Periodic Reporting for period 1 - LDTSing (Low Dimensional Topology and Singularity Theory)
Reporting period: 2021-11-01 to 2023-10-31
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.
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.
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.
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.
By their very nature, all results described in the previous section are beyond the state of the art. From a wider perspective, we expect our work on non smoothability of some surface singularities to be of interest in the community of singularity theory as we make use of topological tools but obtain analytic conclusions.
Our work on line arrangements and configurations brings tools from smooth 4-manifold topology in a very classical setting and we expect refinements of our work with other techniques in the future.
Overall we have emploied 4-manifold topology techniques for various problems inspired from algebraic and analytic geometry as prescribed by our main objective with the expectation that these results will bring the two communities closer and mootivate further investigations.
Our work on line arrangements and configurations brings tools from smooth 4-manifold topology in a very classical setting and we expect refinements of our work with other techniques in the future.
Overall we have emploied 4-manifold topology techniques for various problems inspired from algebraic and analytic geometry as prescribed by our main objective with the expectation that these results will bring the two communities closer and mootivate further investigations.