Negative-definite fillings of plumbed manifolds

Motivated by the study of smoothings of rational surface singularities as well as symplectic f illings of plumbed 3-manifolds, we consider an analogue problem in a purely topological setting. The question of when a rational surface singularity admits a unique smoothing is of particular interest and has led to a conjecture of Kollár which has been proved in some cases. We look at smooth, definite fillings of certain plumbed manifolds and consider the question of which intersection forms can be realized by such fillings. We describe various constructions and an obstruction based on Donaldson’s diagonalization theorem. Finally, we present a couple of uniqueness results and discuss their relevance for Kollár’s conjecture as well as the problem of embedding lens spaces in certain 4-manifolds. While the main motivation lies in problems from singularity theory and symplectic topology, our results are purely topological in nature and the main techniques used are algebro-combinatorial. This is joint work with Duncan McCoy and JungHwan Park.