\def\R{{\bf R}}
\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\T{{\bf T}}
\def\Q{{\bf Q}}
\def\Crit{\mathop{\rm Crit}}
\def\id{\mathop{\rm id}}
\def\ind{\mathop{\rm ind}}
\def\dom{\mathop{\rm dom}}
\def\SU{\mathop{\rm SU}}
\def\SO{\mathop{\rm SO}}
\def\Sp{\mathop{\rm Sp}}
\def\U{\mathop{\rm U}}
\def\Nov{\mathop{\rm Nov}}
\def\Area{\mathop{\rm Area}}
\def\ev{\mathop{\rm ev}}
\def\coker{\mathop{\rm coker}}
\def\im{\mathop{\rm im}}
\def\SF{\mathop{\rm SF}}
\def\Fix{\mathop{\rm Fix}}
\def\Symp{\mathop{\rm Symp}}
\def\CZ{\mathop{\rm CZ}}
\def\Spec{\mathop{\rm Spec}}
\def\CF{\mathop{\rm CF}}
\def\HF{\mathop{\rm HF}}
\def\CH{\mathop{\rm CH}\nolimits}
\def\HC{\mathop{\rm HC}}
\let\section\beginsection
\leftline{\font\title=cmr17\title Michael Hutchings.  Floer theory.}
\bigskip
\leftline{Notes by Dmitri Pavlov, pavlov@math.}

\section Morse theory.

Suppose $X$ is a finite-dimensional smooth manifold and $f\colon X\to\R$ is a Morse function.
Morse theory relates the topology of~$X$ and critical points and gradient flow of~$f$.
For example, if $f$ is a Morse function, then the number of critical points is at least
$\sum_i\dim H_i(X)$.
We can also describe the cup product, characteristic classes, Reidemeister torsion etc.

We are interested in this because this theory has infinite-dimensional generalizations.
A classical example is the loop space of~$X$: $LX$.
Choose a Riemannian metric on~$X$.
Then we can define a function $f\colon LX\to\R$: $f(\gamma)=\int_{S^1}|\gamma'(t)|^2dt$.
This function is called energy.
Critical points of~$f$ are closed geodesics parametrized at constant speed.
It follows that for any metric on~$S^2$ there are at least three closed geodesics.

Floer theory has many versions:
symplectic Floer theory studies invariants of symplectomorphisms of a symplectic manifold.

\proclaim Theorem.  (Arnold Conjecture.)
If $M$ is a closed symplectic manifold and $\phi$ is a Hamiltonian symplectomorphism
with nondegenerate fixed points, then the number of fixed points is at least $\sum_i\dim H_i(M)$.

Much weaker Lefschetz theorem says that the number of fixed points is at least
$\left|\sum_i(-1)^i\dim H_i(M)\right|$.

The strategy for proving Arnold conjecture is to define Floer homology of a symplectomorphism
as the homology of some chain complex generated by fixed points.

The second step is to show invariance under Hamiltonian isotopies.
And the third step is to show that Floer homology of $\id_M$ is $H_*(M)$.

First we shall do finite-dimensional Morse theory.  Then we shall discuss symplectic Floer
homology.  Then we shall apply this theory to obtain some topological invariants of 3-manifolds.
We take a 3-dimensional manifold, construct a functional on some infinite-dimensional space
and take Floer homology of it.  An interesting example is Seiberg-Witten homology.
There is a (mostly equivalent) Hegaard-Floer homology.  A third example is embedded contact homology.

Symplectic field theory is related to this stuff.  SFT gives invariants of contact manifolds
and Legendrian knots.  A contact form on a closed oriented 3-manifold is a 1-form~$\lambda$
such that $\lambda\wedge d\lambda\ne0$.
The kernel of~$\lambda$ is called a contact structure.
Associated to~$\lambda$ is the Reeb vector field~$R$ defined by $d\lambda(R,x)=0$ and $\lambda(R)=1$.

\proclaim Weinstein Conjecture.  For any closed oriented 3-manifold $Y$ and any contact form
$R$ has a closed orbit.  Proved by Taubes using Floer homology ideas.

End of overview.

Finite-dimensional Morse theory.

Suppose that $X$ is a closed smooth manifold, $f\colon X\to\R$ is a smooth function.
A critical point of~$f$ is a point~$p\in X$ such that $df(p)=0$.  Denote the set of all
critical points by $\Crit(f)$.
If $p$ is a critical point, then we define the Hessian $H(f,p)\colon T_pX\otimes T_pX\to\R$
as follows: if $\nabla$ is a any connection on cotangent bundle, then 
$H(f,p)=\nabla(df)_p\colon T_pX\to T_p^*X$.
Equivalently, we can regard $df$ as a section $df\colon X\to T^*X$.
Then $H(f,p)$ is the composition $T(df)_p\colon T_pX\to T_{(p,0)}(T^*X)=T_pX\oplus T_p^*X\to T_p^*X$.
Critical point~$p$ is called nondegenerate if $H(f,p)$ is nondegenerate.
Equivalently, the graph of~$df$ is transverse to the zero section at~$(p,0)$.
The index of~$p$ is the number of negative eigenvalues counted with multiplicities.
Note that nondegenerate critical points are isolated.
Morse lemma says that if $p$ is a nondegenerate critical point of index~$i$, then
there is a chart centered at~$p$ such that $f=f(p)+H(f,p)$ in this chart.

A function $f$ is called a Morse function if all of its critical points are nondegenerate.

\proclaim Proposition.  The set of Morse functions is open and dense in the space of smooth functions
$X\to\R$ in the smooth topology.

\proclaim Example.  Height function on torus.

\proclaim Non-example.  Height function on torus lying horizontally.

A Morse-Bott function is a function whose critical points have nondegenerate Hessian
and the set of critical functions is a nice submanifold.

\proclaim Classical Morse theory.  If $f$ is a Morse function and the values of~$f$
at critical points are all distinct.
If $a$ is not a critical value, define $X_a=f^{-1}(-\infty,a]$.
This is a smooth manifold with boundary $\partial X_a=f^{-1}(a)$.

\proclaim Basic lemmas.  If the interval $[a,b]$ contains no critical values, then $X_a$
is diffeomorphic to $X_b$.  If the interval $[a,b]$ contains a single critical value
and its index is~$i$, then $X_b$ is obtained from~$X_a$ by attaching an $i$-handle
$D^i\times D^{n-i}$ along $S^{i-1}\times D^{n-i}$.

The simplest application of this theory is Morse inequalities: if $c_i$ is the number
of critical points of index~$i$ and $b_i=\dim H_i(X)$, then
for each~$i$ we have $c_i-c_{i-1}+\cdots+(-1)^ic_0\ge b_i-b_{i-1}+\cdots+(-1)^ib_0$.

Morse complex and Morse homology.

Suppose that $X$ is a closed smooth manifold and $f\colon X\to\R$ is a Morse function.
Choose a generic Riemannian metric on~$X$.  Hence we have $\nabla f$, the gradient of~$f$.
Idea: Define a chain complex which is generated by critical points of~$f$ and whose
differential counts flow lines of $-\nabla f$ between critical points.  This complex
leads to Morse homology $H^M_*(X,f,g)$.

\proclaim Fundamental theorem.  $H^M_*(X,f,g)=H^s_*(X)$.

\proclaim Exercise.  If $C$ is a chain complex, $c_i=\dim C_i$ and $b_i=\dim H_i$,
then $c_i-\cdots+(-1)^ic_0\ge b_i-\cdots+(-1)^ib_0$ for all~$i$.

Let $\xi=-\nabla f$, let $\phi_s\colon X\to X$ for $s\in\R$ denote the flow
generated by~$\xi$.
If $p$ is a critical point, define the descending manifold of~$p$ as $D(p)=\{x\in X\mid\lim_{s\to-\infty}\phi_s(x)=p\}$ and
the ascending manifold of~$p$ as $A(p)=\{x\in X\mid\lim_{s\to\infty}\phi_s(x)=p\}$.

\proclaim Proposition.  Let $p$ be an index~$i$ critical point.  Then $D(p)$ is an embedded
open disc of dimension~$i$.  Also $T_pD(p)$ is the negative eigenspace of
$H(f,p)\colon T_pX\to T^*_pX\to T_pX$.
Likewise, $A(p)$ is an embedded open disc of dimension $\dim(X)-i$ and $T_pA(p)$ is the
positive eigenspace.

\proclaim Definition.  The pair $(f,g)$ is called Morse-Smale if $f$ is a Morse function and
for any two critical points $p$~and~$q$ the manifold $D(p)$ is transversal to $A(q)$.

\proclaim Proposition.  Fix a Morse function~$f$.  If $g$ is generic, then $(f,g)$ is Morse-Smale.
Generic means open and dense.

From now on we assume that $(f,g)$ is Morse-Smale.

If $p$ and $q$ are critical points a flow line of $\xi=-\nabla f$ from~$p$ to~$q$ is a path
$\gamma\colon\R\to X$ such that $\lim_{s\to-\infty}\gamma(s)=p$ and $\lim_{s\to\infty}\gamma(s)=q$
and $\gamma'=\xi$.  Flow lines are not unique.

Note: Morse cohomology is obtained by replacing $-\nabla f$ with $\nabla f$.

Note that $\R$ acts on the set of flow lines from~$p$ to~$q$ by precomposition with translations,
Define $m(p,q)$ as the set of all flow lines modulo this action.

The map $\gamma\to\gamma(0)$ identifies the set of all flow lines from~$p$ to~$q$ with
$D(p)\cap A(q)$.  The Morse-Smale condition implies that if $p\ne q$ then $m(p,q)$ is a manifold
of dimension $\ind(p)-\ind(q)-1$.

Orientation of $m(p,q)$.  Choose an orientation of $D(p)$ for each~$p$.  Given $[\gamma]\in m(p,q)$
let $x=\gamma(0)\in D(p)\cap A(q)$.  We have $T_xD(p)=T_x(D(p)\cap A(q))\oplus(T_xX/T_xA(q))$
(follows from transversality).  The last space is isomorphic to
$T_{[\gamma]}m(p,q)\oplus T_x\Im(\gamma)\oplus T_qD(q)$.
Orient $T_{[\gamma]}m(p,q)$ so that this isomorphism is orientation preserving.

\proclaim Definition of Morse complex $C_*(X,f,g)$.  Denote by $C_i$ the free abelian
group generated by index~$i$ critical points.
A differential $\partial_i\colon C_i\to C_{i-1}$ is defined as follows: $\partial p
=\sum_qq\cdot\#m(p,q)$.  Note that $m(p,q)$ has dimension~0.  The number $\#m(p,q)$ counts
points with orientation (positive orientation gives 1, negative gives $-1$).

\proclaim Theorem.  If $X$ is closed and $(f,g)$ is Morse-Smale then for any two critical
points $p$~and~$q$ the moduli space $m(p,q)$ has a natural compactification to a manifold
with corners $\bar m(p,q)$ whose codimension~$k$ stratum is $\bar m(p,q)_k
=\cup_{r_i\in\Crit(f)}m(p,r_1)\times m(r_1,r_2)\times\cdots m(r_{k-1},r_k)\times m(r_k,q)$.
Here $p$, $r_1$, \dots, $r_k$, and $q$ must be all different.

\proclaim Corollary.  If $\ind(p)-\ind(q)=1$, then $m(p,q)$ is compact and has dimension~0,
hence it is finite and the Morse differential is well-defined.

\proclaim Corollary.  If $\ind(p)-\ind(q)=2$, then $\bar m(p,q)$ is a compact oriented
manifold with boundary $$\bigcup_{r\colon\ind(p)-\ind(r)=1}m(p,r)\times m(r,q).$$

Now we shall prove that $\partial^2=0$.  Suppose that $\ind(p)-\ind(q)=2$.  Then $\bar m(p,q)$
is a compact oriented manifold with boundary $\cup_{r\colon\ind(p)-\ind(r)=1}m(p,r)\times m(r,q)$.
Then $\langle\partial^2p,q\rangle=\sum_{r\colon\ind(p)-\ind(r)=1}\langle\partial p,r\rangle\cdot\langle\partial r,q\rangle=\#\partial\bar m(p,q)=0$.

In general, in good case we can get a compactified moduli space of flow lines by adding broken
flow lines.

Decomposition into (compactifications of)
descending boundary manifolds gives us a CW-decomposition and
the Morse boundary map is the cellular boundary map.
Note that the closure of descending manifold need not to be a closed ball.
The sign of the differential depends on choice of orientations of descending manifolds.

First we shall show that Morse homology does not depend on $f$~and~$g$.
If $(f',g')$ is another Morse-Smale pair, then there is canonical isomorphism
between two Morse homologies.

We can define Morse complex without making any orientation choices.
For every critical point we add two different elements in the set of generators,
which correspond to different orientations.  Then we say that their sum is zero.
Henceforth we shall usually omit orientation choices from the notation.

There are two ways to prove that Morse homology is independent of metric and Morse function.
One of them is called continuation maps.
Suppose $(f_0,g_0)$ and $(f_1,g_1)$ are two Morse-Smale pairs on~$X$.
Let $T=\{(f_t,g_t)\mid t\in[0,1]\}$ be a generic smooth path from $(f_0,g_0)$ to $(f_1,g_1)$.
Define the continuation map between two Morse complexes as follows.
Fix a nonnegative smooth function $\beta\colon[0,1]\to\R$ such that $\beta^{-1}(0)=\{0,1\}$,
$\beta'(0)>0$ and $\beta'(1)<0$.

Define a vector field~$V$ on~$[0,1]\times X$ by $V=\beta(t)\partial_t+\zeta_t$,
where $\zeta_t$ is the negative of $g_t$-gradient of~$f_t$.
Note that $\Crit(V)=\{0\}\times\Crit(f_0)\cup\{1\}\times\Crit(f_1)$.

If $P$ and $Q$ are critical points of~$V$, then denote by $m^V(P,Q)$ the moduli
space of flow lines of~$V$ (modulo reparametrization) from~$P$ to~$Q$.
Orient descending manifolds of critical points of~$V$ as follows:
$(0,p)$: use $[0,1]$ direction first, then chosen orientation of~$D(p)$ in~$X$,
$(1,q)$: use chose orientation of~$D(q)$ in~$X$.
If $(f_t,g_t)$ is generic, then $m^V(P,Q)$ is an oriented manifold.
Note that $m^V((0,p),(0,q))=(-1)^{\ind(p)+\ind(q)}m_0(p,q)$,
$m^V((1,p),(1,q))=m_1(p,q)$ and $\dim m^V((0,p),(1,q))=\ind(p)-\ind(q)$.
Now for $p\in\Crit_i(f_0)$ define $\Phi(p)=\sum_{q\in\Crit_i(f_1)}\#m^V((0,p),(1,q))q$.

