|
Research
On s'habitue, c'est tout.
I'm mostly interested in higher dimensional contact geometry. In
contrast to the 3-dimensional case, this is a relatively unexplored
area with many basic questions unanswered. Recently though this
direction has attracted a lot of attention by several new and
unexpected discoveries (e.g. the notion of loose Legendrian knots and the existence of contact structures on any
almost contact manifold).
From 2010 to 2013, my research has been partially funded by the ANR
project Higher
dimensional contact topology.
Video lectures: I have recorded a few video lectures on the
proof of non-displaceability of exact Lagrangian submanifolds given
by Gromov in his paper from 85. I would be interested in any
feedback. You can watch them here:
Trivia: My research area is a subdomain of
Symplectic geometry. Apart from the mathematical
definition, I do not have any clue what "symplectic"
actually means, but on a trip to Asia, I learned that the Chinese
character for "symplectic" is
"辛", because that
character is pronounced sin-plectic. The standard
meaning of "辛" is "hot, spicy", so that at least in China I'm a hot
geometer...
Nonfillability and Overtwistedness
For many years my work was centered around non-fillability
questions, and ideas on how to generalize the notion of
overtwisted to higher dimensions.
The bLob
(being an acronym of "bordered Legendrian open book") was
such a conceivable generalization of the overtwisted
disk. First examples of
closed manifolds containing a bLob were found by
Francisco Presas, and soon afterwards it was discovered that
any contact manifold can
be modified to admit such a submanifold.
By now due to the work of
Borman-Eliashberg-Murphy a definition of
overtwistedness is known that
implies flexibility which settles the question
of overtwistedness in higher dimensions at least on the
theoretical level. An
article by Casals-Murphy-Eliashberg then shows that
many of the previous conjectural definitions of
overtwistedness are in fact equivalent to the one by
Borman-Eliashberg-Murphy.
The general opinion these days is that the most workable
definition of overtwistedness consists in regarding
stabilizations of overtwisted 3-manifolds
first introduced in the joint
paper with Fran Presas.
|
|
Group actions on contact manifolds, symplectic orbifolds, open
books
A second part of my work uses more geometric methods like group
actions, branched covers, resolution of orbifolds, and open
books.
Articles: 2019
|
The Bourgeois construction associates to every contact open book on a manifold V a contact structure on V×T2. We study in this article some of the properties of V that are inherited by V×T2 and some that are not. Giroux has provided recently a suitable framework to work with contact open books. In the appendix of this article, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this language.
|
2016
|
We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which are smoothly conjugate to each other but not by contactomorphisms.
|
|
By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group ΩSO∗(W) of any symplectically aspherical filling W, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in H∗(W). Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called "Lobs") as boundary conditions.
|
2013
|
We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n+1>3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two non-isomorphic contact structures become isomorphic after connect-summing with a manifold containing a plastikstufe.
|
|
For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher-dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.
|
2011
|
We prove several results on weak symplectic fillings of contact 3-manifolds,
including: (1) Every weak filling of any planar contact manifold can
be deformed to a blow up of a Stein filling. (2) Contact manifolds
that have planar torsion with an extra homological condition are not
weakly fillable -this gives many new examples of contact manifolds
without Giroux torsion that have no weak fillings. (3) Weak
fillability is preserved under splicing of contact manifolds along
symplectic pre-Lagrangian tori - this gives many new examples of
contact manifolds without Giroux torsion that are weakly but not
strongly fillable.
We establish the obstructions to weak fillings via two parallel
approaches using holomorphic curves. In the first approach, we
generalize the original Gromov-Eliashberg "Bishop disk" argument to
study the special case of Giroux torsion via a Bishop family of
holomorphic annuli with boundary on an "anchored overtwisted
annulus". The second approach uses punctured holomorphic curves, and
is based on the observation that every weak filling can be deformed
in a collar neighborhood so as to induce a stable Hamiltonian
structure on the boundary. This also makes it possible to apply the
techniques of Symplectic Field Theory, which we demonstrate in a
test case by showing that the distinction between weakly and
strongly fillable translates into contact homology as the
distinction between twisted and untwisted coefficients.
|
|
Helmut Hofer introduced in '93 a novel technique based on holomorphic
curves to prove the Weinstein conjecture. Among the cases where these
methods apply are all contact 3-manifolds (M,ξ) with
π2(M) ≠ 0. We modify Hofer's argument to
prove the Weinstein
conjecture for some examples of higher dimensional contact manifolds.
In particular, we are able to show that the connected sum with a real
projective space always has a closed contractible Reeb orbit.
|
2010
|
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".
|
2009
|
Symplectic reduction is a technique that can be used to decrease the dimension of Hamiltonian manifolds. Unfortunately, this only works under strong assumptions on the group action, and in general, even for a compact Lie group, the reduction at a coadjoint orbit that is transverse to the moment map will only yield a symplectic orbifold.
