Research
9. On existence and construction of non-loose knots ( Joint with Rima Chatterjee, John Etnyre and Hyunki Min ) (arXiv)
9. On existence and construction of non-loose knots ( Joint with Rima Chatterjee, John Etnyre and Hyunki Min ) (arXiv)
Abstract: In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling
Abstract: In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling
8. One stabilization is not enough for closed knotted surfaces ( Joint with Kyle Hayden and Sungkyung Kang )
8. One stabilization is not enough for closed knotted surfaces ( Joint with Kyle Hayden and Sungkyung Kang )
Abstract: In this brief note, we show that there exist smooth 4-manifolds (with nonempty boundary) containing pairs of exotically knotted 2-spheres that remain exotic after one (either external or internal) stabilization. It follows that the ``one is enough'' theorem of Auckly-Kim-Melvin-Ruberman-Schwartz does not hold for closed surfaces whose homology classes are characteristic.
Abstract: In this brief note, we show that there exist smooth 4-manifolds (with nonempty boundary) containing pairs of exotically knotted 2-spheres that remain exotic after one (either external or internal) stabilization. It follows that the ``one is enough'' theorem of Auckly-Kim-Melvin-Ruberman-Schwartz does not hold for closed surfaces whose homology classes are characteristic.
7. Exotic codimension-1 submanifolds in 4-manifolds and stabilizations ( Joint with Hokuto Konno and Masaki Taniguchi) (arXiv) [ YouTube video ]
7. Exotic codimension-1 submanifolds in 4-manifolds and stabilizations ( Joint with Hokuto Konno and Masaki Taniguchi) (arXiv) [ YouTube video ]
Abstract: In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-1 submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by S^2\times S^2. Thus we give the first example of exotic phenomena of 4-dimensional topology that does not respect Wall's principle: various types of exotica for orientable 4-manifolds disappear after finitely many stabilizations. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.
Abstract: In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-1 submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by S^2\times S^2. Thus we give the first example of exotic phenomena of 4-dimensional topology that does not respect Wall's principle: various types of exotica for orientable 4-manifolds disappear after finitely many stabilizations. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.
Abstract: We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on S^3.
Abstract: We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on S^3.
5. Diffeomorphisms of 4-manifolds with boundary and exotic embeddings ( Joint with Nobuo Iida, Hokuto Konno and Masaki Taniguchi) (arXiv)
5. Diffeomorphisms of 4-manifolds with boundary and exotic embeddings ( Joint with Nobuo Iida, Hokuto Konno and Masaki Taniguchi) (arXiv)
Abstract: We define family versions of the invariant of 4-manifolds with contact boundary due to Kronheimer and Mrowka and using this to detect exotic diffeomorhisms on 4-manifold with boundary. Further we developed some techniques that has been used to show existence of exotic codimension-2 submanifolds and exotic codimension-1 submanifolds in 4-manifolds.
Abstract: We define family versions of the invariant of 4-manifolds with contact boundary due to Kronheimer and Mrowka and using this to detect exotic diffeomorhisms on 4-manifold with boundary. Further we developed some techniques that has been used to show existence of exotic codimension-2 submanifolds and exotic codimension-1 submanifolds in 4-manifolds.
4. Family Bauer-Furuta invariant, Exotic Surfaces and Smale's conjecture (Joint with Jianfeng Lin) (arXiv) [ YouTube video ]
4. Family Bauer-Furuta invariant, Exotic Surfaces and Smale's conjecture (Joint with Jianfeng Lin) (arXiv) [ YouTube video ]
Abstract: We establish the existence of a pair of exotic surfaces in a puncture K3 which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer-Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the Tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the S^1-equivariant family Bauer-Furuta invariant of any orientation-preserving diffeomorphism on S^4 is trivial and that the Pin(2)-equivariant family Bauer-Furuta invariant for a diffeomorphism on S^2 x S^2 is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on S^4 or S^2 x S^2. En route, we observe a curious element in the Pin(2)-equivariant stable homotopy group of spheres which could be used to detect an exotic diffeomorphism on S^4.
Abstract: We establish the existence of a pair of exotic surfaces in a puncture K3 which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer-Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the Tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the S^1-equivariant family Bauer-Furuta invariant of any orientation-preserving diffeomorphism on S^4 is trivial and that the Pin(2)-equivariant family Bauer-Furuta invariant for a diffeomorphism on S^2 x S^2 is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on S^4 or S^2 x S^2. En route, we observe a curious element in the Pin(2)-equivariant stable homotopy group of spheres which could be used to detect an exotic diffeomorphism on S^4.
3. An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology (Joint with Nobuo Iida and Masaki Taniguchi) (arXiv) [ YouTube video ]
3. An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology (Joint with Nobuo Iida and Masaki Taniguchi) (arXiv) [ YouTube video ]
Abstract: We give infinitely many knots in S^3 that are not smoothly H-slice (that is, bounding a null-homologous disk) in many 4-manifolds but they are topologically H-slice. In particular, we give such knots in punctured elliptic surfaces E(2n). In addition, we give obstructions to codimension-0 embedding of weak symplectic fillings with b_3=0 into closed symplectic 4-manifolds with b_1=0 and b_2^+=3 mod 4. We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures. All of these results follow from our main result that gives an adjunction inequality for embedded surfaces in certain 4-manifolds with contact boundary under a non-vanishing assumption on Bauer-Furuta type invariants.
Abstract: We give infinitely many knots in S^3 that are not smoothly H-slice (that is, bounding a null-homologous disk) in many 4-manifolds but they are topologically H-slice. In particular, we give such knots in punctured elliptic surfaces E(2n). In addition, we give obstructions to codimension-0 embedding of weak symplectic fillings with b_3=0 into closed symplectic 4-manifolds with b_1=0 and b_2^+=3 mod 4. We also show that any weakly symplectically fillable 3-manifold bounds a 4-manifold with at least two smooth structures. All of these results follow from our main result that gives an adjunction inequality for embedded surfaces in certain 4-manifolds with contact boundary under a non-vanishing assumption on Bauer-Furuta type invariants.
Journal: Algebraic & Geometric Topology.
Journal: Algebraic & Geometric Topology.
Abstract: We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4-manifolds.
Abstract: We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4-manifolds.
- On 3-manifolds that are boundaries of exotic 4-manifolds (Joint with John Etnyre and Hyunki Min) (arXiv)
Journal: Transactions of the American Mathematical Society.
Journal: Transactions of the American Mathematical Society.
Abstract: We give several criteria on a closed, oriented 3–manifold that will imply that it is the boundary of a (simply connected) 4–manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3–manifold, or contact 3–manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4–manifolds that admit infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.
Abstract: We give several criteria on a closed, oriented 3–manifold that will imply that it is the boundary of a (simply connected) 4–manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3–manifold, or contact 3–manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4–manifolds that admit infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.