Non-loose torus knots
We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on $S^3$.
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arXiv subjects
Anubhav Mukherjee
Publications and source records attributed to Anubhav Mukherjee.
Corks for exotic diffeomorphisms
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 $S^2 \times S^2$, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of $S^2 \times S^2$, 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.
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Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed $4$-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist.
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The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory
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 $\mathbb{Z}/p$-equivariant Seiberg--Witten--Floer homology of 3-manifolds.
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On four-dimensional Dehn twists and Milnor fibrations
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 $\mathbb{R}^4$'s and contractible manifolds.
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Diffeomorphisms of 4-manifolds with boundary and exotic embeddings
We define family versions of the invariant of 4-manifolds with contact boundary due to Kronheimer and Mrowka and use these to detect exotic diffeomorphisms of 4-manifolds with boundary. Further, we show the existence of the first example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
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Complete Riemannian 4-manifolds with uniformly positive scalar curvature
We obtain topological obstructions to the existence of a complete Riemannian metric 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 $\mathbb{R}^4$'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.
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Existence and construction of non-loose knots
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.
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One stabilization is not enough for closed knotted surfaces
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.
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Exotic codimension-1 submanifolds in 4-manifolds and stabilizations
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$. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.
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An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology
Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, 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 the boundaries of the punctured elliptic surfaces $E(2n)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed symplectic 4-manifolds with $b_1=0$ and $b_2^+\equiv 3$ mod $4$. From here we prove a Bennequin type inequality for symplectic caps of $(S^3,ξ_{std})$. We also show that any weakly symplectically fillable $3$-manifold bounds a $4$-manifold with at least two smooth structures.
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On 3-manifolds that are boundaries of exotic 4-manifolds
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 admits 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.
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Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture
We establish the existence of a pair of exotic surfaces in a punctured $K3$ 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 $\mathrm{Pin}(2)$-equivariant family Bauer--Furuta invariant for a diffeomorphism on $S^{2}\times S^{2}$ is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on $S^{4}$ or $S^{2}\times S^{2}$. Furthermore, our theorem also applies to certain exotic loops of diffeomorphisms on $S^{4}$ (as recently discovered by Watanabe) and show that these loops have trivial family Bauer--Furuta invariants. En route, we observe a curious element in the $\mathrm{Pin}(2)$-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on $S^{4}$.
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A note on embeddings of 3-manifolds in symplectic 4-manifolds
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.
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