\proclaim Lemma.  $\Phi$ is a well-defined chain map.

\proclaim Proof.  Usual arguments show that if $p\in\Crit_i(f_0)$ and $q\in\Crit_i(f_1)$,
then $m^V((0,p),(1,q))$ is compact (finite), hence $\Phi$ is well-defined.
If $p\in\Crit_i(f_0)$ and $q\in\Crit_{i-1}(f_1)$, then $m^V((0,p),(1,q))$ has a compactification
to a 1-manifold with boundary $\partial\overline{m^V((0,p),(1,q))}=\cup_{r\in\Crit_{i-1}(f_0)}m^V((0,p),(0,r))\times m^V((0,r),(1,q))\cup\cup_{r\in\Crit_i(f_1)}m^V((0,p),(1,r))\times m^V((1,r),(1,q))$.

One can now easily see that chain map conditions are satisfied.
Another way to look at this: $V$ has a well-defined Morse complex with $C_i=C_{i-1}(X,f_0,g_0)\oplus C_i(X,f_1,g_1)$ and $\partial=\pmatrix{-\partial_0&0\cr\Phi&\partial_1\cr}$, therefore
$0=\partial^2=\pmatrix{\partial_0^2&0\cr-\Phi\partial_0+\partial_1\Phi&\partial_1^2\cr}$.

\proclaim Example.  If $(f_t,g_t)$ does not depend on~$t$, then $\Phi=\id$.

\proclaim Proof.  Since $V=\beta(t)\partial_t+\zeta$, any flow line of~$V$ on $[0,1]\times X$
projects to a flow line of $\zeta$ on~$X$.  If $p,q\in\Crit_i(f)$, then $m_\zeta(p,q)=\emptyset$
when $p\ne q$ and one-element set consisting of constant map if~$p=q$.
It follows that $\Phi=\pm\id$ and it is easy to check that $\Phi=\id$.

\proclaim Lemma.  Suppose we have two generic paths between a pair of Morse-Smale pairs.
Then associated continuation maps $\Phi_0$ and $\Phi_1$ are chain homotopic.

\proclaim Proof.  The space of Morse-Smale pairs is contractible, hence
we can choose a generic homotopy between two given generic paths.
Hence we have a map of $[0,1]^2$ to Morse-Smale pairs.
We define a vector field $\tilde V$ on $[0,1]^2\times X$
by $\tilde V=\beta(t)\partial_t+\zeta_{s,t}$.
Define $K\colon C^M_*(X,f_0,g_0)\to C^M_*(X,f_1,g_1)$ by counting flow lines of~$\tilde V$
with appropriate signs.
If $p\in\Crit_i(f_0)$, then $K(p)=\sum_{q\in\Crit_{i+1}(f_1)}\#\cup_{s\in[0,1]}m^{\tilde V}(((s,0),p),((s,1),q))q$.
Trivial computations show that $K$ is a chain homotopy.

\proclaim Corollary.  The map $\Phi_*\colon H^M_*(X,f_0,g_0)\to H^M_*(X,f_1,g_1)$
does not depend on generic path~$T$ from~$(f_0,g_0)$ to~$(f_1,g_1)$
because the space of Morse-Smale pairs is contractible.

\proclaim Warning/Cool Thing.  For other kinds of Floer theory continuation map
depends on homotopy class of path.

\proclaim Lemma.  $\Phi$ is a functor from fundamental groupoid of piecewise-smooth
Morse-Smale pairs to chain complexes.  (Composition of paths gets mapped to the composition of maps.)

\proclaim Theorem.  Given $(f_0,g_0)$ and $(f_1,g_1)$, let $\Gamma$ be a generic path from
$(f_0,g_0)$ to $(f_1,g_1)$ and let $\Delta$ be the reverse path.
Obviously $\Phi_\Delta\Phi_Gamma\sim\id_{C^M_*(X,f_0,g_0)}$.
Likewise for $\Phi_\Gamma\Phi_\Delta$.  Hence $\Phi_\Gamma$ induces isomorphism on Morse homology.
Hence we can define $H^M_*(X)=H^M_*(X,f,g)$.

Note that if $f_t$ is Morse for all~$t$, then we can identify critical points
of $f_0$~and~$f_1$.  If $(f_t,g_t)$ is Morse-Smale for all~$t$, this identification is an
isomorphism of chain complexes.  This isomorphism is chain homotopic to continuation map.
If we replace $\beta$ by $\epsilon\beta$ for sufficiently small~$\epsilon$, then
they become equal.
Continuation maps for different~$\beta$ are chain homotopic.

Bifurcations.  For a generic family $(f_t,g_t)$ there are times $t_1<\cdots<t_k$
such that at each~$t_m$ one of the following happens:
(1)~Descending and ascending manifolds of two critical points whose indices differ by~1
do not intersect transversally.  Morse complex does not change.
(2)~Handleslide: if $q,q'\in\Crit_i$ and $q\ne q'$.  A flow line from~$q'$ to~$r\in\Crit_{i-1}$
gets glued to a new flow line from~$q$ to~$r$ either before or after the bifurcation.
Thus, $\partial_+q=\partial_-q\pm\partial q'$.
Also $\partial_+p=\partial_-p\mp\langle\partial_-p,q\rangle q'$ for~$p\in\Crit_{i+1}$.
Define an isomorphism $\Phi\colon C_-\to C_+$ as $\Phi(q)=q\pm q'$ and $\Phi(s)=s$ for all other
critical points~$s$.
(3)~Birth or death: two critical points~$p\in\Crit_i$ and $q\in\Crit_{i-1}$ cancel.
Algebraically: $\partial_+=\partial_-p'\pm\langle\partial_-p',q\rangle\partial_-p$ for all $p'\ne p$.
Define chain homotopy $\Phi\colon C_-\to C_+$ as $\Phi(p)=0$, $\Phi(q)=\pm\partial_-p\mp\langle\partial_-p,q\rangle q$ and $\Phi(r)=r$ for all $r\ne p,q$.

\proclaim Exercise.  With corrections as necessary, $\Phi$ induces an isomorphism on homology.

Disadvantages of this approach: Analysis gets more complicated; Not clear that the
isomorphism on homology is canonical.

\proclaim Conjecture.  For a given family $(f_t,g_t)$ if we replace $\beta$ by $\epsilon\beta$
for sufficiently small $\epsilon$, then $\Phi'=\Phi$.

\proclaim Morse homology versus singular homology.

Idea: Define a map $C^M_*\to C_*$ by sending a critical point to its descending manifold.

\proclaim Proposition.  For each critical point~$p$ the descending manifold~$D(p)$ has
a compactification to a manifold~$\bar D(p)$ with corners such that codimension~$k$ stratum is a union
of $m(p,q_1)\times m(q_1,q_2)\times\cdots\times m(q_{k-1},q_k)\times D(q_k)$
over all distinct $q_i\in\Crit(f)$ distinct from~$p$.
The endpoint map to~$D(q_k)$ is continuous.

\proclaim Claim.  In general $\bar D(p)$ is homeomorphic to a closed ball with dimension~$\ind(p)$.
Hence the compactified manifolds $\bar D(p)$ together with the maps $e\colon\bar D(p)\to X$
give~$X$ the structure of CW-complex.  Now the cellular complex coincides with Morse complex,
hence Morse homology is isomorphic to cellular (therefore, singular) homology.

\proclaim Other approach.  Define chain map $C^M_*(X,f,g)\to C_*(X)$.  Roughly $p\to\bar D(p)$.

\proclaim Another approach.  If things are sufficiently smooth, we can use currents.

We shall not take these approaches.

\proclaim One more approach.  Pseudocycles.  We shall not take this approach, but it is probably
the best one.

\proclaim Proof of the Proposition.  We have $$\partial\bar D(p)
=\bigcup_{q\in\Crit(f)\setminus\{p\}}(-1)^{\ind(p)+\ind(q)+1}m(p,q)\times D(q)$$
and $$\partial\bar m(p,q)=\bigcup_{p\ne q\ne r\ne p}(-1)^{\ind(p)+\ind(r)+1}m(p,r)\times m(r,q).$$

\proclaim Lemma.  For each $p\ne q$ there is a cubical singular chain $m_{p,q}\in C^s_{|p|-|q|-1}(\bar m(p,q))$ such that
$m_{p,q}$ represents the relative fundamental chain in $H_{|p|-|q|-1}(\bar m(p,q),\partial\bar m(p,q))$ and
at the chain level $\partial m_{p,q}=\sum_{p\ne q\ne r\ne p}(-1)^{|p|+|r|+1}m_{p,r}\times m_{r,q}$.
Here $\times$ is the cross product on cubical chains.

\proclaim Proof.  By induction on $|p|-|q|$.  Given $p$~and~$q$ the right hand side represents
the fundamental class of $\partial\bar m(p,q)$.  It is easy to see that $\partial$ of right
hand side equals zero.  So right hand side is a cycle, hence it clearly represents
the fundamental class of~$\partial$.  

\proclaim Lemma.  For each critical point~$p$ we can choose $d_p\in C^s_{|p|}(\bar D(p))$ such
that $d_p$ represents the relative fundamental class in~$H_*(\bar D(p),\partial\bar D(p))$
and $\partial d_p=\sum_{p\ne q}(-1)^{|p|+|q|+1}m_{p,q}\times d_q$.

\proclaim Proof.  User previous lemma and induction.

\proclaim Rest of the proof of the Proposition.
Define a map $D\colon C^M_*(X,f,g)\to C^s_*(X)$ by $D(p)=e_\#d_p$.
Now $D$ is a chain map: $\partial^sD=D\partial^M$.
Proof: Let $p\in\Crit_i(f)$.  We have $\partial^sd_p=\sum_{q\in\Crit(f)\setminus\{p\}}
(-1)^{i+|q|+1}e_\#(m_{p,q}\times d_q)$.  If $\ind(q)\ge i$, then $m(p,q)=\emptyset$,
if $\ind(q)\le i-2$, then the corresponding cube is degenerate.
Hence we have $\sum_{q\in\Crit_{i-1}(f)}\#m(p,q)e_\#(dq)=D(\partial^Mp)$.

Hence we have a map $H^M_*(X,f,g)\to H^s_*(X)$.
It is easy to show that this map is independent on the choice of $f$~and~$g$.

Now we show that this map is an isomorphism.
Fix $(f,g)$ as before.  A singular cube $\sigma\colon[-1,1]^i\to X$ will be called
admissible if it is smooth and transverse to the ascending manifolds of the critical points.
The subcomplex generated by admissible cubes is a deformation retract,
hence induces canonical isomorphism on homology.
Given an admissible chain~$\sigma$ and $p\in\Crit(f)$, define $m(\sigma,p)=\{(t,\gamma)\mid
t\in\dom\sigma\land\gamma\colon[0,\infty)\to X\land\gamma(0)=\sigma(t)\land\gamma'(s)=\xi(\gamma(s))\land\lim_{s\to\infty}\gamma(s)=q\}$.
Now $T_t[-1,1]^i$ is isomorphic to $T_\gamma m(\sigma,p)\oplus T_pD(p)$.
We have $\dim m(\sigma,p)=i-\ind(p)$.

Define $A\colon C^a_*(X)\to C^M_*(X,f,g)$ as $A(\sigma)=\sum_{p\in\Crit_i(f)}\#m(\sigma,p)p$.
We can compactify $m(\sigma,p)$.  The boundary consists of the boundary of the cube
times the broken lines.
We have $\partial\bar m(\sigma,p)=m(\partial\sigma,p)\cup\bigcup_{q\in\Crit(f)\setminus\{p\}}
(-1)^{i+\ind(q)}m(\sigma,q)\times m(q,p)$.
Lemma: $A$ is a chain map: $A\partial^s=\partial^MA$.  Proof: If $p\in\Crit_{i-1}(f)$,
then $\bar m(\sigma,p)$ is a compact 1-manifold with boundary.  Obvious computation completes
the proof of the lemma.
Exercise: The induced map $A_*\colon H^s_*(X)\to H^M_*(X)$ is defined.
Lemma: If the metric $g$ is nice near the critical points (so that moduli spaces are
smooth manifolds with corners), then $AD=\id_{C^M_*}$ and $DA$ is chain homotopic
to $\id_{C^a_*(X)}$.  Proof: $AD=\id$ is clear because if $p\in\Crit_i$ and $q\in\Crit_i$,
then $D(p)\cap A(q)$ is empty if $p\ne q$ and $\{p\}$ if $p=q$.
Chain homotopy $K\colon C^a_*(X)\to C^a_{*+1}(X)$ is defined by flowing a cube down.
More precisely, define the forward orbit of~$\sigma$ to be the space $F(\sigma)=[-1,1]^i\times[0,\infty)$ with the map $e\colon F(\sigma)\to X$: $e(t,s)=\phi_s(\sigma(t))$ where $\phi_s$ is the gradient
flow.  Compactify to $e\colon\bar F(\sigma)\to X$, choose appropriate singular chains on~$\bar F(\sigma)$.

More algebraic topology (on closed smooth manifolds) using Morse theory.

\proclaim Poincar\'e duality.  If $X^n$ is oriented, then $H^M_i(X)=H_M^{n-i}(X)$.

\proclaim Definition of Morse cohomology.  Replace chains by additive functions on
chains.  Replace the differential by its dual.

\proclaim Proof of Poincar\'e duality.  Replace $f$ with $-f$.  Note that orientations
for descending manifolds of~$f$ together with orientation of~$X$ induce orientations
of descending manifolds of~$-f$ (which are the same as ascending manifolds of~$f$)
because $T_pX=T_pD(p,f)\oplus T_pA(p,f)$ and $A(p,f)=D(p,-f)$.

\proclaim Definition.  A local coefficient system on a topological space~$X$ is a functor
from fundamental groupoid of~$X$ to the category of abelian groups.
On a locally path connected~$X$ this is the same as a flat $G$-bundle.

Now we define a chain complex~$C$ generated by pairs $(\sigma,g)$ where
$\sigma$ is a cube (or a simplex) in~$X$ and $g\in G_{\sigma(t_0)}$, where $t_0$
is the center of the cube.  The differential is defined in an obvious way
(use obvious path from the center of the cube to the center of its face).

If $X$ is path connected and simply connected, then any local coefficient system
is constant.
If $X^n$ is a manifold, define a local coefficient system $O_x$ on~$X$ by
$O_x=H_n(X,X\setminus\{x\})$ (an orientation of~$X$ at~$x$ is a generator of~$O_x$).

\proclaim Theorem.  If $X$ is a closed manifold and $G$ is any local
coefficient system on~$X$, then $$H_i(X,G)=H^{n-i}(X,G\otimes O).$$

Suppose $X$ is a closed smooth manifold and $(f,g)$ is a Morse-Smale pair
and $L$ is a local coefficient system on~$X$.  Define $C^M_i(X,f,g,L)=\oplus_{p\in\Crit_i(f)}L_p$
and $\partial(p,l)=\sum_{q\in\Crit_{i-1}(f)}\sum_{\gamma\in m(p,q)}\epsilon(\gamma)(q,\phi_\gamma(l))$.

