Low Dimensional Topology and Singularity Theory

The aim of the project is two-fold.

One goal is to employ techniques from smooth 4-dimensional topology in the study of deformations of isolated surface singularities.

More specifically the project aims at advancing in the study of smoothings of rational surface singularities by means of gauge-theoretic invariants as well as lattice-theoretic combinatorial techniques. A conjecture of Kollar regarding a class of rational surface singularities with a unique smoothing will be considered.

The conjecture has natural symplectic and topological counterparts.

The plan consists in proving the topological version and investigating the extent to which this version of the problem can lead to advancements in the original conjecture.Another primary goal is to investigate properties of the 3-dimensional rational homology sphere group, such as n-divisibility and torsion, via constructions involving rational cuspidal curves in possibly singular homology planes.

In this context a first specific goal is producing examples of 3-manifolds which are either Seifert fibered spaces or obtained via Dehn surgery on an algebraic knots which are 2-divisible in the rational homology sphere group.

In a similar setting it will be investigated the extent to which rational homology balls bounded by integral surgeries on torus knots can be realized algebraically.