Floer and Novikov homology, contact topology and related topics
April 21 -- 24, 2014
Seminar Room B
Kavli IPMU, The University of Tokyo
Access to Kavli IPMU
Poster
List of speakers :
Manabu Akaho (Tokyo Metropolitan University)
Dan Burghelea (Ohio State University)
River Chiang (National Cheng Kung University, Taiwan)
Vincent Colin (Université de Nantes)
Mihai Damian (Université de Strasbourg)
Urs Frauenfelder (Seoul National University)
Kei Irie (Kyoto University)
Tetsuya Ito (Kyoto University)
Otto van Koert (Seoul National University)
Fabiola Manjarrez-Gutierrez (CIMAT)
Sheila Sandon (Université de Strasbourg)
Yasha Savelyev (ICMAT, Madrid)
Tadayuki Watanabe (Shimane University)
Program
[PDF file]
Monday, April 21
10:30 -- 12:00 Frauenfelder
13:30 -- 15:00 Akaho
15:30 -- 17:00 Damian
Tuesday, April 22
10:30 -- 12:00 Burghelea
13:30 -- 15:00 Manjarrez-Gutierrez
15:30 -- 16:30 Ito
16:40 -- 17:40 Watanabe
19:00 -- Workshop Dinner
Wednesday, April 23
10:30 -- 12:00 van Koert
13:30 -- 15:00 Sandon
15:30 -- 17:00 Irie
Thursday, April 24
10:30 -- 12:00 Savelyev
13:30 -- 15:00 Chiang
15:30 -- 17:00 Colin
Organizers:
Toshitake Kohno, Kaoru Ono, Andrei Pajitnov, Kyoji Saito
Titiles and Abstracts
Manabu Akaho (Tokyo Metropolitan University)
title: On Morse homology of manifolds with boundary
abstract: In this talk we explain Morse homology of manifolds with
boundary, motivated by Floer theory of Lagrangian submanifolds with
concave end. First we observe Riemannian metrics and Morse functions on
manifolds with boundary whose gradient vector fields are tangent to the
boundary. Then we discuss their unstable manifolds and define our Morse
homology, which is isomorphic to the absolute singular homology.
Moreover we consider product on our Morse complexes, which satisfies the
Leibnitz rule. Finally we mention some application to Floer theory of
Lagrangian submanifolds.
Dan Burghelea (Ohio State University)
title: A (computer friendly) alternative to Morse Novikov theory for angle
valued maps.
abstract:
Morse-Novikov theory is a useful tool to analyze the dynamics of a large class
of vector fields
(which admit a Lyapunov closed one form) and relate rest points,
trajectories between rest points and closed trajectories to the algebraic topology
of the underlying manifold, at least in generic situation. A generic vector
field
which admits a closed one form as Lyapunov admits also an angle valued map as
Lyapunov.
We present a computer friendly alternative to the Morse-Novikov theory for an
angle valued map and implicitly of Morse theory which works for a much larger
class of spaces and maps (compact ANR and tame maps) which, from the point of
view of algebraic topology, does as much as the smooth Morse-Novikov. It is based
on invariants (in case of simplicial complexes computable by algorithms of the same
complexity as of the ones which calculate Betti numbers).
The invariants proposed provide refinements of the familiar topological invariants
(Novikov and standard Betti numbers, monodromy) and reveal stability properties
invisible in the smooth theory.
The lecture is based on joint work with T. Dey and S. Haller and was influence by
the work on Morse-Novikov theory and closed trajectories of Pajitnov, Hutchings-Lee
and Burghelea-Haller.
River Chiang (National Cheng Kung University, Taiwan)
title: Examples of higher dimensional non-fillable contact manifolds
abstract: A bordered Legendrian open book, introduced by Massot, Niederkruger, and Wendl, is a higher dimensional analog of an overtwisted disk. Its existence in a contact manifold obstructs fillability. In this talk, we would discuss an extension of their construction of such objects using equivariant means. This is a joint work in progress with Y. Karshon.
Vincent Colin (Univ. de Nantes)
title: Higher-dimensional Heegaard Floer homology
abstract: In a work in progress with Ko Honda, we extend the definition of
the hat version of Heegaard Floer homology to contact manifolds of
arbitrary odd dimension using higher-dimensional
open book decompositions and the theory of Weinstein domains. This also
suggests a reformulation and an extension
of Symplectic Khovanov homology to links in arbitrary 3-manifolds.
Mihai Damian (Univ. de Strasbourg)
title: Lifted Floer homology and topology of monotone Lagrangian
submanifolds
abstract:
We establish a new version of Floer homology for monotone Lagrangian submanifolds
which is related to the homology of the universal cover and more generally to
Novikov homology. We get some constraints on the topology of monotone Lagrangian
submanifolds in C^n and in CP^n. In particular we show that there are some
Lagrangians in these manifolds which do not admit monotone embeddings.