\proclaim Proof.  We have $\partial^2(p,l)=\sum_{r\in\Crit_{i-2}}(r,\sum_{q\in\Crit_{i-2}}\sum_{\gamma_1\in m(p,q)\atop\gamma_2\in m(q,r)}\epsilon(\gamma_1)\epsilon(\gamma_2)\phi_{\gamma_2}(\phi_{\gamma_1}(l)))=0$, because two corresponding paths are homotopic.

\proclaim Poincar\'e duality.  $C^M_i(X,f,g,L)=C_M^{n-i}(X,-f,g,L\otimes O)$ is an isomorphism
of chain complexes.

\proclaim Cup product.  Let $X$ be a closed smooth manifold.
Choose a generic triple of Morse-Smale pairs $(f_1,g_1)$, $(f_2,g_2)$, $(f_3,g_3)$.
Define $*\colon C_M^i(X,f_1,g_1)\otimes C_M^j(X,f_2,g_2)\to C_M^{i+j}(X,f_3,g_3)$
as follows: $\langle p*q,r\rangle$ is the number of flow lines from~$r$ to~$s$,
from~$s$ to~$p$ and from~$s$ to~$q$, where $s$ is an arbitrary point of~$X$.
Codimension counting and transversality and compactness shows that there is only
a finite number of such points.
Why is~$*$ a chain map?  Want $\delta(p*q)=(\delta p)*q+p*(\delta q)$.
Trivial configuration counting show that this is true.

We can also do this for general coefficient systems: $H_M^i(X,G_1)\otimes H_M^j(X,G_2)\to H_M^{i+j}(X,G_1\otimes G_2)$.

\proclaim Theorem.  Under the canonical isomorphism $H_M^*=H_s^*$ the product~$*$
agrees with the cup product.

\proclaim Proof.  Idea: $\langle p*q,r\rangle=\#(A(p)\cap A(q)\cap D(r))$.

Spectral sequences in Morse theorem.

Let $F\to E\to B$ be a smooth fiber bundle where $F$,~$E$,~and~$B$ are closed
smooth manifolds.  Define $E^2_{i,j}=H_i(B,H_j(E_b))$.  Here $H_j(E_b)$ is a local
coefficient system.  For $k\ge2$ we have maps $\partial_k\colon E^k_{i,j}\to E^k_{i-k,j+k-1}$
such that $\partial_k^2=0$ and $E^{k+1}=H(E^k)$.
We define $E^\infty_{i,j}=E^k_{i,j}$ for $k>i$ and $k>j$.

$H_*(E)$ has a filtration: $\alpha\in F_iH_*(E)$ iff $\alpha$ can be represented
by a sum of singular cubes such that their projections to~$B$ depend
on only $i$ of the coordinates on $[-1,1]^*$.

We have associated graded groups $G_iH_*(E)=F_iH_*(E)/F_{i-1}H_*(E)$.  Note that
$G_mH_n(E)=E^\infty_{m,n-m}$.

Example: $\SU(2)\to\SU(3)$ is a fibration with base $S^5$ and fiber $S^3$.  We have $E^2=E^\infty$.
Hence $H_*(\SU(3))=H_*(S^5\times S^3)$.

Construction of Leray-Serre spectral sequence for Morse theory.

Suppose $B$,~$E$,~and~$F$ are closed manifolds.
Choose a generic family of pairs $\{(f_b,g_b)\mid b\in B\}$ where $f_b\colon E_b\to\R$
is a smooth function on~$b$, and $g_b$ is a metric on~$E_b$.  Choose a Morse-Smale
pair $(f^B,g^B)$ on~$B$ such that $(f_b,g_b)$ is Morse-Smale whenever $b\in\Crit(f^B)$.
Choose a connection~$\nabla$ on~$E$.  For $b\in B$ denote by $\xi_b$ the negative gradient
of~$f_b$ with respect to~$g_b$.  Denote by $\xi^B$ the negative gradient of~$f^B$ with respect
to~$g^B$.
Define a vector field~$V$ on~$E$ by $V(b,e)=\xi_b+H(\xi^B)$, where $H(\xi^B)$ is the
horizontal lift with respect to~$\nabla$.
Now $\Crit(V)=\cup_{b\in\Crit(f^B)}\Crit(f_b)$.  Define a chain complex
$C_*=\oplus_{i+j=*}\oplus_{b\in\Crit_i(f^B)}\Crit_j(f_b)$.
Define $\partial\colon C_*\to C_{*-1}$ by counting flow lines of~$V$ in the
usual manner.  Usual arguments show that $\partial$ is well defined,
$\partial^2=0$, and $H_*(C_*,\partial)=H_*(E)$.

In general, this chain complex has a filtration defined by $F_iC_*=\oplus_{b\in\Crit(f^B)\atop\ind(b)\le i}\Z\{\Crit_{*-\ind(b)\}(f_b)}$.

This filtered chain complex gives us a spectral sequence.
We have $E^0_{i,j}=G_iC_{i+j}=\oplus_{b\in\Crit_i(f^B)}\Crit_j(f_b)$.
The differential $\partial_0\colon E^0_{i,j}\to E^0_{i,j-1}$ is induced by~$\partial$.
We have $E^1_{i,j}=\oplus_{b\in\Crit_i(f^B)}H^M_j(E_b,f_b,g_b)$.

It is easy to see that $E^2_{i,j}=H^M_i(B,f^B,g^B,\{H^M_j(E_b,f_b,g_b)\})$.

\proclaim Theorem.  For $k\ge2$ Morse theory spectral sequence agrees with Leray-Serre
spectral sequence.

\proclaim Idea of proof.  Leray-Serre spectral sequence comes from filtration on
$C^s_*(E)$ defined by $F_iC_*(E)$ spanned by cubes whose projection depends on at most~$i$ coordinates.

A chain map that preserves filtration defines a map of spectral sequences.
If it is an isomorphism on $E^2$~term, then it is an isomorphism on all higher terms.

The idea is to define $D\colon C^M_*\to C_*(E)$ by sending a critical point
of~$V$ to its descending manifold, compactified and made into a singular chain,
taking care to use cubes that project to cubes of the correct dimension in~$B$.
Previous discussion shows that this is an isomorphism on $E^2$ terms.

\proclaim Definition.  A 1-form~$\omega$ on~$X$ is Morse if 
it is closed and is locally a differential of Morse function.

Let $p$~and~$q$ be critical points of~$\omega$.  Let $\gamma_n$ be a sequence of flow
lines from~$p$ to~$q$.  Let $r_0$, \dots, $r_{k+1}$ be critical points such that
$r_0=p$~and~$r_{k+1}=q$.  Let $\eta_i$ be a flow line from~$r_i$ to~$r_{i+1}$.
Say that $\gamma_n$ converges to $(\eta_0,\ldots,\eta_k)$ if there are real
numbers $s_{n,0}<\cdots<s_{n,k}$ such that $\gamma_n$ restricted to corresponding
interval converges to~$\eta_i$ in~$C^\infty$ on compact sets as $n\to\infty$.
Also we require that homology class of $\gamma_n-\sum_i\eta_i$ is zero for sufficiently
large~$n$.
Define the energy of a flow line~$\gamma$ by $E(\gamma)=\int|\gamma'^2|=\int\gamma^*\omega$.

\proclaim Proposition.  If $\gamma_n$ is a sequence of flow lines with $E(\gamma_n)<C$,
then there is a subsequence converging to a broken flow line.

\proclaim Lemma.  Let $\gamma\colon\R\to X$ such that $\gamma'(s)$ is dual to $-\omega(\gamma(s))$
(a)~$E(\gamma)\ge0$ and equality holds iff $\gamma$ is a constant map to a critical point.
(b)~If $E(\gamma)<\infty$, then there are critical points $p$~and~$q$ such that $\gamma$
is a flow line from~$p$ to~$q$: $\lim_{s\to-\infty}\gamma(s)=p$ and $\lim_{s\to\infty}\gamma(s)=q$.
(c)~There is $\delta>0$ such that if $\gamma$ is nonconstant then $E(\gamma)>\delta$.

\proclaim Proof of Proposition.  Suppose $\gamma_n$ is a sequence of flow lines
from~$p$ to~$q$ with energy less than~$C$.  Without loss of generality we can
assume that $\lim_nE(\gamma_n)=D$.  Choose $\epsilon>0$ such that $\epsilon$-balls
around critical points are disjoint.  Define $s_{n,0}=\inf\{s\in\R\mid d(\gamma_n(s),\omega^{-1}(0))>\epsilon\}$.
Pass to a subsequence so that $\gamma_n$ restricted to $[s_{n,0},s_{n+1,0}]$ converges
to~$\eta_9$ in $C^\infty$ on compact sets.  Must have $E(\eta_0)\le C_0$.
If $E(\eta_0)=C_0$, then $\gamma_n\to(\eta_0)$ in the defined sense.  If $E(\eta_0)<C_0$,
do the following.  Let $t_0=\sup\{t\mid d(\eta_0(t),\omega^{-1}(0))>\epsilon\}$.
Define $s_{n,1}=\inf\{s\mid s>s_{n,0}+t_0+1\land d(\gamma_n(s)\omega^{-1}(0))>\epsilon\}$.
Pass to a subsequence so that $\gamma_n$ restricted to $[s_{n,1},s_{n+1,1}]$ converges
to~$\eta_1$.  And so on.
Note that $\gamma_n$ is homotopic rel endpoints to $\eta_0\ldots\eta_k$.

\proclaim Morse theory for $f\colon X\to S^1$.  (Simplest version.)
Assume $f$ is Morse, choose generic metric.
Let $\Sigma\subset X$ be a level set of~$f$ not containing any critical points.
Define $C_*$ over $\Lambda=\{\sum_{n\ge n_0}a_nt^n\}$ for $a_n\in\Z$.
Let $C_i$ be a free abelian group generated by index~$i$ critical points.
If $p\in\Crit_i(f)$, then $\partial p=\sum_{q\in\Crit_{i-1}(f)}\sum_{n\ge0}t^nz_n$, where
$z_n$ is the number of flow lines from~$p$ to~$q$ that cross~$\Sigma$ $n$~times.
Usual argument shows that $\partial^2=0$, homology as a $\Lambda$-module is a topological
invariant of~$X$ and the homotopy class of $f\colon X\to S^1$ in~$H^1(X,\Z)$.

Let $X$ be a closed smooth manifold, $\omega$ be a Morse 1-form ($d\omega=0$ and locally
$\omega$ is a differential of a Morse function), $g$ be a generic metric on~$X$.
Choose $K\subset H_1(X)$ such that $\int\omega=0$ for all $\alpha\in K$.
(Usual choices: $K=\{0\}$ or $K=\ker(\int)$.)
Let $H=H_1(X)/K$.

Idea: Classify flow lines modulo the following equivalence relation:
If $\gamma$ and $\gamma'$ are elements of $m(p,q)$, then $\gamma\sim\gamma'$ iff
$[\gamma-\gamma']\in K$.  Note: if $\gamma\sim\gamma'$, then $E(\gamma)=E(\gamma')$.
So if $\ind(p)-\ind(q)=1$, then each equivalence class contains only finitely many flow lines.

Novikov ring: Let $H$ be an abelian group and $N\colon H\to\R$ be a homomorphism.
Define $\Nov(H,N)$ as formal sums of elements of~$H$ with integral coefficients
such that for any neighborhood of~0 there are only finitely many elements~$h\in H$ with nonzero
coefficient and $N(h)$ in the neighborhood.  Multiplication
is defined as in group ring.  There is an obvious injection $\Z[H]\to\Nov(H,N)$.
If $N=0$, then is an iso.

\proclaim Definition of Novikov complex.  Fix a base point $x_0\in X$.
An anchored critical point is a pair $(p,\eta)$ where $p\in\omega^{-1}(0)$
and $\eta$ is a path from~$p$ to~$x_0$, modulo~$\sim$.
Define the energy $E(p,\eta)=\int_\eta-\omega$.
(Alternatively: Let $\pi\colon(\tilde X,\tilde x_0)\to(X,x_0)$ be the covering space
determined by~$K$.  The pair $(p,\eta)$ corresponds to a zero of~$\pi^*\omega$ on~$\tilde X$.
We have $\pi^*\omega=d\tilde f$, where $\tilde f(\tilde x_0)=0$ and $E(p,\eta)=\tilde f(\tilde p)$.

Let $C_i$ be a submodule of Novikov ring generated by all points~$\tilde p$ with index~$i$
such that only finite number of terms have nonzero coefficients and energy lying
outside given neighborhood of~0.

\proclaim Exercise.  $C_i$ is a module over~$\Lambda=\Nov(H,-\omega)$ where $H$ acts
by $h(p,\eta)=(p,\eta+h)$.  Moreover this is a free module with one generator
for each index~$i$ critical point of~$\omega$ in~$X$.

Define $\partial\colon C_i\to C_{i-1}$ by $\partial(p,\eta)=\sum_{q\in\Crit_{i-1}(\omega)}\sum_{\gamma\in m(p,q)}\epsilon(\gamma)(q,\eta-\gamma)$.

\proclaim Exercise.  $\partial$ is well-defined.

\proclaim Theorem.  $H^N_*(X,x_0,\omega,g,K)$ depends only on~$X$, $[\omega]$, and~$K$.
(The isomorphisms are canonical only up to multiplication by elements of~$H$.)

Usually we cannot vary $[\omega]$ except by positive scaling.

Interpretation in classical topology: Chose a cell decomposition of~$X$, lift
it to a cell decomposition of~$\tilde X$.  Then $C^c_*(\tilde X)$ is a module over~$\Z[H]$.

\proclaim Theorem.  (Novikov.)  $H^N_*=H_*(C^c_*(\tilde X)\otimes_{\Z[H]}\Lambda)=H_*(C^{\infty/2}_*(\tilde X))$.
Here $C^{\infty/2}_*(\tilde X)$ is the half-infinite singular chain complex of~$\tilde X$
(formal sums of simplices such that for any real~$c$ only finitely many simplices
intersect $\{\tilde x\in\tilde X\mid\tilde f(\tilde x)>c\}$.

Often $H^N_*$ is torsion.  Suppose $H$ has no torsion so that the quotient ring
of~$\Lambda$ is a field.  Suppose $H^N_*\otimes Q(\Lambda)=0$.

\proclaim Theorem.  The Reidemeister torsion of $C^c_*(\tilde X)\otimes_{\Z[H]}Q(\Lambda)$
is equal to the Reidemeister torsion of $C^N_*\otimes_\Lambda Q(\Lambda))$ times
a certain count of closed orbits of the gradient flow.

