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Tobias Ekholm

Publications and source records attributed to Tobias Ekholm.

At least 55 records · Page 3Linked to original sources
arXiv2012

Legendrian knots and exact Lagrangian cobordisms

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $Λ_+$ and $Λ_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $Λ_+$ to that of $Λ_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

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arXiv2012

Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.

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arXiv2012

Knot contact homology

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $Λ_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $Λ_K$, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization $\R \times U^*\R^3$ with Lagrangian boundary condition $\R \times Λ_K$. We perform an explicit and complete computation of the Legendrian homology of $Λ_K$ for arbitrary links $K$ in terms of a braid presentation of $K$, confirming a conjecture that this invariant agrees with a previously-defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.

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arXiv2011

Effect of Legendrian Surgery

The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown that the results of this paper imply P. Seidel's conjecture equating symplectic homology with Hochschild homology of a certain Fukaya category.

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arXiv2010

Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n. As an application, we determine the classes represented by g_n in the cobordism group of immersions which is naturally identified with the stable 3-stem. It follows in particular that g_n represents a generator of the stable 3-stem if and only if n is divisible by 3.

math.GT↗
arXiv2009

A Duality Exact Sequence for Legendrian Contact Homology

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class αin CH^* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].

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arXiv2009

Regular homotopy and total curvature

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8πand we establish a non-injectivity result for local minima.

math.GT↗
arXiv2009

Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology

We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology. More precisely, if $L\subset X$ is an exact Lagrangian submanifold of an exact symplectic manifold with convex end $Λ\subset Y$, where $Y$ is a contact manifold and $Λ$ is a Legendrian submanifold, and if $L$ has empty concave end, then the linearized Legendrian contact cohomology of $Λ$, linearized with respect to the augmentation induced by $L$, equals the rational SFT of $(X,L)$. Following ideas of P. Seidel, this equality in combination with a version of Lagrangian Floer cohomology of $L$ leads us to a conjectural exact sequence which in particular implies that if $X=\C^{n}$ then the linearized Legendrian contact cohomology of $Λ\subset S^{2n-1}$ is isomorphic to the singular homology of $L$. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [6] in terms of the resulting isomorphism.

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arXiv2007

Isotopies of Legendrian 1-knots and Legendrian 2-tori

We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.

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arXiv2007

Cobordisms of fold maps and maps with prescribed number of cusps

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if there are cobordisms between their source- and target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism groups of fold maps of $(2k-1)$-manifolds into $(3k-2)$-space. Analogous cobordism semi-groups for arbitrary closed $(3k-2)$-dimensional target manifolds are endowed with Abelian group structures and described. Fold bordism groups in the same dimensions are described as well.

math.GT↗
arXiv2006

Rational Symplectic Field Theory over Z_2 for exact Lagrangian cobordisms

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positive end of a pair $(X'',L'')$, then there are natural morphisms from the spectral sequences of $(X',L')$ and of $(X'',L'')$ to the spectral sequence of $(X,L)$. As an application, we show that if $Λ\subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times\R,Λ_k^s\times\R)$, where $Y\times\R$ is the symplectization of $Y$ and where $Λ_k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $Λ$ subdivided into $k$ subsets, give Legendrian isotopy invariants of $Λ$.

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arXiv2005

Morse flow trees and Legendrian contact homology in 1-jet spaces

Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks in $T^\ast M$, with boundary on the projection of $L$ and asymptotic to the double points of this projection at punctures, provided $n\le 2$, or provided $n>2$ and the front of $L$ has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of $L$ in terms of Morse theory.

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arXiv2005

Legendrian Contact Homology in P X R

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of $\R^n$ and, more generally, invariants of self transverse immersions into $\R^n$ up to restricted regular homotopies. When $n=3$, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng

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arXiv2004

Invariants of Knots, Embeddings and Immersions via Contact Geometry

This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersions and embeddings using the Legendrian contact homology of the associated Legendrian submanifold.

math.GT↗
arXiv2004

Orientations in Legendrian Contact Homology and Exact Lagrangian Immersions

We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n+1)-space from Z_2 to Z. We demonstrate how the Z-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into C^n and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in Z_2 otherwise.

math.SG↗
arXiv2002

Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology

Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to be very rich. For example, infinite families of pairwise non-isotopic Legendrian $n$-spheres and $n$-tori, which are indistinguishable by means of previously known invariants, are constructed. In a sense, the definition of contact homology presented in this paper is a high dimensional analog of the work of Chekanov and others on Legendrian 1-knots in 3-space.

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