson's diagonalization theorem.
A draft of this paper can be provided upon request. A preprint is expected to be posted on the ArXiv in a few months. This is a joint project with Duncan McCoy and JungHwan Park.
2) This project is still at an earlier stage so we refrain from giving precise statements which may change in the future. The upshot can be summarized as follows. Given a Gorenstein surface singularitywith link Y, it is possible to fix some analytic invariants so that any Milnor fiber coming from a smothing has a prescribed second Betti number (say k>0) and is positive-definite. By changing the orientation we obtain a neg-def filling for -Y. This filling can be glued to the canonical plumbing P for Y giving a closed neg-def 4-manifold. By Donaldsons diagonalization theorem, this forces the intersection lattice of P to embed in the standard lattice in codimension k. We can provide various examples which shows that such an embedding does not exist, thus showing that the corresponding singularities are not smoothable. We also recover previously known examples.
This is a joint project with Duncan McCoy and JungHwan Park.
3) Theorem: Any smooth, complex realization of an (n,k) configuration with k>4 satisfies n>k(k-1)+3
The proof consists in examining separately the three cases n=k(k-1)+i with i=1,2, or 3. In each case one blows up all points of the configuration and then blows down (topologically) a generic line not in the configuration. The resulting arrangement of spheres has a regular neighbourhood that is a neg-def 4-manifold, lattice theoretic arguments then exclude such a configuration to be embedded in a standard lattice. The argument gets increasingly more complicated as i gets bigger, but conceptually they all rely on the same idea.
A draft of this paper can be provided upon request. A preprint is expected to be posted on the ArXiv in a few months. This is a joint project with Duncan McCoy and JungHwan Park.
4) For this project the main results are a bit technical so we provide an overall summary with no precise statements. Given a knot one can consider the sliceness problem in 4-manifolds other than the 4-ball. When looking at the punctured projective plane this question is related to the topological types of cusps for rational plane curves. We employ several tools from 4-manifold topology such as: correction terms from Heegaard Floer homology and their variants in involutive Floer homology, the 10/8 theorem, Donaldson's theorem C. We find constraints for some algebraic knots to bound such disks in a punctured CP^2 or in a sum of two copies of CP^2.
This is joint work with Andras Stipsicz, Maggie Miller and JungHwan Park.