\section Pseudoholomorphic curves in symplectic manifolds.

Note: we shall call them holomorphic curves.

\proclaim Definition.  A symplectic manifold is a smooth manifold with a closed nondegenerate 2-form
(nondegenerate means that it defines an isomorphism $TM\to T^*M$, equivalently
$\omega^n\ne0$ everywhere, where $2n$ is the dimension of the manifold,
which must be even).

\proclaim Definition.  A morphism of symplectic manifolds is a smooth map
such that 2-form of the second manifold pulls back to the 2-form of the first manifold.
An isomorphism of symplectic manifolds is called a {\it symplectomorphism}.

\proclaim Darboux's Theorem.  Every symplectic manifold has an open cover
such that restriction to each element is symplectomorphic to $R^{2n}$ with
standard symplectic form.

\proclaim Note.  Every symplectomorphism preserves volume form~$\omega^n$,
hence it preserves volume.

\proclaim Question.  How different is a symplectomorphism from a volume-preserving
diffeomorphism?

\proclaim Answer.  For $n=1$ these notions are the same.

\proclaim Gromov nonsqueezing theorem.  Suppose there is a symplectic
embedding~$\phi\colon B^{2n}(r)\to B^2(R)\times\R^{2n-2}$.  Then $r\le R$.

\proclaim Recognition of~$\R^4$.  Suppose $M$ is a symplectic manifold
such that $\tilde H_*(M)=0$ and $M$ is asymptotically symplectomorphic to $\R^4$,
more precisely, there are compact sets $K_1\subset M$ and $K_2\subset\R^4$
and a symplectomorphism $\phi\colon M\setminus K_1\to\R^4\setminus K_2$.
Then $M$ is symplectomorphic to~$\R^4$, by a symplectomorphism agreeing with~$\phi$
outside a compact set.

\proclaim Symplectomorphisms of~$S^2\times S^2$.  Let $\omega_1$ and $\omega_2$
be two symplectic forms on~$S^2$ with $\int\omega_1=\int\omega_2$.
Then ${\rm Symp}(S^2\times S^2,\omega_1\oplus\omega_2)$ has two
connected components.  The identity component is homotopy equivalent to~$\SO(3)\times\SO(3)$.

\proclaim Definition.  An {\it almost complex structure\/} on~$M$ is a bundle
map $J\colon TM\to TM$ such that $J^2=-1$.  Example: $M$ is a complex manifold,
and $J$ is multiplication by~$i$.  In such a case $J$ is called {\it integrable}.

\proclaim Definition.  An almost complex structure is $\omega$-tame if $\omega(v,Jv)>0$
whenever $v\ne0$.  It is $\omega$-compatible if it is $\omega$-tame and $\omega(Jv,Jw)=\omega(v,w)$.

\proclaim Note.  If $J$ is $\omega$-tame, then it defines a Riemannian metric~$g$
on~$M$ by $g(v,w)=(\omega(v,Jw)+\omega(w,Jv))/2$.
If $J$ is $\omega$-compatible, then $g(v,w)=\omega(v,Jw)$.

\proclaim Idea.  Lots of complex geometry generalizes to symplectic manifolds
with tame almost complex structure.

\proclaim Proposition.  There is no obstruction to finding or extending
tame or compatible almost complex structures.  More precisely, let $\Omega$
be the space of symplectic forms on~$\R^{2n}$, $J$ be the space of complex
structures on~$\R^{2n}$, $C$ be the set of compatible pairs,
and $T$ be the set of tame pairs.
Then $C$ and $T$ are fibrations over~$\Omega$ with contractible fibers.

\proclaim Proof.  We only show that these fibrations have contractible,
more precisely, convex fibers.  This is obvious.

\proclaim Exercise.  Let $\omega$ be a linear symplectic form and $g$ a linear
metric on~$\R^4$.  Then $g$ comes from $\omega$ and $\omega$-compatible~$J$
iff $\omega$ is self-dual with respect to~$g$ and $|\omega|=\sqrt2$.

\proclaim Definition.  A Riemann surface is a closed surface~$\sigma$ is a closed
surface~$\Sigma$ with an almost complex structure~$J$ (necessarily integrable).
$m_{g,n}$ is the space of genus~$g$ Riemann surfaces with $n$ marked points.

\proclaim Fact.  $m_{g,n}$ is a smooth orbifold with dimension $3g-3+n$ for $g\ge2$
or $g=1$ and $n\ge1$ or $g=0$ and $n\ge3$.
For $g=1$ and $n=0$ the dimension is~1 and for $g=0$ and $n\le2$.

\proclaim Definition.  Let $M$ be an almost complex manifold.  A (pseudo)holomorphic
curve in~$M$ is a smooth map $u\colon(\Sigma,j)\to(M,J)$ such that $J\circ du=du\circ j$.
(This means that $du$ respects almost complex structure.)
Two curves are equivalent if there is a biholomorphic map from~$(\Sigma,j)$
to~$(\Sigma',j')$ such that the obvious diagram commutes.

\proclaim Note.  If $u$ is an embedding then the equivalence class of~$u$ is determined
by its image in~$M$.  Hence embedded holomorphic curves are the same as
embedded surfaces~$\Sigma$ such that $J$ maps $T\Sigma$ to itself.

\proclaim Proposition.  If $u\colon\Sigma\to M$ is a holomorphic curve, then its area is equal to
$\int u^*\omega=\langle[\omega],u_*[\Sigma]\rangle$.

\proclaim Proof.  To prove the first part
it is enough to show that $\Area(v,Jv)=\omega(v,Jv)$ for all~$v\in TM$.
This follows from the fact that $\Area(v,Jv)=\omega(v,Jv)$.
To prove the second part it is enough to show that for all $v$~and~$w$ in~$T_pM$
we have $\Area(v,w)\ge\omega(v,w)$ with equality iff $w$ is in the span of $v$~and~$Jv$.
Since $J$ is $\omega$-compatible, we can find a basis $e_i$ and $f_i$
for $T_pM$ such that at~$p$ we have $\omega=\sum_ie_i^*\wedge f_i^*$ and $J(e_i)=f_i$.
Wlog $v=e_1$.  Write $w=\sum_i a_ie_i+b_if_i$.  Now $\Area(v,w)=\left(\sum_i(a_i^2+b_i^2)-a_1^2\right)^{1/2}\ge b_1$.  We have equality iff $a_i=b_i=0$ for $i\ne 1$.

\proclaim Trivial example of holomorphic curves.

\item{1.} The zero set of a homogeneous polynomial over~$\C$ in 3 variables is a holomorphic
curve in $\C P^3$.
\item{2.} If $u\colon(\Sigma,j)\to(M,J)$ is $J$-holomorphic (we assume that $J$ is $\omega$-tame)
and $u_*[\Sigma]=0\in H_2(M)$, then $u$ is constant.  (Proof: $\Area(u)=0$.)
\item{3.} Suppose $M=\Sigma\times V$, where $\Sigma$ is a Riemann surface and $V$ is a symplectic
manifold.  Take $\omega$~and~$J$ be product symplectic and almost-complex structure.
For any $p\in V$ the map $\Sigma\to\Sigma\times V$ sending $x\to(x,p)$ is holomorphic.
Claim: These are they only holomorphic curves in the homology class $[\Sigma]\times[\cdot]$.
Proof: Let $u'\colon(\Sigma',j')\to(\Sigma\times V,J)$ be another holomorphic curve
in this homology class.  The projection $\Sigma'\to\Sigma\times V\to V$ is $J_V$-holomorphic.
Its homology class is~0 in~$H_2(V)$.  So it is constant and the image of~$u$ is contained
in~$\Sigma\times\{p\}$ for some~$p\in V$.  The projection $\Sigma'\to\Sigma\times V\to\Sigma$
is also holomorphic.  A non-constant holomorphic map $\Sigma'\to\Sigma$ is a branched cover of degree
at least~1.  Since homology class is~$[\Sigma]$ the degree of this map is~1 and it is
an isomorphism.

\proclaim Gromov Non-Squeezing Theorem.  If $\phi\colon B^{2n}(r)\to B^2(R)\times\R^{2n-2}$
is symplectomorphism, then $r\le R$.

\proclaim Proof.  (Modulo some stuff.)  Choose $c>0$ such that $\Im(\phi)\subset B^2(R)\times[-c,c]^{2n-2}$.  We get a map $\phi\colon B^{2n}(r)\to S^2(R+\epsilon)\times\T^{2n-2}$.
Choose an $\omega$-tame almost complex structure on the last manifold in such a way that
on $\Im(\phi)$ it agrees with $\phi$-pushforward of standard complex structure on~$\R^{2n}$.
The key is to show the existence of $J$-holomorphic sphere $u\colon S^2\to S^2(R+\epsilon)\times
\T^{2n-2}$ such that $\Im(u)\ni\phi(0)$ and $u_*[S^2]=[S^2(R+\epsilon)]\times[\cdot]$.
Let $\Sigma$ denote $\phi^{-1}(\Im(u))\subset B^{2n}(r)$.  We have $0\in\Sigma$.
If $t<r$ and if we replace $\Sigma\cap B^{2n}(t)$ with another surface with the same boundary,
area does not decrease.  Monotonicity lemma for minimal surfaces: Above conditions on~$\Sigma$
imply $\Area(\epsilon)\ge\pi r^2$.  Proof: Define $\Sigma_t=\Sigma\cap B^{2n}(t)$,
$\theta(t)=\Area(\Sigma_t)/(\pi t^2)$, and $l(t)$ as the length of $\Sigma\cap S^{2n}(t)$.
We have $\lim_{t\to0}\theta(t)=1$ and $\theta'(t)\ge0$.
Also $\Area'(\Sigma_t)\ge l(t)$ and $\Area(\Sigma_t)\le tl(t)/2$.
We have $\theta(r)\ge\theta(0)=1$.  (Monotonicity.)
Now $\pi r^2\le\Area(\sigma)\le\Area(u)=\pi(R+\epsilon)^2$.  This is true
for any $\epsilon>0$, hence $r\le R$.

How do we show that the desired holomorphic curve exists?
(1)~Define some kind of count of holomorphic spheres containing a given point.
(2)~Show that this count is a topological invariant,
(3)~In the case of interest, this invariant equals~1.

Let $(X,\omega)$ be a symplectic manifold, $J$ an $\omega$-tame almost complex structure,
$m_{g,n}(X,A)$ be the space of genus~$g$ $J$-holomorphic curves in~$X$ with $n$ marked
points in the homology class~$A$.  There are evaluation maps $\ev_i\colon m_{g,n}(X,A)\to X$
that send a $J$-holomorphic curve to its corresponding marked point.

Last time: deduced Gromov non-squeezing theorem from the
following statement: Let $(V,\omega)$ be a compact symplectic manifold with $\pi_2(V)=0$.
Let $X=(S^2\times V,\omega_0\oplus\omega)$.  Let $A=[S^2]\times[\cdot]\in H_2(X)$.
Let $J$ be any $\omega$-tame almost complex structure on~$X$.
Then $\ev_1\colon m_{0,1}(X,A)\to X$ is surjective.
Outline of the proof: Suppose $V$ has dimension $2n-2$.  We prove that if $J$
is generic, then $m_{0,1}(X,A)$ is a compact oriented manifold of dimension~$2n$.
If $\{J_t\mid t\in[0,1]\}$ is a generic homotopy then $\bigcup_t\{t\}\times m_{0,1}^{J_t}(X,A)$
is a cobordism from~$m_{0,1}^{J_0}(X,A)$ to~$m_{0,1}^{J_1}(X,A)$.
If $J$ is product almost complex structure, then $m_{0,1}^J(X,A)=X$.
This implies that for generic~$J$ the degree of~$\ev_1$ is~1.

\proclaim Genericity and transversality.

\proclaim Definition.  A $J$-holomorphic curve is multiply covered if it can
be factored through a branched cover of surface of degree greater than~1.
Otherwise it is called simple.

Let $m^*_{g,n}(X,A)$ be the space of simple curves in $m_{g,n}(X,A)$.
Terminology: generic means Baire (countable intersection of open dense sets).

\proclaim Theorem.  For generic~$J$ the manifold $m^*_{g,k}(X,A)$ is oriented (not necessarily
compact) and we have $\dim m^*_{g,k}(X,A)=(n-3)(2-2g)+2\langle c_1(TX),A\rangle+2k$.

\proclaim Sard-Smale theorem.  Let $f\colon X\to Y$ be a smooth map of separable Banach manifolds
whose differential at each point is Fredholm and has index~$l$.  Assume the map is $C^k$,
where $k\ge1$ and $k\ge l+1$.  Then a generic $y\in Y$ is a regular value of~$f$
so that $f^{-1}(y)$ is a manifold of dimension~$l$.

\proclaim Theorem.  Let $Z$ be a smooth finite-dimensional manifold.  Let $k\ge2$.
Then a generic $C^k$ function $f\colon Z\to\R$ is Morse.

\proclaim Proof.  Let $Y=C^k(Z,\R)$.  This is a smooth Banach manifold (a Banach space).
Let $E$ be a vector bundle over $Y$ such that $E_{f,z}=T^*_zZ$.
Define a section $\psi\colon Y\times Z\to E$ by $\psi(f,z)=df_z$.
Then $\psi^{-1}(0)=\bigcup_f\{f\}\times\Crit(f)$.
Claim: For $(f,z)\in\psi^{-1}(0)$ we have $d\psi\colon T_{f,z}Y\times Z\to T_{f,z,0}E$
is surjective, therefore $\psi^{-1}(0)$ is a Banach manifold.
Actually we show that $T_{f,z}Y\times Z\to T_{f,z,0}E\to E_{f,z}=T^*_zZ$ is surjective.
If $f_1\in C^k(Z,\R)$ and $v\in T_zZ$, then $\nabla\psi(f_1,v)=df_1(z)+\nabla_v(df)$
can be anything.
Claim: The projection $\psi^{-1}(0)\to Y$ has $d\pi_{f,z}$ Fredholm so that Sard-Smale applies.
We have $\ker(d\pi_{f,z})=\ker(\nabla\psi\colon T_zZ\to E_{f,z}$.
There is an automorphism $\nabla\psi\colon\coker(d\pi\colon T_{f,z}\psi^{-1}(0)\to T_fY)
\to\coker(\nabla\psi\colon T_zZ\to E_{f,z})$.
It follows that a generic $f\in C^k(Z,\R)$ is a regular value of~$\pi\colon\psi^{-1}(0)\to C^k(Z,\R)$.
If $f$ is a regular value of~$\pi$ then for every $z\in\Crit(f)$ we have
$d\pi\colon T_{f,z}\psi^{-1}(0)\to T_fC^k(Z,\R)$ is surjective.

