Searcharxiv⌕ Search

arXiv subjects

Frédéric Bourgeois

Publications and source records attributed to Frédéric Bourgeois.

14 recordsLinked to original sources
arXiv2025

On torsion in (bi)linearized Legendrian contact homology in dimension 3

Linearized Legendrian contact homology (LCH) and bilinearized LCH are important homological invariants for Legendrian submanifolds in contact geometry. For legendrian knots in $\mathbb{R}^3$, very little was previously known about the possibility of having torsion in these invariants when they are defined over integer coefficients. In this paper, we give properties of torsion that can appear in linearized LCH with integer coefficients, and also give the full geography of bilinearized LCH with integer coefficients.

math.SG↗
arXiv2025

Contact homology of contact manifolds and its applications

This is a survey of contact homology and its applications to the study of contact manifolds. It is a small tribute to Yasha Eliashberg's huge generosity with his countless explanations of his deep mathematical insights all along his career. It is also the author's wishful thinking that this text could be useful to students and young mathematicians for learning about some of the holomorphic curves based invariants in contact geometry.

math.SG↗
arXiv2022

Geography of bilinearized Legendrian contact homology

We study the geography of bilinearized Legendrian contact homology for closed, connected Legendrian submanifolds with vanishing Maslov class in 1-jet spaces. We show that this invariant detects whether the two augmentations used to define it are DGA homotopic or not. We describe a collection of graded vector spaces containing all possible values for bilinearized Legendrian contact homology and then show that all these vector spaces can be realized.

math.SG↗
arXiv2014

$S^1$-equivariant symplectic homology and linearized contact homology

We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduce a rigorously defined substitute for cylindrical/linearized contact homology based on an $S^1$-equivariant construction.

math.SG↗
arXiv2013

The Gysin exact sequence for $S^1$-equivariant symplectic homology

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.

math.SG↗
arXiv2013

Bilinearised Legendrian contact homology and the augmentation category

In this paper we construct an $\mathcal{A}_\infty$-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(Λ)$ and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Legendrian contact homology. Those are constructed as a generalisation of linearised Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order compositions map in the category and generalises the idea of [6] where an $\mathcal{A}_\infty$-algebra was constructed from one augmentation. This category allows us to define a notion of equivalence of augmentations when the coefficient ring is a field regardless of its characteristic. We use simple examples to show that bilinearised cohomology groups are efficient to distinguish those equivalences classes. We also generalise the duality exact sequence from [12] in our contest, and interpret geometrically the bilinearised homology in terms of the Floer homology of Lagrangian fillings (following [8]).

math.SG↗
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.

math.SG↗
arXiv2009

Fredholm theory and transversality for the parametrized and for the $S^1$-invariant symplectic action

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^1$-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^1$-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn's unique continuation theorem to a class of elliptic integro-differential inequalities of order two.

math.SG↗
arXiv2009

Towards a good definition of algebraically overtwisted

Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We give a proof that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish "algebraically overtwisted".

math.SG↗
arXiv2008

An exact sequence for contact- and symplectic homology

A symplectic manifold $W$ with contact type boundary $M = \partial W$ induces a linearization of the contact homology of $M$ with corresponding linearized contact homology $HC(M)$. We establish a Gysin-type exact sequence in which the symplectic homology $SH(W)$ of $W$ maps to $HC(M)$, which in turn maps to $HC(M)$, by a map of degree -2, which then maps to $SH(W)$. Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of $M$.

math.SG↗
arXiv2008

Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.

math.SG↗
arXiv2005

Contact homology and homotopy groups of the space of contact structures

Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higher dimensional contact manifolds and higher order homotopy groups.

math.SG↗
arXiv2003

Coherent Orientations in Symplectic Field Theory

We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. The orientations are determined by a certain choice of orientation at each closed Reeb orbit, that is similar to the orientation of the unstable tangent spaces of critical points in finite--dimensional Morse theory.

math.SG↗