...
Cotangent
...
bundles
...
and
...
Viterbo
...
Functoriality
...
In
...
the
...
morning,
...
we
...
will
...
go
...
through
...
one
...
method
...
of
...
computing
...
the
...
symplectic
...
cohomology
...
of
...
cotangent
...
bundles.
...
- Show that the unit cotangent bundle is a Liouville domain. Explain the correspondence between closed geodesics and Reeb orbits. State the isomorphism between symplectic cohomology and the homology of the free loop space (Theorem 3.5 in [Seidel]). Give the statement only with
Latex $\mathbb{Z} / 2\mathbb{Z}$
...
coefficients.
...
- Define
...
- the
...
- Morse
...
- homology
...
- of
...
- the
...
- free
...
- loop
...
- space
...
- following
...
- Section
...
- 2
...
- of
...
- [Abbondandolo-Schwarz
...
- ].
...
- Explain
...
- the
...
- Abbondandolo-Schwarz
...
- construction
...
- of
...
- an
...
- isomorphism
...
- between
...
- symplectic
...
- cohomology
...
- and
...
- the
...
- homology
...
- of
...
- the
...
- free
...
- loop
...
- space,
...
- following
...
- Section
...
- 3 of [Abbondandolo-Schwarz
...
- ].
...
- Make
...
- sure
...
- to
...
- draw
...
- the
...
- pictures
...
- that
...
- are
...
- associated
...
- to
...
- these
...
- maps
...
- (i.e.
...
- counting
...
- pseudo-holomorphic
...
- half-cylinders
...
- in
...
- the
...
- cotangent
...
- bundle
...
- followed
...
- by
...
- gradient
...
- flow
...
- lines
...
- in
...
- the
...
- base).
...
In
...
the
...
afternoon,
...
- Define
...
- the
...
- notion
...
- of
...
- a
...
- Liouville
...
- subdomain.
...
- Give
...
- examples
...
- coming
...
- from
...
- exact
...
- Lagrangian
...
- embeddings,
...
- and
...
- from
...
- Weinstein
...
- handle
...
- attachment.
...
- Explain
...
- the
...
- construction
...
- of
...
- Section
...
- 2
...
- in
...
- [Viterbo
...
- ]
...
- associating
...
- maps
...
- on
...
- symplectic
...
- cohomology
...
- to
...
- Liouville
...
- subdomains.
...
- Warning:
...
- what
...
- Viterbo denotes
Latex $FH^*$
...
is
...
- the
...
- dual
...
- of
...
- the
...
- theory
...
- Seidel
...
- denotes
...
Latex
...
$SH^*$
...
. To
...
- keep
...
- this
...
- talk
...
- consistent
...
- with
...
- the
...
- others,
...
- you
...
- should
...
- dualise
...
- Viterbo's
...
- statements.