We say that $E\to X$ is a Banach vector bundle if $E$~and~$X$ are Banach manifolds,
$\pi^{-1}(x)$ is a Banach space for any~$x\in X$, and $X$ has an open cover
such that the corresponding restrictions are trivial bundles.

\proclaim Theorem.  If $\nabla\psi\colon T_xX\to E_x$ is surjective for all $x\in\psi^{-1}(0)$,
then $\psi^{-1}(0)$ is a Banach submanifold of~$X$.

\proclaim Proposition.  Let $Y$~and~$Z$ be separable Banach manifolds,
$\pi\colon E\to Y\times Z$ a Banach vector bundle,
$\psi\colon Y\times Z\to E$ a smooth section.
Suppose that for all $(y,z)\in\psi^{-1}(0)$ the following holds:
(1)~$\nabla\psi\colon T_{y,z}(Y\times Z)\to E_{y,z}$ is surjective.
(2)~$\nabla\psi\colon T_zZ\to E_{y,z}$ is Fredholm of index~$l$.
Then for generic $y\in Y$ the set $\{z\in Z\mid\psi(y,z)=0\}$ is an $l$-dimensional
submanifold of~$Z$ (and moreover, at each point in this set $\nabla\psi$ is surjective on
tangent space to~$Z$).

\proclaim Proof.  It follows from~(a) that $\psi^{-1}(0)\subset Y\times Z$ is a Banach
manifold (by implicit function theorem).
Let $\pi\colon\psi^{-1}(0)\to Y$ denote the projection.
By Sard-Smale theorem, it is enough to show that for all $(y,z)\in\psi^{-1}$ the
differential of~$\pi$, $D=d\pi\colon T_{y,z}\psi^{-1}(0)\to T_yY$ is Fredholm.
By~(b) it is sufficient to prove these two statements: (1)~$\ker(D)=\ker(\nabla\psi\colon T_zZ\to E_{y,z})$ and (2)~The map $\nabla\psi\colon T_yY\to E_{y,z}$ induces an isomorphism
$\coker(D)\to\coker(\nabla\psi\colon T_zZ\to E_{y,z})$.
Proof of~(1): $\ker(D)=\{(0,\dot z)\mid(0,\dot z)\in T_{y,z}\psi^{-1}(0)\}
=\ker(\nabla\psi\colon T_zZ\to E_{y,z})\}$.
Proof of~(2): Check that $\nabla\psi\colon T_yY\to E_{y,z}$ sends $\im(D)\to\im(\nabla\psi\colon T_zZ\to E_{y,z})$.
Let $(\dot y,\dot z)\in T_{y,z}\psi^{-1}(0)$.  Need to show $\nabla\psi(D(\dot y,\dot z))\in
\im(\nabla\psi\colon T_zZ\to E_{y,z})$.
Check $\nabla\psi\colon\coker(D)\to\coker(\nabla\psi\colon T_zZ\to E_{y,z})$ is injective.
Have $\nabla\psi(\dot y,0)=\nabla\psi(0,\dot z)$ for some $\dot z\in T_zZ$.
Then $(\dot y,-\dot z)\in T_{y,z}\psi^{-1}(0)$ and $\dot y=D(\dot y,-\dot z)$.
Check $\nabla\psi\colon\coker(D)\to\coker(\nabla\psi\colon T_zZ\to E_{y,z})$ is surjective.
Let $e\in E_{y,z}$.  By hypothesis~(a), $e=\nabla\psi(\dot y,\dot z)$ for some $(\dot y,\dot z)
\in T_{y,z}Y\times Z$.  We have $e=\nabla\psi(\dot y,0)+\nabla\psi(0,\dot z)=0$
in $\coker(\nabla\psi\colon T_zZ\to E_{y,z})$.

\proclaim Example.  Suppose $Z$ is a closed smooth manifold,
$Y=C^k(Z,\R)$, $E_{f,z}=T^*_zZ$, and $\psi(f,z)=df_z$.  It follows that
for generic $f\in C^k(\R)$ and for any $z\in df^{-1}(0)=\Crit(f)$
we have $\nabla\psi\colon T_zZ\to E_{f,z}$ is surjective.

\proclaim Spectral flow.  Reference: Robbin and Salaman, Spectral flow and the Maslov index.
Let $H$ be a Hilbert space and let $A_s\colon H\to H$ be a continuous family of unbounded operators
parametrized by~$s\in\R$.  Assume there are invertible self-adjoint operators
$A_+$~and~$A_-$ such that $\lim_{s\to\infty}A_s=A_+$ and $\lim_{s\to-\infty}A_s=A_-$
in the norm topology.  More technical assumptions.
(All technical assumptions are satisfied if $H$ is finite dimensional.)
Can define spectral flow of~$A$ (an integer number).
Idea: count the number of eigenvalues of~$A_s$ that cross~0 as $s$ goes from~$-\infty$ to~$\infty$.
If $H$ is finite-dimensional, then $\SF(A)$ is the difference
between the number of positive eigenvalues of $A_+$~and~$A_-$.

\proclaim Theorem.  Under certain assumptions $\partial_s+A_s\colon L^2_1(\R,H)\to L^2(\R,H)$
is Fredholm and its index is the spectral flow of~$A$.

\proclaim Proof.  Assume $H$ is finite-dimensional.
For each $h\in H$ by fundamental theorem of ODE's there is a function $f\colon\R\to H$
such that $(\partial_s+A_s)f_h(s)$ and $f_h(0)=h$.
Define $H^+=\{h\in H\mid\lim_{s\to\infty}f_h(s)=0\}$ and $H^-=\{h\in H\mid\lim_{s\to-\infty}
f_h(s)=0\}$.  Then $\ker(\partial_s+A_s)$ is isomorphic to~$H^+\cap H^-$.
Claim: $H^+$ is the negative eigenspace of~$A^+$ and $H^-$ is the positive eigenspace of~$A^-$.
Claim: $\partial_s+A_s$ is bounded and has closed image.
Then $\coker(\partial_s+A_s)=\ker(-\partial_s+A_s^*)=\ker(\partial_s-A_s^*)$.
Similarly to above, $\ker(\partial_s-A_s^*)=H^{+\perp}\cap H^{-\perp}$.
We have $\ind(\partial_s+A_s)=\dim(H^+\cap H^-)-\dim(H^{+\perp}-H^{-\perp})
=\dim(H^+\cap H^-)+\dim(H^++H^-)-\dim(H)
=\dim H^++\dim H^--\dim H=\SF(A)$.

\proclaim Proposition.  If $X$ is a closed smooth manifold, $f\colon X\to\R$ is a Morse function.
then for a generic $C^k$-metric~$g$ on~$X$, the pair $(f,g)$ is Morse-Smale.

\proclaim Proof.  Fix distinct critical points $p$~and~$q$ of~$f$.
Let $Y$ be the space of $C^k$-metrics~$g$ on~$X$.  Let $Z$ be the space of locally~$L^2_1$
maps $\gamma\colon\R\to X$ such that $\lim_{s\to-\infty}\gamma(s)=p$ and $\lim_{s\to\infty}\gamma(s)=q$, and $\gamma(-\infty,-R]$ and $\gamma[R,\infty)$ are $L^2_1$ for sufficiently large~$R$.

\proclaim Proposition.  Let $Y$~and~$Z$ be $C^k$ separable Banach manifolds,
$E\to Y\times Z$ a Banach space bundle, $\psi\colon Y\times Z\to E$ a $C^k$ section
such that $k\ge1$ and $k\ge l+1$.
Suppose for all $(y,z)\in\psi^{-1}(0)$ we have
(a)~$\nabla\psi\colon T_{(y,z)}(Y\times Z)\to E_{(y,z)}$ is surjective
and (b)~restriction $\nabla\psi\colon T_zZ\to E_{(y,z)}$ is Fredholm of index~$l$.
Then for generic $y\in Y$ the set $\{z\in Z\mid\psi(y,z)=0\}$ is an $l$-dimensional $C^k$
submanifold of~$Z$ (and on this set $\nabla\psi$ is surjective onto tangent space to~$Z$).

\proclaim Proposition.  Let $X$ be a closed smooth manifold, $f\colon X\to\R$ a Morse
function, and $k$ a sufficiently large integer.
Then for a generic $C^k$ metric~$g$ on~$X$, the pair $(f,g)$ is Morse-Smale.

\proclaim Proof.  Fix $p$ and $q$ in $\Crit(f)$.  Let $Y$ be the space of $C^k$ metrics on~$X$.
Let $Z$ be the space of locally $L_1^2$ maps $\gamma\colon\R\to X$ such that
$\lim_{s\to-\infty}\gamma(s)=p$, for $R$ sufficiently small so that $\gamma(-\infty,R]\subset U_p$,
where $U_p$ is a coordinate chart around~$p$, we have $\gamma\colon(-\infty,R]\to\R^n$ is $L_1^2$,
Also we must have $\lim_{s\to\infty}=q$ and an analogous statement for $U_q$.
Exercise: $Z$ is a $C^\infty$ Banach manifold, and $T_\gamma Z=L_1^2(\gamma^*TX)$.
Let $E_{g,\gamma}=L^2(\gamma^*TX)$.  This is a $C^\infty$ Banach space bundle.
Let $\psi$ be a section of~$E$ such that $\psi(g,\gamma)(s)=\gamma'(s)-\xi(\gamma(s))$.
Thus $\psi(g,\gamma)=0$ iff $\gamma$ is a $C^{k+1}$ flow line with respect to~$g$ from~$p$ to~$q$.
Claim: hypotheses of previous proposition are satisfied.
Fix some torsion free connection on $TX\to X$.
If $\psi(g,\gamma)=0$ then $\nabla\psi(\dot g,\dot\gamma)=\nabla_{\gamma'}\dot\gamma
-\nabla_{\dot\gamma}\xi-\dot\xi$.  Given $(g,\gamma)$, choose a trivialization of~$\gamma^*TX$,
which is parallel with respect to connection on~$TX$.
We have $\nabla\psi(\dot g,\dot\gamma)=\partial_s\dot\gamma-A_s\dot\gamma-\dot\xi$,
$A_s=\nabla\xi\colon T_{\gamma(s)}X\to T_{\gamma(s)}X$.
We also have $\lim_{s\to-\infty}A_s=H(f,p)$ and $\lim_{s\to\infty}A_s=H(f,q)$.
(a)~We check $\nabla\psi$ is surjective.  Can show $\nabla\psi$ has closed range.  (Skip.)
Enough to show if $\omega\in L^2(\gamma^*TX)$ is orthogonal to $\im(\nabla\psi)$ then $\omega=0$.
Such an $\omega$ satisfies $\int_\R\langle\dot\xi,\omega\rangle ds=0$ for every $\dot g$.
At any given point in the image of~$\gamma$ can find a $\dot g$ such that $\dot\xi=\omega$ there.
Choosing $\dot g$ supported near that point gives $\omega=0$ there.
(b)~$L^2_1(\gamma^*TX)\to L^2(\gamma^*TX)$ sends $\dot\gamma$ to $\partial_s\dot\gamma-A_s\dot\gamma$ and is a Fredholm operator by theory from last time.
The index is equal to the spectral flow, which is equal to $\ind(p)-\ind(q)$.
Previous proposition says for generic~$g$ the manifold $m(p,q)$ has dimension $\ind(p)-\ind(q)$
and for all $\gamma\in m(p,q)$, operators $\partial_s-A_s$ is surjective.
Observe $H_+=T_{\gamma(0)}D(p)$ and $H_-=T_{\gamma(0)}A(q)$.
Recall $\coker(\partial_s-A_s)$ is the intersection of orthogonal complements to $H_+$ and $H_-$,
which is zero, hence the span of $H_+$ and $H_-$ is $\R^n$, hence
$D(p)$ intersects $A(q)$ transversally.  This completes the proof.

Now let $(M,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost
complex structure on~$M$, $A$ be an element of~$H_2(X)$,
$m_{g,k}(X,A)$ be the space of genus~$g$ $J$-holomorphic curves in~$X$ in class~$A$ with $n$~marked
points,
$m_{g,k}^*(X,A)$ be the space of simple (not multiply covered) curves in $m_{g,k}(X,A)$.

\proclaim Theorem.  For generic~$J$ the manifold $m_{g,n}^*(X,A)$ is oriented.
Its dimension is $$(n-3)(2-2g)+2\langle c_1(TX),A\rangle+2k.$$

\proclaim Theorem.  Fix a complex structure $j$ on $\Sigma_g$.
Let $\Sigma=(\Sigma_g,j)$,
$m(\Sigma,X,A)$ be the space of $J$-holomorphic maps $(\Sigma_g,j)\to(X,J)$ in class~$A$,
$m^*(\Sigma,X,A)$ the space of simple curves in $m(\Sigma,X,A)$.
For generic~$J$ the manifold $m^*(\Sigma,X,A)$ is oriented and has dimension
$n(2-2g)+2\langle c_1(TX),A\rangle$.

\proclaim Proof Sketch.  Ignore the issue of Banach space completions.
Let $Y$ be the space $\omega$-tame almost complex structures~$J$ on~$X$,
$Z$ be the space of smooth maps $u\colon\Sigma\to X$ representing the class~$A$.
Let $E_{J,u}=T(T^{0,1}\Sigma\otimes_\C u^*TX)$ is the space of $\C$-antilinear bundle maps
$T\Sigma\to u^*TX$.  Let $\psi(J,u)$ be the map $T\Sigma\to u^*TX$ defined
by $\psi(J,u)=du+J\circ du\circ j$.
We have $\psi(J,u)=0$ iff $J\circ du=du\circ j$, i.e., $u$ is holomorphic.
We say that $u$ is somewhere injective whenever there is a $z\in\Sigma$ such that
$u^{-1}(u(z))=\{z\}$ and $du\colon T_z\Sigma\to T_{u(z)}X$ is injective.
Fact: If $u$ is holomorphic then $u$ is simple iff $u$ is somewhere injective
(McDuff-Salamon, Chapter~2).
Need to show: for $(J,u)\in\psi^{-1}(0)$:
(a)~$\nabla\psi\colon T_{J,u}Y\times Z\to E_{J,u}$ is simple; (b)~$\nabla\psi\colon
T_uZ\to E_{J,u}$ is Fredholm of index $n(2-2g)+2\langle c_1(TX),A\rangle$.
We see that $T_JY$ is the space of bundle maps $\dot J\colon TX\to TX$ such that $\dot JJ+J\dot J=0$.
and $T_uZ=T(u^*TX)$.  We have $\psi(J,u)=du+J\circ du\circ j$.
(b)~We have $\nabla\psi(0,\dot u)=(\partial_su+J\partial_tu)(ds-idt)$ plus zero order term.
We can deform the zeroth order term so that this is $\bar\partial$ operator
on holomorphic vector bundle $u^*TX$ over~$\Sigma$.
Riemann-Roch theorem implies that the index is $n(2-2g)+2c_1(u^*TX)$.
To complete the proof, we use somewhere injective to prove~(a).

