arXiv subjects

Cecilia Karlsson

Publications and source records attributed to Cecilia Karlsson.

5 recordsLinked to original sources

arXiv2024

To compute orientations of Morse flow trees in Legendrian contact homology

Let $Λ$ be a closed, connected Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. Associated to $Λ$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $Λ$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $Λ$, which allows us to compute the Legendrian contact homology of $Λ$ via Morse theory. If $Λ$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $Λ$ can be defined with coefficients in $\mathbb Z$. In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of $Λ$ is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case $n=1$.

math.SG↗

arXiv2024

Orientations of Morse flow trees in Legendrian contact homology

Let $Λ$ be a closed, connected, spin Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. We give a coherent orientation scheme for the moduli space of rigid Morse flow trees of $Λ$, implying that the Legendrian contact homology of $Λ$ with integer coefficients can be computed using Morse flow trees. If $n>1$ then this orientation scheme can be computed with an algorithm which uses intersections of oriented flow manifolds in $M$ together with combinatorial data coming from the trees.

math.SG↗

arXiv2021

Legendrian contact homology for attaching links in higher dimensional subcritical Weinstein manifolds

Let $Λ$ be a link of Legendrian spheres in the boundary of a subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some geometrical assumptions, the computation of the Legendrian contact homology of $Λ$ can be reduced to a computation of Legendrian contact homology in 1--jet spaces. Since the Legendrian contact homology in 1--jet spaces is well studied, this gives a simplified way to compute the Legendrian contact homology of $Λ$. We restrict to the case when the attaching spheres of the subcritical handles of $X$ do not interact with each other, and we assume that there are no handles of index $n-1$. Moreover, we will only consider mod 2 coefficients for now. The more general situation will be addressed in a forthcoming paper. As an application we compute the homology of the free loop space of $\mathbb{CP}^2$.

math.SG↗

arXiv2019

A note on coherent orientations for exact Lagrangian cobordisms

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $Λ_\pm \subset J^1(M)$. It is well known that the Legendrian contact homology of $Λ_\pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization $\mathbb R \times J^1(M)$, and that $L$ induces a morphism between the $\mathbb Z_2$-valued DGA:s of the ends $Λ_\pm$ in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators.

math.SG↗

arXiv2016

Area preserving isotopies of self transverse immersions of S^1 in R^2

Let C and C' be two smooth self transverse immersions of S^1 into R^2. Both C and C' subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C' induces a 1-1 correspondence between the disks of C and C'. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C' is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C'. In this paper we show that this is also a sufficient condition.

math.SG↗