LDTSing · Low Dimensional Topology and Singularity Theory

Low Dimensional Topology and Singularity Theory

The general aim of the project is to investigate problems in 4-manifold topology inspired by questions in complex analytic geometry and, morespecifically in the context of deformations of complex surface singularities and in the study of rational cuspidal curves. The problem of classification of deformations of rational surface singularities naturally leads to a smooth topological analogue of understanding negative-definite fillings of the corresponding link of the singularity. The study of rational cuspidal plane curves is related to certain Dehn surgeries on the connected sum of the links of the singularity of the curve which bound rational homology balls. Exploring these problems in a purely topological setting can sometimes show how certain classification results, when appropriately reformulated, have purely topological counterparts. Moreover, in some cases, one can recover analytic results with topological arguments. Comparing problems and results across different categories is a very important aspect of mathematical research as it often leads to a deeper understanding of known results and can generate new, more general, outcomes which in turn lead to deeper questions. In the context of links of rational surface singularities we have shown that for minimally rational ones there is a unique smoothing component precisely when the corresponding link bounds a unique negative-definite form. Thus showing a topological manifestation of a uniqueness result in the analytic setting. For nonrational singularities, we have shown that, in some cases, it is possible to prove nonsmoothability with topological arguments. When the second Betti number and the signature of a Milnor fiber are prescribed by analytic invariants, one can provide exmples where the link of the singularity does not bound a smooth 4-manifold with those invariant thus showing that the corresponding singularity is not smoothable. Inspired by the classical study complex line arrangements and configurations one can consider the natural equivalent question in a purely topological setting. Following previous work by Ruberman-Starkston, we improve their results by obstructing more configurations to be realized at the cost of switching from the topological locally flat category to the smooth category. Inspired by the classification problem for rational cuspidal plane curves, we have considered the question of which Dehn surgeries along a connected sum of torus knots bound rational homology balls. By employing various topological invariants almost a full classification can be obtained for alternating sums of torus knots. This shows, as expected, a substantial difference with the algebraic setting.

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