Suppose we have a symplectic manifold~$(M,\omega)$, $J$ is $\omega$-tame almost
complex structure.  Fix $(\Sigma,j)$ and $u\colon\Sigma\to M$.
We have $\bar\partial(u)=du+J\circ du\circ j\in\Omega^{0,1}(\Sigma,u^*TM)=\Gamma(T^{0,1}\Sigma\otimes_\C u^*TM)$.
The map~$u$ is holomorphic iff $\bar\partial(u)=0$.  If $u$ is holomorphic, derivative of~$\bar\partial$ defines an operator
$D_u\colon T(u^*TM)\to\Gamma(T^{0,1}\Sigma\otimes_\C u^*TM)$.

\proclaim Definition.  $u$ is transverse if $D_u$ is surjective (on~$C^\infty$
on appropriate Banach space completions).

Last time we proved that if $J$ is generic then all simple holomorphic curves are transverse.
Note: If $u$ is transverse, then $m(\Sigma,M)$ is a manifold near~$u$, and
$T_um(\Sigma,M)=\ker(D_u)$.  Also $\dim=\ind(D_u)=n(2-2g)+2\langle c_1(TM),A\rangle$.
To complete the proof we need to show that for any
pair $(J,u)$ where $u$ is $J$-holomorphic, the following operator is surjective:
$T_Jg\oplus\Gamma(u^*TM)\to\Gamma(T^{0,1}\Sigma\otimes u^*TM)$,
$(\dot J,\xi)\to D_u\xi+\dot J\circ du\circ j$.
Can show closed range.
More precisely, suppose $\eta$ is perpendicular to image.
Since $u$ is somewhere injective, find $z\in\Sigma$ such that $u$ embeds a neighborhood of~$z$
into~$M$, disjoint from the rest of~$u(\Sigma)$.
It follows that $\eta=0$ in a neighborhood of~$z$.
So we have $\langle D(\dot J,0),\eta\rangle=\int_\Sigma\langle\dot J\circ du\circ j,\eta\rangle
=\int_{u^{-1}(U)}\langle\dot J\circ du\circ j,\eta\rangle=\int_N\langle\dot J\circ du\circ j,\eta\rangle>0$.
Since $\eta\perp\im D_u\rangle$ we have $D_u^*\eta=0$.
Unique continuation theorem shows that if $D_u^*\eta=0$ and $\eta$ vanishes
to infinite order at a point, then $\eta=0$.

\proclaim Back to the Gromov nonsqueezing theorem.  Take $M=S^2\times V$, $\omega=\omega_{S^2}\oplus\omega_V$,
$J=J_{S^2}\oplus J_V$.  Claim: The holomorphic spheres $S^2\times\{v\}$ are transverse. 

\proclaim Proof.  Let $u$ be the map $S^2\to S^2\times\{v\}$.
We have the operator $D_u\colon\Gamma(u^*TM)\to\Gamma(T^{0,1}S^2\otimes u^*TM)$.
We have splitting $u^*TM=TS^2\oplus T_vV=TS^2\oplus\C^{n-1}$.
$D_u$ respects this splitting.
$D_u$ is a sum of operators of the form $\Gamma(L)\to\Gamma(T^{0,1}S^2\otimes L)$
where $L$ is a line bundle.
Suppose an operator of the latter type has a nonzero cokernel.
Carleman Similarity Principle: If you have a solution of Cauchy-Riemann type equation
with a zero-order perturbation, then you can perform a change of coordinates
such that the solution becomes holomorphic.  In our case $\eta\in\Omega^{0,1}(S^2,L)$ satisfies
such an equation.  It follows that any zero of~$\eta$ has negative multiplicity.
From Carleman it follows that $\deg(T^{0,1}S^2\otimes L)\le0$.

\proclaim Remark.  Consider $M=\Sigma_g\times V$.
Are the curves $\Sigma\times\{v\}$ transverse?  No, if $g>0$.

Also: moduli spaces are canonically oriented.

\proclaim Gromov Compactness Theorem.  (Simplest version.)
$M$, $\omega$, $J$ as usual.  Let $A\in H_2(M)$.  Suppose there is no $B\in\pi_2(M)$
such that $0<\int_B\omega<\int_A\omega$.  Then $m(S^2,M,A)$ is compact.
Moreover, if $J_t$ is a family of $\omega$-tame almost complex structures,
then $\cup_t\{t\}\times m_{J_t}(S^2,M,A)$ is compact.

\proclaim Idea of proof.  Consider a sequence of maps $u_k\colon S^2\to M$.
The energy $E(u_k)=\int_{S^2}|du_k|^2=\int_AC\omega$.  If you also have
$|du_k|<c$ then we can pass to a convergent subsequence.
If not, reparametrize and pass to a subsequence so that $|du_k(0)|>k$.
Rescale the maps near~0.  A holomorphic sphere magically appears with energy less than $\int_A\omega$.

\proclaim Intersection property.  Let $(M,\omega)$ be a 4-dimensional symplectic manifold,
$C_1$~and~$C_2$ two distinct simple connected $J$-holomorphic curves.
Then the intersection of $C_1$ and $C_2$ are isolated.
Also each intersection point has positive multiplicity.
Multiplicity is~1 iff intersection is transverse.

Easy part: transverse intersection must have positive sign.

\proclaim Adjunction formula.  Let $C$ be a simple $J$-holomorphic curve in~$X$.
Define $c_1(C)=c_1(TX)$, where $TX$ is restricted to~$C$.
Let $\chi(C)$ be the Euler characteristic of domain of~$C$.
Then $c_1(C)=\chi(C)+C\cdot C-2\sum\delta(p)$, where the sum is taken over all
singular points of~$C$,
and $\delta(p)$ is a positive integer, $\delta(p)=1$ iff $C$ has a transverse self-intersection at~$p$.

\proclaim Theorem.  (Recognition of~$\R^4$.)
Let $(M,\omega)$ be (noncompact) symplectic manifold, $\tilde H_*(M)=0$.
Suppose there is a compact subset~$K$ of~$M$ such that $(M\setminus K,\omega)$ is symplectomorphic
to $(\R^4\setminus B,\omega_s)$.  Then $(M,\omega)$ is symplectomorphic to $\R^4$ with standard
symplectic structure.

\proclaim Proof.
We see that $\tilde M\setminus K$ is symplectomorphic to $S^2\times S^2\setminus B$.
Also $H_*(\tilde M)=H_*(S^2\times S^2)$.
Choose $\omega$-tame $J$ on~$\tilde M$ which is product structure outside of~$K$.
Take $A=(1,0)\in H_*(\tilde M)$ and $B=(0,1)\in H_*(\tilde M)$.
Look at $\ev\colon(\tilde M,A)\to\tilde M$.
Similarly to proof of nonsqueezing, this evaluation map has degree~1.
Consider $(p,q)$ as shown.  There is a holomorphic sphere $S^2\times\{q\}\subset m_{0,1}$.
We claim that $S^2\times\{q\}$ is the only holomorphic sphere in the class~$(1,0)$
containing~$(p,q)$.  Suppose $c'$ is another one.  Then it is simple as above.
By intersection positivity $c\cdot c'>0$.  But $(1,0)\cdot(1,0)=0$.  Contradiction.
Also $C$ is transverse by arguments from last time.
So $(p,q)$ is a regular value of~$\ev$, and $\#\ev^{-1}(p,q)=1$.
Claim: The holomorphic spheres in the class~$(1,0)$ give a foliation of~$\tilde m$.
There is a unique such sphere through each point.
Any such sphere is embedded: $2=c_1(C)=\chi(C)+C\cdot C-2\delta(C)=2+0-0$.
Different spheres don't intersect.

Now we know that every point in~$\tilde M$ is in a unique embedded holomorphic sphere in class
$(1,0)=A$ or $(0,1)=B$ respectively.
Define $\phi\colon S^2\times S^2\to\tilde M$ as follows.
Given $(x,y)\in S^2\times S^2$ let $C_1$ be the sphere through $(p,x)$ in class~$A$,
let $C_2$ be the sphere through $(y,q)$ in class~$B$.
Define $\phi(x,y)$ to be the intersection of $C_1$ with $C_2$.
(Unique by intersection positivity.)
Claim: $\phi$ is a diffeomorphism.  Surjective because  every point in~$\tilde M$ is contained
in an $A$ sphere and a $B$ sphere.
Injective because different $A$ spheres are disjoint and different $B$ spheres are disjoint.
Smoothness omitted.

Also $\phi$ is the identity at infinity.
Claim: $\phi$ is isotopic to a symplectomorphism that is still identity at infinity.
Proof: Let $\omega$ denote symplectic forms.  Claim: $\omega\wedge\phi^*\omega>0$.
We have $(\omega\wedge\phi^*\omega)(\partial_{x_1},\partial_{y_1},\partial_{x_2},\partial_{y_2})
=\omega(\partial_{x_1},\partial_{y_1})\phi^*\omega(\partial_{x_2},\partial_{y_2})+\omega(\partial_{x_2},\partial_{y_2})\phi^*\omega(\partial_{x_1},\partial_{y_1})$.
Let $\omega_t=t\omega+(1-t)\phi^*\omega$ on $S^2\times S^2$ for $t\in[0,1]$.
Since $\omega\wedge\phi^*\omega>0$ it follows that $\omega_t\wedge\omega_u>0$ for all $t$~and~$u$.
Hence all $\omega_t$'s are all in the same cohomology class.
Choose 1-form $\beta_t$ with $d\beta_t=(d/dt)\omega_t$ and $\beta_t=0$ at infinity.
Look for $\phi_t\colon S^2\times S^2\to\tilde M$ such that $\phi_t^*\omega=\omega_t$ and
$\phi_0=\phi$ and $\phi_t$ depends smoothly on~$t$.
Then $\phi_1^*\omega=\omega_1$.
To get $\phi_t^*\omega=\omega_t$ for all~$t$ it is enough to get $(d/dt)\phi_t^*\omega=(d/dt)\omega_t$.
We have $(d/dt)\phi_t^*\omega=L_{X_t}\omega=(d/dt)\omega_t=d\beta_t$.
$X_t$ is a vector field on $S^2\times S^2$ determining the isotopy.
Since $\omega$ is nondegenerate there is $X_t$ with $I_{X_t}=\beta_{t'}$
$X_t=0$ at infinity so $\phi_1=\phi_0$ at infinity.
Conclusion: we have a symplectic diffeomorphism between $S^2\times S^2$ and $\tilde M$,
which is identity at infinity.

\proclaim Theorem.  The group of symplectic diffeomorphisms of $S^2\times S^2$ is $\SO(3)\times\SO(3)$.

\proclaim Proof.  Key point is that for any $\omega$-tame $J$ on~$S^2\times S^2$ we have two
foliations by $A$-spheres and $B$-spheres.
Note: The claim is false for $S^2\times S^2$ with symplectic structure $\omega_1\oplus\omega_\lambda$, where $\int_{S^2}\omega_1=1$, $\int_{S^2}\omega_\lambda=\lambda\in(1,2]$.
Trouble is that $(0,1)=(1,0)+(-1,1)$.  So in constructing $B$ foliation the compactness
argument fails.

Suppose $(X,\omega)$ is a closed symplectic manifold.  A symplectic isotopy
is a smooth family of symplectomorphisms of~$X$.
We say that isotopy $\phi$ is generated by vector fields~$X_t$ if
$(d/dt)\phi_t(x)=X_t(\phi(x))$.  Differentiating $\phi_t^*\omega=\omega$ we get
$0=L_{X_t}\omega=di_{X_t}\omega+i_{X_t}d\omega$.
Conclusion: $X_t$ generates a symplectic isotopy iff $\omega(X_t,\cdot)$ is a closed 1-form.

\proclaim Definition.  The isotopy $\phi_t$ is Hamiltonian if $\omega(X_t,\cdot)$ is exact,
i.e., $\omega(X_t,\cdot)=dH_t$, where $H_t\colon X\to\R$.

\proclaim Degenerate Arnold Conjecture.  If $\phi\colon(X,\omega)\to(X,\omega)$ is Hamiltonian
isotopic to $\id_X$ then $|\Fix(\phi)|\ge\min\{|\Crit(f)|\mid f\colon X\to\R\}$.  It is still open!

\proclaim Nondegenerate Arnold Conjecture.  If $\phi\colon(X,\omega)\to(X,\omega)$ is Hamiltonian
isotopic to $\id_X$ then $|\Fix(\phi)|\ge\sum_i\dim(H_i(X,\Q))$.  Proved using Floer homology.

A fixed point~$p$ of~$f$ is nondegenerate if $1-df_p\colon T_pX\to T_pX$ is invertible.
Equivalently, graph $T(f)=\{(x,f(x))\}\subset X\times X$ is transverse to the diagonal $\Delta=\{(x,x)\}\subset X\times X$ at $(p,p)$.

\proclaim Lefschetz Fixed Point Theorem.  If $X$ is a closed smooth manifold and $f\colon X\to X$
is a smooth map with nondegenerate fixed points then
$\sum_{p\in\Fix(f)}\mathop{\rm sign}\det(1-df_p)=\sum_i(-1)^i\mathop{\rm tr}(f_*\colon H_i(X,\Q)\to H_i(X,\Q))$.
It follows that if $f\sim\id_X$ then $|\Fix(f)|\ge\left|\sum_i(-1)^i\dim H_i(X,\Q)\right|=\left|\chi(X)\right|$.

\proclaim Note.  Arnold conjecture is false if we only require $\phi$ to be symplectically
isotopic to $\id_X$.

\proclaim Counterexample.  The map $\phi\colon(T^2,\omega)\to(T^2,\omega)$
such that $\phi(x,y)=(x+1/2,y)$ is symplectically isotopic to identity via the map
$\phi_t(x,y)=(x+t/2,y)$.  This symplectic isotopy is not Hamiltonian because
it is generated by $X_t=\partial_x/2$ and $i_{X_t}\omega=dy/2$, which is not exact.
In fact, there is no Hamiltonian isotopy from~$\phi$ to~id.
If $\phi$ is a symplectic isotopy, define the flux of $\phi$ to be an element of $H^1(X,\R)$
such that its value on $\gamma\colon S^1\to X$ is $\int_{[0,1]\times S^1}f^*\omega$,
where $f\colon[0,1]\times S^1\to X$ is the map such that $f(t,0)=\phi_t(\gamma(0))$.

\proclaim Fact.  This gives a well-defined element of~$H^1(X,\R)$
and $\phi$ is homotopic relative to endpoints to a Hamiltonian isotopy iff flux of~$\phi$ is zero.