In this article, we show how to construct resolutions of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularize generic symplectic quotients for compact Lie group actions. More precisely, if a point in the Lie coalgebra is regular, that is, its stabilizer is a maximal torus, then we may apply our desingularization result. Regular elements of the Lie coalgebra are generic in the sense that the singular strata have codimension at least three.
Additionally, we show that even though the result of a symplectic cut is an orbifold, it can be modified in an arbitrarily small neighborhood of the cut hypersurface to obtain a smooth symplectic manifold.
|
|
In this paper we present a method to obtain resolutions of symplectic orbifolds
arising as quotients of pre-symplectic semi-free S1-actions. This includes, in
particular, all orbifolds arising from symplectic reduction of Hamiltonian S1-manifolds
at regular values, as well as all isolated cyclic orbifold singularities. As an
application, we show that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold. DISCLAIMER: Contrary to what is claimed in "Symplectic resolutions, Lefschetz property and formality" by G. Cavalcanti, M. Fernández, and V. Muñoz, their result does not render our construction useless. The main innovation in their resolution method consists in only studying orbifolds with isolated singularities and ignoring more difficult singularities.
|
2007
|
Recently Francisco Presas Mata constructed the first examples of closed contact
manifolds of dimension larger than 3 that contain a plastikstufe,
and hence are non-fillable. Using contact surgery on his examples we
create on every sphere S2n-1, n>1, an exotic contact
structure ξ- that allows us to embed a
plastikstufe. As a consequence, every closed contact
manifold M (with exception of S1) can
be converted by taking the connected sum of (M,ξ)#
(S2n-1,ξ-) into a
contact manifold that is not (semi-positively) fillable.
|
2006
|
In this article, we give a first prototype-definition of overtwistedness in higher
dimensions. According to this definition, a contact manifold is
called overtwisted if it contains a plastikstufe, a submanifold
foliated by the contact structure in a certain way. In three
dimensions the definition of the plastikstufe is identical to the
one of the overtwisted disk. The main justification for this
definition lies in the fact that the existence of a plastikstufe
implies that the contact manifold does not have a (semipositive)
symplectic filling.
|
|
In the first part of this paper the five-dimensional contact SO(3)-manifolds are
classified up to equivariant coorientation preserving contactomorphisms. The construction
of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an
application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn
manifold with exponents (k,2,2,2). The standard contact
structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined
in the article gives left-handed twists. In an appendix we also describe the classification of
five-dimensional contact SU(2)-manifolds.
|
2005
|
In this paper, we give an open book decomposition for the contact structures on some
Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition
uses right-handed Dehn twists as conjectured by Giroux.
|
All articles may also be found on
the
arXiv .
Other
-
|
The aim of this text is to give an accessible overview to some recent results
concerning contact manifolds and their symplectic fillings. In particular, we work out the
weakest compatibility conditions between a symplectic manifold and a contact structure on
its boundary to still be able to obtain a sensible theory (Chapter II), furthermore we prove
two results (Theorem A and B in Section I.4) that show how certain submanifolds inside a
contact manifold obstruct the existence of a symplectic filling or influence its topology. We
conclude by giving several constructions of contact manifolds that for different reasons do not
admit a symplectic filling.
|
-
|
The aim of this notes is to explain some non-fillability results in higher dimensional contact topology, which are closely related to the question of how to define overtwistedness. We start with an overview of some basic examples and theorems known so far, comparing them with analogous results in dimension three. We will also describe an easy construction of non-fillable manifolds by Fran Presas. Then we will explain how to use holomorphic curves with boundary to prove the non-fillability results stated earlier. No a priori knowledge of holomorphic curves will be required; though many properties will only be quoted.
|
-
|
The first part of the thesis contains an exposition of S1-actions on
3-manifolds, including the classification of 3-dimensional contact S1-manifolds.
The known classification results of 4-dimensional symplectic SO(3)- and SU(2)-manifolds are
reproved, before the main result, the classification of 5-dimensional contact SO(3)-manifolds is worked out.
|
-
|
My diploma thesis consisted in giving a (finite) set of generators for the Poisson
algebra on the sphere, and a set of necessary relations (similarly
to a finitely presented group).
|
People
References
-
Yakov Eliashberg
Classification of
overtwisted contact structures on 3-manifolds
Invent. Math. 98 (1989), no. 3, 623-637.
-
Francisco Presas
A class of
non-fillable contact structures
Geom. Topol. 10 (2006), 1373-1389.
-
Emmy Murphy
Loose Legendrian Embeddings in High Dimensional Contact Manifolds
arXiv (2012) 1201.2245
-
Matthew Strom Borman
Yakov Eliashberg
Emmy Murphy
Existence and classification of overtwisted contact structures in all dimensions
Acta Math. 215 (2015), no. 2, 281-361
-
Roger Casals
Emmy Murphy
Francisco Presas Mata
Geometric criteria for overtwistedness
arXiv math.SG:1503.06221
|