Leipzig/Berlin Symplectic Homology Learning Seminar
Summer Semester 2010
Time and place
We plan to meet normally every two to three weeks on
Thursdays at the MaxPlanckInstitut
für Mathematik in den Naturwissenschaften,
Inselstrasse
22, Leipzig. There will also be one meeting at the
HumboldtUniversität, Adlershof campus,
Rudower
Chaussee 25, Berlin.
For most sessions there will be two talks, followed by a
dinner outing, to which all are welcome.
Schedule of talks (approximate)
Thursday April 29, 2010
15:1518:00
MPI, Room A01 
 Chris Wendl: Introduction to Symplectic Homology
(overview and discussion of topics to be covered)
expanded notes

Thursday May 20, 2010
14:3017:15
MPI, Room A01 
 Felix Schmäschke: Viterbo functoriality
 Klaus Mohnke: Autonomous Hamiltonians and MorseBott moduli
spaces

Thursday June 3, 2010
13:1517:30
HU Berlin, Rudower Chaussee 25 Room 2.009 
 Klaus Mohnke: Autonomous Hamiltonians and MorseBott moduli
spaces (conclusion)
 Matthias Schwarz: Product structures
 Stephan Mescher: Wrapped Floer cohomology

Thursday June 24, 2010
15:1518:30
MPI, Room A01 
 Slava Matveev: Computations via Lefschetz fibrations
 Oliver Fabert: The exact sequence for symplectic and
contact homology

Thursday July 8, 2010
13:1518:00
MPI, Room G10 
 Slava Matveyev: Computations via Lefschetz fibrations
(conclusion)
 Alex Krestiachine: The isomorphism for subcritical handle
attaching
 Chris Wendl: The effect of Legendrian surgery

Overview of the seminar
The term symplectic homology (or also cohomology, depending
on sign preferences) refers to a family of closely related
theories that adapt ideas from Floer homology into the setting of
symplectic manifolds with contact type boundary or noncompact symplectic
manifolds with cylindrical ends. The original formulation, introduced by
Floer, Hofer, Cieliebak and Wysocki in the early 1990s, was of a
"quantitative" nature, in that it
associated numerical invariants (defined via filtrations on Floer
homology) to compact domains in symplectic manifolds, inspired in part by
the theory of symplectic capacities. This theory has applications to
symplectic embedding questions, and was used for instance to give a
symplectic classification of polydisks in the standard Euclidean space,
and to show that the interior of a symplectic manifold with contact
boundary "sees the boundary"
in some sense. In recent years, a more "qualitative" theory has
become increasingly popular: first introduced by Viterbo, symplectic
homology in this form is an invariant of the symplectic "completion"
obtained by adding a cylindrical end to an exact symplectic manifold with
contact type boundary. Its definition leads naturally to an algebraic
variant of the Weinstein conjecture (and also its proof in many cases),
and it has more recently been shown to have deep relations to other
invariants of related objects, such as the linearized contact homology of
the boundary. Applications include dynamical results related to the
Weinstein conjecture, obstructions to exact Lagrangian embeddings, and the
existence of exotic Stein structures, among others.
The goal of our seminar will be essentially to learn what symplectic
homology is and what it's good for. More specifically, we will begin at
the beginning (with the quantitative version) and hope by the end to
understand some of the most recent developments, notably the preprint
by BourgeoisEkholmEliashberg computing the effect of Legendrian
surgery on symplectic homology, and the work of Seidel, Smith, McLean et
al expressing symplectic homology in terms of Lefschetz fibrations.
We assume the target audience for this seminar to be familiar with the
main ideas of Hamiltonian Floer homology on closed symplectic manifolds.
For other basic notions such as contact manifolds, Stein domains, contact
surgery and Lefschetz fibrations, we will attempt to introduce and explain
them as needed.
Literature list
The following is a (not very exhaustive) list of articles on
symplectic homology, with an attempt to group them into vaguely sensible
categories. Some of these are not well suited for seminar talks, but
nonetheless contain interesting ideas.
 Survey articles
 "Quantitative" symplectic homology (definitions and
applications related to symplectic capacities, embedding problems etc.)

Sketch in Section 6.6 of H. Hofer and E. Zehnder, Symplectic Invariants
and Hamiltonian Dynamics, Birkhäuser 1994.

A. Floer and H. Hofer,
Symplectic homology I: Open sets in C^{n}

K. Cieliebak, A. Floer and H. Hofer,
Symplectic homology II: a general construction

A. Floer, H. Hofer and K. Wysocki,
Applications of symplectic homology I

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki,
Applications of symplectic homology II: Stability of the action
spectrum

P. Biran, L. Polterovich and D. Salamon,
Propagation in Hamiltonian dynamics and relative symplectic
homology

K. Cieliebak, V. Ginzburg and E. Kerman,
Symplectic homology and
periodic orbits near symplectic submanifolds

D. Hermann,
Holomorphic
curves and Hamiltonian systems in an open set with restricted contacttype
boundary
 Basics of "qualitative" symplectic homology (definitions and
computations, relation to the algebraic Weinstein conjecture,
wrapped Floer cohomology)
 Relations to other invariants (contact homology, Rabinowitz Floer
homology, string topology)

F. Bourgeois and A. Oancea,
Symplectic homology,
autonomous Hamiltonians, and MorseBott moduli spaces

F. Bourgeois and A. Oancea,
An exact sequence for contact
and symplectic homology

F. Bourgeois and A. Oancea,
The Gysin exact sequence for
S^{1}equivariant symplectic homology

K. Cieliebak, U. Frauenfelder and A. Oancea,
Rabinowitz Floer homology and symplectic homology

A. Abbondandolo, M. Schwarz,
On the Floer homology of
cotangent bundles

A. Abbondandolo, M. Schwarz,
Floer homology of cotangent
bundles and the loop product

D. Salamon, J. Weber,
Title: Floer homology and the
heat flow
 Relations to Lefschetz fibrations, computations, applications to
exotic symplectic structures
 Computations via surgery
 Other (recent developments)
Videos of related talks
Here are some links to the downloadable videos of a few onehour talks of
interest from recent MSRI workshops:
 Mark McLean, Symplectic
Homology
A general introduction, including applications to exotic symplectic
structures on C^{2n} and exotic contact spheres.
 Kai Cieliebak, Some remarks on
symplectic and contact homology
Touches upon various topics including the relationship between Symplectic Homology and Contact Homology, Rabinowitz Floer
Homology, and (in the last 10 minutes) the isomorphism for subcritical handle attaching and its analogue in Contact
Homology.
 Tobias Ekholm, Legendrian
contact homology and symplectic homology in dimension four
Summarizes the main results of the recent preprint by BourgeoisEkholmEliashberg on Legendrian surgery, including some
examples of how to compute Legendrian contact homology.
For more information contact me,
Chris Wendl by sending
email to my surname (at) math (dot) huberlin (dot) de