If $\phi$ is symplectically isotopic to identity, then the flux of~$\phi$ is defined
and is zero iff $\phi$ is Hamiltonian isotopic to identity.
If $X=T^2$, the flux of $\phi$ in the previous example is $(0,1/2)\ne0$.

Now we enter the world of Floer homology.
A.~Floer, Symplectic fixed points and holomorphic spheres, Communications in Mathematical Physics.

\proclaim Symplectic action functional.  Let $L$ be the space of contractible smooth loops in~$X$.
Let $\tilde L$ be the space of pairs~$(\gamma,[u])$, where $\gamma\in L$ and $[u]$ is homotopy
class of $u\colon D^2\to X$ relative boundary such that $\gamma$ is the restriction of~$u$
to the boundary.
Consider the map $A\colon\tilde L\to\R$ such that $A(\gamma,[u])=\int_{[0,1]}H_t(\gamma(t))dt
+\int_{D^2}u^*\omega$.

\proclaim Lemma. $(\gamma,[u])\in\Crit(A)$ iff $\gamma'(t)=X_{H_t}(\gamma(t))$ iff
$\gamma(t)=\phi_t(\gamma(0))$.
Hence $\Crit(A)=\Fix(\phi)$.

\proclaim Proof.  Let $\xi\in T_{(\gamma,[u])}\tilde L=\Gamma(\gamma^*TX)$.
Now $$\eqalign{\quad dA_\gamma(\xi)=(d/ds)\left(\int_{[0,1]}H_t(\gamma_s(t))dt+\int_{u_s}\omega\right)
=\int_{[0,1]}dH_t(\xi(t))dt+\int_{[0,1]}\omega(\xi,\gamma'(t))dt\hfil\cr
\hfil{}=\int_{[0,1]}(\omega(X_{H_t},\xi(t))+\omega(\xi(t),\gamma'(t))dt
=\int_{[0,1]}\omega(\xi(t),\gamma'(t)-X_{H_t})dt.\quad\cr}$$
$(\gamma,[u])\in\Crit(A)$ iff the above integral is zero for all~$\xi$ iff $\gamma'(t)-X_{H_t}=0$
for all~$t$.

\proclaim Key example.  Suppose $H_t=H\colon X\to\R$ is a Morse function,
$X_{H_t}=X$ and $\omega(X,\cdot)=dH$.
Now $\Crit(H)\subset\Fix(\phi)$.  Arnold conjecture is trivial in this case.

Let $J_t$, where $t\in S^1$ be a family of $\omega$-compatible almost complex structures
on~$X$.  Each $J_t$ defines a metric~$g_t$ on~$X$ by $g_t(v,w)=\omega(v,J_tw)$.
This defines a metric on~$\tilde L$ as follows: If $(\gamma,[u])\in\tilde L$
and $\xi_1,\xi_2\in T_{(\gamma,[u])}$, then $\langle\xi_1,\xi_2\rangle=\int_{S^1}g_t(\xi_1(t),
\xi_2(t))dt$.
Consider a path $\tilde u\colon\R\to\tilde L$ such that $\tilde u(s)=(u(s,\cdot),[D_s])$.
Lemma: $\tilde u$ is an upward gradient flow line of~$A$ iff
$\partial_su+J_t(\partial_tu-X_{H_t})=0$.
Note: This is almost the equation for~$u$ to define a pseudoholomorphic map $\R\times S^1\to X$.
(Only almost because $J$ depends on~$t$ and there is $X_{H_t}$ term.)
Proof: Need to check that $(d/ds)\tilde u(s)=\nabla A(u(s))$, i.e.,
fix~$s$, let $\gamma(t)=u(s,t)$, want for any $\xi\in T_\gamma L=\Gamma(\gamma^*TX)$
the following: $dA_\gamma(\xi)=\langle(d/ds)\tilde u(s),\xi\rangle$,
i.e., $\int_{[0,1]}\omega(\xi,\gamma'(t)-X_{H_t})=\langle J_t(-\gamma'(t)+X_{H_t}),\xi\rangle$,
i.e., $\int_{[0,1]}\omega(\xi,\gamma'(t)-X_{H_t})=\int_{[0,1]}\omega(J_t(-\gamma'(t)+X_{H_t}),J_t\xi)dt$.  (Tame ok.)

\proclaim Key example.  $H_t=H\colon X\to\R$ is a Morse function,
$J_t=J$ is an $\omega$-compatible almost complex structure,
therefore $g_t=g$, $g(v,w)=\omega(v,Jw)$.
Suppose $\eta\colon\R\to X$ is an upward gradient flow line of~$H$, i.e., $\eta'(s)=\nabla H(\eta(s))$.
Define $u(s,t)=\eta(t)$.  This satisfies $\partial_s\eta=JX_H$ because $JX_H=\nabla H$ because
$\langle JX_H,v\rangle=dH(v)$ because $\langle JX_H,v\rangle=\omega(JX_H,Jv)=\omega(X_H,v)=dH(v)$.

\proclaim Conclusion.  When $H_t$ and $J_t$ do not depend on~$t$ we have
$\Crit(H)\subset\Crit(A)/\pi_2(X)$.
All gradient flow lines of~$H$ are gradient flow lines of~$A$,
therefore Morse homology of~$A$ is isomorphic to the Morse homology of~$H$.

Recall that $(M,\omega)$ is a closed symplectic manifold,
$H$ is a 1-periodic Hamiltonian: $H\colon S^1\times M\to\R$,
$X_{H_t}$ is the corresponding vector field,
and $\phi_t$ is a family of symplectomorphisms such that 
$\phi_0=\id_M$, $(d/dt)\phi_t(x)=X_{H_t}(\phi_t(x))$.
Let $\phi=\phi_1$.
We say that $\phi_t$ is a Hamiltonian isotopy from~$\id_M$ to~$\phi$.

Arnold Conjecture: $\phi$ has at least as many fixed points as the minimum number of critical points
of a function $M\to\R$.
Nondegenerate version: If fixed points of~$\phi$ are nondegenerate then
the number of fixed points is at least $\sum_i\dim H_i(M,\Q)$.

Now let $L$ be the space of contractible loops in~$M$.
Define a symplectic action functional $A\colon L\to\R$ (if $\langle\omega,\pi_2(M)\rangle=0$)
$A(\gamma)=\int_{S^1}H_t(\gamma(t))dt+\int_D\omega$, where $D$ is a disc in~$M$ with boundary~$\gamma$.
We have $\Crit(A)=\Fix(\phi)=\{\gamma\colon S^1\to M\mid\gamma'(t)=X_{H_t}(\gamma(t))$.
If $J_t$ is an $\omega$-compatible almost complex structure for $t\in S^1$ then we have
a metric on~$L$.

An upward gradient flow line of~$A$ from~$\gamma_-$ to~$\gamma_+$ is a map
$u\colon\R_s\to S^1_t\to M$ satisfying $\partial_su+J_t(\partial_tu-X_{H_t})=0$.
We have $\lim_{s\to-\infty}u(s,t)=\gamma_-(t)$ and $\lim_{s\to\infty}u(s,t)=\gamma_+(t)$.

Floer homology: define Morse homology for~$A$ generated by fixed points of~$\phi$
differential counts flow lines as above.

If $H_t$ does not depend on~$t$, this should recover ordinary Morse homology of~$H$.

Technical issues: transversality, grading on chain complex (dimension of moduli
space of flow lines), compactness (counting), gluing ($\partial^2=0$), orientations
(counting with signs), removing symplectically aspherical assumption.

Let $\phi\colon(M,\omega)\to(M,\omega)$ be any symplectomorphism (not necessarily Hamiltonian
isotopic to identity).  Define the mapping torus $Y_\phi=[0,1]\times M/(1,x)\sim(0,\phi(x))$.
$Y_\phi$ fibers over~$S^1$ with fiber~$M$.  There is a vector field~$\partial_t$ on~$Y_\phi$.
Fixed points of~$\phi$ correspond to circles in~$Y$ that are tangent to~$\partial_t$
and go once around the $S^1$ direction.
$\R\times Y_\phi$ has a symplectic form $\Omega=\omega+ds\wedge dt$.
Choose an $\Omega$-tame almost complex structure~$J$ on $\R\times Y$ such that
$J\colon TM\to TM$, $J(\partial_s)=\partial_t$ and $J$ does not depend on~$s$.
Equivalently, choose $\omega$-tame almost complex structure $J_t$ on~$M$ for each $t\in\R$
such that $J_{t+1}=\phi_*\circ J_t\circ\phi_*^{-1}$.  If $X_+$~and~$X_-$ are fixed points of~$\phi$
corresponding to circles $\gamma_+$~and~$\gamma_-$ in~$Y_\phi$, define a flow line from~$X_+$
to~$X_-$ to be a $J$-holomorphic cylinder $C\subset\R\times Y_\phi$ such that $C$ is asymptotic
to $\R\times Y_t$ as $s\to\infty$.

This is more general version of Floer homology for any $\phi\in\Symp(M,\omega)$.
Chain complex is generated by~$\Fix(\phi)$.  Differential counts holomorphic cylinders
in~$\R\times Y_\phi$.  Any flow line as above is a section of $\R\times Y_\phi\to\R\times S^1$.

Why this generalizes previous setup?  Suppose $\phi=\phi_1$ comes from $H\colon S^1\times M\to\R$.
$Y_\phi\leftrightarrow S^1\times M$, $(t,x)\leftrightarrow(t,\phi_t(x))$,
$\gamma\colon S^1\to Y_\phi\leftrightarrow\gamma\colon S^1\to M$.

\proclaim Grading.  Given two fixed points $x_+$~and~$x_-$, what is the dimension
of the moduli space $m(x_+,x_-)$ of flow lines from~$x_+$ to~$x_-$?
Dimension may be different for different components of~$m(x_+,x_-)$.
Let $u$ be a flow lines from~$x_+$ to~$x_-$.
We have $u\colon\R\times\R\to M$ such that $u(s,t+1)=\phi^{-1}(u(s,t))$,
$\partial_su+J_t\partial_tu=0$ and $\lim_{s\to\pm\infty}u(s,t)=\gamma_\pm(t)$.
What is the dimension of~$m(x_+,x_-)$ near~$u$?  Assume everything is transverse.
$u$ corresponds to a cylinder $C\subset\R\times Y$.

Deformation operator $D\colon L^2_1(C,TM)\to L^2(C,T^{0,1}C\otimes_\C TM)$.
If we choose some trivialization of~$TM$ over~$C$, then $D$ has the form
$D\colon L^2_1(\R\times S^1,\R^{2n})\to L^2(\R\times S^1,\R^{2n})$
such that $D\xi=\partial_s\xi+J_0\partial_t\xi+A(s,t)\xi)$.  $J_0$ is the standard
complex structure on~$\R^{2n}$.

$\dim m(x_+,x_-)$ near~$u$ is in $d(D)$.  What is the index of~$D$?
$A_\pm(t)=\lim_{s\to\pm\infty}A(s,t)$ is a symmetric matrix.
We have trivialized $TM$ over $\gamma_\pm$.
Linearization of the flow $\partial_t$ along $\gamma_\pm$ from~$t=0$
to a given~$t$ defines a pair of symplectic linear maps $\psi^\pm_t\colon T_xM\to T_xM$ such that
$\psi^\pm_0=\id_{T_xM}$ and $\psi^\pm_1=d\phi\colon T_xM\to T_xM$.
With respect to trivialization we get two paths of symplectic matrices 
$\psi^\pm_t$.  Assume fixed points are nondegenerate, so 1 is not an eigenvalue of~$\psi^\pm_1$.
$A_\pm(t)$ is given by $(d/dt)\psi^\pm_t=J_0A_t\psi^\pm_t$.

\proclaim Theorem.  Let $A(s,t)$ be matrices parametrized by $(s,t)\in\R\times S^1$
such that $\lim_{s\to\pm\infty}A(s,t)=A_\pm(t)$ is symmetric and 1 is not an eigenvalue
of~$\psi^pm_1$.  Then the operator~$D\colon L^2_1(\R\times S^1,\R^{2n})\to L^2(\R\times S^1,\R^{2n})$
defined by $D\xi=\partial_s\xi+J_0\partial_t\xi+A(s,t)\xi$ is Fredholm and $\ind(D)=\CZ(\psi^+_t)-\CZ(\psi^-_t)$.
CZ is the integer Conley-Zehnder index associated to a path of symplectic matrices.

Recall that we have a symplectic manifold~$(M,\omega)$ with an automorphism~$\phi$.
Consider the mapping torus of~$\phi$: $Y_\phi=[0,1]\times/(1,x)\sim(0,\phi(x)=\R\times M/(t+1,x)\sim(t,\phi(x))$.
Fixed points of~$\phi$ correspond to parallel sections.  Choose an $\omega$-tame almost
complex structure on~$E$, or equivalently $J_t$ on~$M$
such that $J_{t+1}=\phi_*J_t\phi_*^{-1}$.

Floer homology of $\phi$: Chain complex generators are fixed points of~$\phi$, regarded
as circles in~$Y_\phi$, differential counts holomorphic sections of~$\R\times Y_\phi$:
If $\gamma_+$~and~$\gamma_-$ are circles in~$Y_\phi$ corresponding to $x_+$~and~$x_-$
in~$\Fix(\phi)$, then $\langle\partial x_+,x_-\rangle$ counts
maps $u\colon\R\times\R\to M$ such that $u(s,t+1)=\phi^{-1}(u(s,t))$,
$\partial_su+J_t\partial_tu=0$ and $\lim_{s\to\pm\infty}u(s,t)=\gamma_\pm(t)=x_\pm$.

The difference between gradings of $x_+$ and $x_-$ equals expected dimension of~$m(x_+,x_-)$.
Compute this.  Given $u\in m(x_+,x_-)$ compute $\ind(D_u)$.
We have $D_u\colon L^2_1(\R\times S^1,u^*E)\to L^2(\R\times S^1,u^*E)$.
and $D_u\xi=\partial_s\xi+J_t\partial_t\xi+A(s,t)\xi$.
Choose Hermitean trivialization~$\tau$ of $u^*E$ over $\R\times S^1$.
We have $u^*E=\R\times S^1\times\R^{2n}$ and $D_u\xi=\partial_s\xi+J_0\partial_s\xi+A(s,t)\xi$,
where $J_0$ is the standard complex structure on~$\R^{2n}$.
$\tau$ restricts to a trivialization $\tau_\pm$ of~$E$ over~$\gamma_\pm$.
For $|s|\gg0$ we have $\partial_s\xi+J_0\partial_t\xi+A(s,t)\xi\approx\partial_s\xi+J_0\nabla_t\xi$.
With respect to $\tau_\pm$ parallel transport along $\gamma_\pm$ from~0 to~$t$
defines a symplectic map $\psi^\pm_t\in\Symp(\R^{2n},\omega_0)$.
The path $\psi^\pm_t$ of symplectic matrices is equivalent to a path $A^\pm_t$ of symmetric
matrices via $(d/dt)\psi^\pm_t=J_0A^\pm_t\psi_t$.
Conclusion: $D_u\xi=\partial_s\xi+J_0\partial_t\xi+A(s,t)\xi$ where $\lim_{s\to\pm\infty}A(s,t)=A^\pm_t$.
Theorem: If $1\notin\Spec(\psi^\pm_1)$ then $D$ is Fredholm and $\ind(D)=\CZ(\psi^+_t)-\CZ(\psi^-_t)$.