Urs Frauenfelder (Seoul National University)
title: Spicy Hopf algebras and growth of Reeb chords
abstract: This is joint work with Felix Schlenk. We considerthe homology growth of
based loop spaces of closed manifolds whose universal cover is not homotopy
equivalent to a finite CW complex.The homology of the based loop space has the
structure of a Hopf algebraand we take advantage of the action of the fundamental
group on it.
Kei Irie (Kyoto University)
title: Hofer-Zehnder capacity, symplectic homology and loop product
abstract:
Hofer-Zehnder(HZ) capacity is a quantitative invariant of symplectic
manifolds, which reflects behavior of Hamiltonian flows on manifolds.
Symplectic homology is a version of Floer homology, which is defined for
convex symplectic manifolds (e.g. cotangent bundles).
We give estimates of HZ capacity using symplectic homology, in particular
its product structure. An application to cotangent bundles which involves
computations of the Chas-Sullivan loop product is also presented.
Tetsuya Ito (Kyoto University)
title: Overtwisted discs in planar open books
abstract:
We show that overtwisted disc in a planar open book can be put in a
topologically nice position so that each the intersection of disc and
pages is essential. This provides a tightness criterion based on
topological method which generalizes Bennequin's proof of the tightness of
the standard contact structure of S^3.
Otto van Koert (Seoul National University)
title: Fractional twists and
invariant contact structures
abstract:
We define fractional twists, a
generalization of Dehn twists, and discuss their role in constructing
contact structures that are invariant under a circle action. We give
some criteria to detect whether these fractional twists are
symplectically isotopic to the identity, and then discuss the
difference between right- and left-handed twists. We shall show that
left-handed twists in an open book often give rise to so-called
algebraically overtwisted contact manifolds. In particular, such
manifolds are not symplectically fillable. This is joint work with River Chiang and
Fan Ding.
Fabiola Manjarrez-Gutierrez (CIMAT)
title: Additivity of Morse-Novikov number of a-small knots
abstract:
A knot is a-small if its exterior does not contain closed incompressible
surfaces disjoint from some incompressible Seifert surface for the knot.
The Morse-Novikov number of a knot is the minimal number of critical points of a
Morse map of the knot exterior to the circle. This concept was introduced
by Pajitnov, Rudolph and Weber, they also proved that the Morse-Novikov
number is subadditive under connected sum of knots.
Given a Morse map of the knot exterior to the circle we can reorganize the
critical points to obtain a decomposition of the knot exterior in such a
way that the the preimages of regular values are Seifert surfaces which
are alternatevely incompressible and weakly incompressible. This notion is
known as circular handle decomposition.
Using circular handle decomposition of knot exteriors we prove that
Morse-Novikov number is additive for a-small knots.
Sheila Sandon (Univ. de Strasbourg)
title: On positive loops of contactomorphisms
abstract: A contact isotopy is said to be positive if it moves every point
in a
direction positively transverse to the contact distribution. In 2000
Eliashberg and
Polterovich noticed that this notion induces for certain contact manifolds
(which
are called orderable) a partial order on the universal cover of the
contactomorphism
group. Orderability of a contact manifold turns out to be sensitive to the
underlying topology (for example the standard real projective space is
orderable
while the standard sphere is not), and was later discovered by Eliashberg,
Kim and
Polterovich to be deeply related to a non-squeezing phenomenon in contact
topology.
In my talk I will review these topics and discuss the fact that
orderability is
equivalent to the non-degeneracy of a natural bi-invariant metric on the
universal
cover of the contactomorphism group (joint work with V. Colin). I will
also present
a recent result about small positive loops on overtwisted contact
manifolds (joint
with R. Casals and F. Presas).
Yasha Savelyev (ICMAT, Madrid)
title: Global Fukaya category
abstract: We introduce a kind of Fukaya category for a smooth Hamiltonian
fibration over a general smooth manifold, and discuss an application to
Hofer geometry. This story is intimately connected to Toen's derived
Morita
theory, and theory of quasi-categories after Lurie and Joyal. This gives a
nice geometric context for some high powered abstraction in category
theory, and raises a number of interesting questions and possibilities.
Tadayuki Watanabe (Shimane University)
title:
Morse theory and Lescop's equivariant propagator
abstract:
For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the gradient $\xi$ of a fiberwise Morse function on $M$, we introduce the notion of ``amidakuji path'' on $M$. An amidakuji path is a piecewise smooth path on $M$ consisting of edges each of which is either a part of a critical locus of $\xi$ or an integral curve of $\xi$. Counting closed amidakuji paths with signs gives the Lefschetz zeta function of $M$. The ``moduli space" of amidakuji paths on $M$ gives Lescop's equivariant propagator, which can be used to define $\mathbb{Z}$-equivariant version of Chern--Simons perturbation theory for $M$.