13. The monodromy diffeomorphism of weighted singularities and Seiberg-Witten theory ( Joint with Hokuto Konno, Jianfeng Lin, and Juan Muñoz-Echániz) (arXiv)
13. The monodromy diffeomorphism of weighted singularities and Seiberg-Witten theory ( Joint with Hokuto Konno, Jianfeng Lin, and Juan Muñoz-Echániz) (arXiv)
Abstract: We prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of Z/p-equivariant Seiberg--Witten--Floer homology of 3-manifolds.
Abstract: We prove that the monodromy diffeomorphism of a complex 2-dimensional isolated hypersurface singularity of weighted-homogeneous type has infinite order in the smooth mapping class group of the Milnor fiber, provided the singularity is not a rational double point. This is a consequence of our main result: the boundary Dehn twist diffeomorphism of an indefinite symplectic filling of the canonical contact structure on a negatively-oriented Seifert-fibered rational homology 3-sphere has infinite order in the smooth mapping class group. Our techniques make essential use of analogues of the contact invariant in the setting of Z/p-equivariant Seiberg--Witten--Floer homology of 3-manifolds.
12. On four-dimensional Dehn twists and Milnor fibrations ( Joint with Hokuto Konno, Jianfeng Lin, and Juan Muñoz-Echániz) (arXiv) [ YouTube video ]
12. On four-dimensional Dehn twists and Milnor fibrations ( Joint with Hokuto Konno, Jianfeng Lin, and Juan Muñoz-Echániz) (arXiv) [ YouTube video ]
Abstract: We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic R^4's and contractible manifolds.
Abstract: We study the monodromy diffeomorphism of Milnor fibrations of isolated complex surface singularities, by computing the family Seiberg--Witten invariant of Seifert-fibered Dehn twists using recent advances in monopole Floer homology. More precisely, we establish infinite order non-triviality results for boundary Dehn twists on indefinite symplectic fillings of links of minimally elliptic surface singularities. Using this, we exhibit a wide variety of new phenomena in dimension four: (1) smoothings of isolated complex surface singularities whose Milnor fibration has monodromy with infinite order as a diffeomorphism but with finite order as a homeomorphism, (2) robust Torelli symplectomorphisms that do not factor as products of Dehn--Seidel twists, (3) compactly supported exotic diffeomorphisms of exotic R^4's and contractible manifolds.
11. Complete Riemannian 4-manifolds with uniform positive scalar curvature (Joint with Otis Chodosh, and Davi Maximo) (arXiv) [ YouTube video ]
11. Complete Riemannian 4-manifolds with uniform positive scalar curvature (Joint with Otis Chodosh, and Davi Maximo) (arXiv) [ YouTube video ]
Abstract: We obtain topological obstructions to the existence of complete Riemannian with uniformly positive scalar curvature on certain (non-compact) 4-manifolds. In particular, such a metric on the interior of a compact contractible 4-manifold uniquely distinguishes the standard 4-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic R^4's that do not admit such a metric and that any (non-compact) tame 4-manifold has a smooth structure that does not admit such a metric.
Abstract: We obtain topological obstructions to the existence of complete Riemannian with uniformly positive scalar curvature on certain (non-compact) 4-manifolds. In particular, such a metric on the interior of a compact contractible 4-manifold uniquely distinguishes the standard 4-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic R^4's that do not admit such a metric and that any (non-compact) tame 4-manifold has a smooth structure that does not admit such a metric.
10. Corks for exotic diffeomorphisms (Joint with Vyacheslav Krushkal, Mark Powell, and Terrin Warren ) (arXiv)
10. Corks for exotic diffeomorphisms (Joint with Vyacheslav Krushkal, Mark Powell, and Terrin Warren ) (arXiv)
Abstract: We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of S^2 X S^2, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of S^2 X S^2, we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.
Abstract: We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of S^2 X S^2, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of S^2 X S^2, we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.
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)
Journal: International Mathematics Research Notices (IMRN).
Journal: International Mathematics Research Notices (IMRN).
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 ) (arXiv)
8. One stabilization is not enough for closed knotted surfaces ( Joint with Kyle Hayden and Sungkyung Kang ) (arXiv)
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)
Journal: Mathematische Annalen.
Journal: Mathematische Annalen.
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 ]
Journal: Journal of Association for Mathematical Research (JAMR).
Journal: Journal of Association for Mathematical Research (JAMR).
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 ] Journal: Advances in Mathematics.
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 ] Journal: Advances in Mathematics.
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.