Let $\{\psi_t\mid t\in[0,1]\}$ be a path of symplectic matrices on~$\R^{2n}$ with
$\psi_0=\id$ and $1\notin\Spec(\psi_1)$.  Define the Conley-Zehnder index
$\CZ(\psi_t)\in\Z$ as follows.  Define the Maslov cycle $M=\{A\in\Sp_{2n}(\R)\mid1\in\Spec(A)\}$.
Roughly speaking, $\CZ(\psi_t)$ is the signed count of $\psi_t$ with~$M$.
Note: $\U(n)=\Sp(2n)\cap O(n)$ is a maximal compact subgroup of~$\Sp(2n)$.
$H^1(\Sp(2n),\Z)=\Z$ is generated by a continuous extension of $\det\colon U(n)\to S^1$.
$M$ is a co-oriented codimension~1 subvariety of~$\Sp(2n)$, Poincar\'e dual to the generator
of $H^1(\Sp(2n),\Z)$.  So if $\{\psi_t\mid t\in S^1\}$ is an arbitrary loop in $\Sp(2n)$ then
$\{\psi_t\}\cap M\in\Z$ is defined.
To define CZ index, declare $\CZ\left(t\to\bigoplus_n\pmatrix{e^t&0\cr0&e^{-t}\cr}\right)=0$.
To defined $\CZ(\psi_t)$, let $\gamma$ be a path from $\bigoplus_n\pmatrix{e^t&0\cr0&e^{-t}\cr}$
to~$\psi_1$, which is transverse to~$M$ such that $\eta+\gamma-\{\psi_t\}=0\in H_1(\Sp(2n))$.
Define $\CZ(\psi_t)=\#(\gamma\cap M)$.

General properties: Naturality: If $\phi\colon[0,1]\to\Sp(2n)$ is an arbitrary path,
then $\CZ(\phi\psi\phi^{-1})=\CZ(\psi)$.  Inverses: $\CZ(\psi^{-1})=-\CZ(\psi)$.
Change of trivialization: If $\phi\colon[0,1]\to\Sp(2n)$ is a path such that
$\phi(0)=\phi(1)=\id$, then $\CZ(\phi\psi)=\CZ(\psi)+2\deg\phi$.  Signature:
If $A$ is a symmetric matrix with $\det(A)\ne0$ and $\|A\|<2\pi$ then
$\CZ(\exp(J_0At))=\sigma(A)/2$.

Why is $\ind(\partial_s+J_0\partial_t+A(s,t))=\CZ(\psi^+_t)-\CZ(\psi^-_t)$?
We omit the proof of Fredholm property.  To compute index we assume that $A(s,t)$ is symmetric.
For each $s\in\R$ take a family of symplectic matrices $\psi_{s,t}$ defined by $\psi_{s,0}=\id$,
$(d/dt)\psi_{s,t}=J_0A_{s,t}\psi_{s,t}$.  Claim: For a given~$s$ we have $0\in\Spec(J_0\partial_t+A(s,t))$ iff $1\in\Spec(\psi_{s,1})$.  $\CZ(\psi^+_t)-\#(\psi_{s,1})\cap M-\CZ(\psi^-_t)=0$.

\proclaim Theorem.  If the almost complex structures~$J_t$ on~$M$ are generic, then
for any $m\in m(x_+,x_-)$, the operator~$D_u$ is surjective, so $m(x_+,x_-)$ is a manifold
near~$u$ of dimension~$\ind(u)$.

\proclaim Proof.  One needs to show that if $u$ is constant, then $D_u$ is always surjective.
If $u$ is nonconstant, then the projection of~$u$ to~$Y_\emptyset$ is somewhere injective.

\proclaim Index theorem for Cauchy-Riemann operators on Riemann surfaces with cylindrical ends.
As\-sump\-tions: $C$ is a Riemann surface with ends identified with $[0,\infty)\times S^1$,
$E$ is a rank~$n$ complex vector bundle on~$C$ (with Hermitean metric),
$D\colon L^2_1(E)\to L^2(T^{0,1}C\otimes E)$ such that in local coordinates and trivialization
$D=\partial_s+i\partial_t+{\rm zeroth\ order\ term}$,
on each end, for some trivialization of~$E$, $D=\partial_s+i\partial_t+A(s,t)$ where
$\lim_{s\to\infty}A(s,t)=A(t)$ symmetric, and if $\psi_0=\id$ and $(d/dt)\psi_t=J_0A(t)\psi_t$,
then $1\notin\Spec(\psi)$.

\proclaim Theorem.  $D$ is Fredholm and $\ind(D)=n\chi(C)+2c_1(E,\tau)+\sum_{\rm ends}\CZ_\tau$.

To compute $c_1(E,\tau)$: take a generic section~$s$ of~$\Lambda^nE$ such that on each end,
$s$ is nonvanishing and constant with respect to trivialization~$\tau$.
Then $c_1(E,\tau)=\#s^{-1}(0)$.  This depends only on~$E$ and homotopy class of~$\tau$.
$\CZ_\tau$ is the CZ index of the path~$\psi_t$ obtained as above.  (Only depends on
homotopy class of~$\tau$.)
We can identify some ends with~$(-\infty,0]\times S^1$ instead, in which case
you subtract the corresponding CZ terms instead of adding them.
Granted that $D$ is Fredholm, prove index formula as follows:
the right hand side of formula is well defined, i.e., does not depend on~$\tau$;
if $C$ is a cylinder, then the theorem is true;
if $C$ has no ends, then the theorem is true (by Riemann-Roch);
index is additive under gluing.

Glue some ends of~$C_1$ to some ends of~$C_2$.
If the operators agree on the global ends, then we can glue $E_1$~and~$E_2$ (using this on ends)
to a bundle~$E_1\#E_2$ over~$C_1\#C_2$ and glue $D_1$~and~$D_2$ to $D_1\#D_2$ over
$C_1\#C_2$.  Then $\ind(D_1\#D_2)=\ind(D_1)+\ind(D_2)$.
Idea of additivity: neck stretching.

Why is $n\chi(C)+2c_1(E,\tau)+\sum_{\rm ends}\CZ_\tau$ independent of~$\tau$?
For any given end, the set of homotopy classes of trivializations is an affine space over
$\pi_1U(n)=\Z$.  If you shift the trivialization by~1, then $c_1$ changes by~$\pm1$.
If $s$ is any generic section of~$E$ which is nonvanishing on ends, then $c_1(E,\tau)=\#s^{-1}(0)
-\sum$.
If you shift trivialization by~1, $\CZ_\tau$ shifts by~$\mp2$.
We now know: index formula is true for cylinders and closed surfaces
and both sides of index formula are additive under gluing.
Next step: deduce index formula when $C$ is a disc.
For an arbitrary~$C$, cap off the ends with discs.
Since formula is true for closed surface and for discs, by additivity it is true for~$C$.

\proclaim Back to Floer homology.  $H\colon S^1\times M\to\R$ generates $\phi\colon(M,\omega)\to(M,\omega)$ and Hamiltonian isotopy $\phi_t$ from~1 to~$\phi$.
Assume $M$ is symplectically aspherical: $\langle c_1(TM),\pi_2(M)\rangle=\langle\omega,\pi_2(M)\rangle=0$.
$\CF_*(H)$ is the free $\Z$-module generated by contractible loops
$\gamma\colon S^1\to M$ with $\gamma'(t)=X_{H_t}$.
(These are some of the fixed points of~$\phi$.)
Grading: Let $\gamma\colon S^1\to M$ be a generator.  Define $\mu(\gamma)\in\Z$ as follows.
Choose a map $\eta\colon D^2\to M$ such that $\eta$ restricted to~$\partial D^2$ is~$\gamma$.
Trivialize $\eta^*TM$ use this to trivialize $\gamma^*TM$.
Homotopy class of trivialization of~$\gamma^*TM$ does not depend on~$\eta$ because
$\langle c_1(TM),\eta-\eta'\rangle=0$.
Linearization of equation, $\gamma'(t)=X_{H_t}$ defines a family of symplectic matrices
$\psi(t)\colon TM_{\gamma(0)}\to TM_{\gamma(t)}$.

\proclaim Floer homology.  We use $\Z/(2)$ coefficients.
Assume $\phi$ has nondegenerate fixed points.
Choose a generic family of $\omega$-tame almost complex structures~$J_t$ on~$M$ for~$t\in S^1$.
Define $(\CF_*(H,J),\partial)$ as follows.
$\Fix(\phi)=\{\gamma\colon S^1\to M\mid\gamma'(t)=X_{H_t}(\gamma(t))\}$.
$\CF_*$ is the free $\Z/(2)$-module generated by fixed points corresponding to contractible~$\gamma$,
with $\Z$-grading.  Given a generator~$\gamma$, let $u\colon D^2\to M$ be a map such that
its restriction to~$S^1$ is~$\gamma$.
This determines a homotopy class of trivialization of~$\gamma^*TM$, independent of~$u$
because $\langle c_1(TM),u-u'\rangle=0$.  With this trivialization, $\{d\phi_t\colon T_{\gamma(0)}M\to T_{\gamma(0)}M\}_{t\in[0,1]}$ is a path of symplectic matrices from~1 to~$\mu(x)=\CZ$.

Given fixed points $x_+$~and~$x_-$ corresponding to $\gamma_+$~and~$\gamma_-$ let
$m(x_+,x_-)=\{u\colon\R_s\times S^1\to M\mid\partial_su+J_t(\partial_tu-X_{H_t})=0\land\lim_{s\to\pm\infty}u(s,\cdot)=\gamma_\pm\}$.
By previous theorem, if $J$ is generic, $\dim m(x_+,x_-)=\mu(x_+)-\mu(x_-)$.
$\R$ acts on $m(x_+,x_-)$ by translating~$s$.

\proclaim Definition.  $\partial\colon\CF_*\to\CF_{*-1}$: $\partial x_+=\sum_{x_-\colon\mu(x_+)-\mu(x_-)=1}\#(m(x_+,x_-)/\R)x_-$.

\proclaim Lemma.  $\partial$ is well defined, i.e., $m(x_+,x_-)/\R$ is finite when $\mu(x_+)-\mu(x_-)$.

\proclaim Gromov compactness.  For any closed~$(M,\omega)$ and any $x_+$~and~$x_-$ a sequence
in $m(x_+,x_-)$ has a subsequence which ``converges'' to a ``broken flow line'' with
``bubble trees'' attached.  If $(M,\omega)$ is symplectically aspherical, then we do not
have any bubbles because any holomorphic sphere has $\int\omega>0$.

\proclaim Theorem.  $\partial^2=0$.

\proclaim Theorem.  $\HF_*(H,J)$ does not depend on $(H,J)$.

\proclaim Proof idea.  Consider generic family $\{(H_s,J_s)\mid s\in\R\}$.
Assume $(H_s,J_s)=(H_+,J_+)$ for large positive~$s$, $(H_s,J_s)=(H_-,J_-)$ for large negative~$s$.
Define $\Phi\colon\HF_*(H_+,J_+)\to\HF_*(H_-,J_-)$.
Choose $x_+$~and~$x_-$ in~$\Fix(\phi_\pm)$ corresponding to~$\gamma_\pm\colon S^1\to M$.
Let $\Phi(x_+)=\sum_{x_-\colon\mu(x_+)=\mu(x_-)}\#m(x_+,x_-)x_-$.
Similarly to Morse theory case, $\Phi$ is a chain map, induces an isomorphism on homology
depending only on the homotopy class of the path from~$(H_+,J_+)$ to~$(H_-,J_-)$.

\proclaim Remark.  $\HF_*(H,J)$ does depend on~$\phi$ in the sense that if $(H_+,J_+)$
and $(H_-,J_-)$ has $\phi_+$~and~$\phi_-$ the map $\Phi\colon\HF_*(H_+,J_+)\to\HF_*(H_-,J_-)$
might be nontrivial.

\proclaim Theorem.  $\HF_*(H,J)=H_{*+n}(M,\Z/(2))$.

\proclaim Proof.  Take $H_t\colon M\to\R$ independent of~$t$, Morse function $H\colon M\to\R$.
(May have to replace $H$ by $\epsilon H$, $\epsilon>0$ small.)
Take $J_t=J$ independent of~$t$, let $g$ be the corresponding metric.
Can arrange that $(H,g)$ is Morse-Smale.

\proclaim Claim.  If we replace $H$ by $\epsilon H$ for $\epsilon>0$ sufficiently small, then
$(\CF_*(H,J),\partial)$ is well-defined and equal to $(C^M_*(H,g),\partial)$.
Need: Generators are the same.  Gradings.  Floer differential is defined.  Differentials agree.
(1)~$\Crit(H)\subset\Fix(\phi)$.  This is an equality if $\epsilon$ is sufficiently small.
(2)~Recall $\CZ\{\exp(J_0At)\mid t\in[0,1]\}=\sigma(A)/2$.
If $x\in\Crit(H)$, then $d\phi_t\colon T_xM\to T_xM$ is $\exp(t\nabla X_H)=\exp(-tJ\cdot{\rm Hess}(H,x))$, therefore $\mu(x)=-\sigma({\rm Hess})/2=-n+\ind(f,x)$.
(3)~and~(4): If $\mu(x_+)-\mu(x_-)=1$, then every $u\in m(x_+,x_-)$ is independent of~$t$.
If $u\in m(x_+,x_-)$ is independent of~$t$, then $D_u$ is surjective.
$D_u\colon L^2_1(\R\times S^1,\R^{2n})\to L^2(\R\times S^1,\R^{2n})$.
$D_u\xi=\partial_s\xi+J_0\partial_t\xi+A(s)\xi$, where $\partial_s+A(s)$ is the Morse theory
deformation operator.
Show: If $\epsilon$ is small enough, then any $\xi\in\ker(D_u)$ is independent of~$t$.
Then the same argument implies that anything in $\coker(D_u)=\ker(D_u^*)$ is $t$-independent.
Let $\xi\in\ker(D_u)$.  Write $\xi=\xi_0+\xi_1$, where
$\xi_0(s,t)=\int_{\tau\in S^1}\xi(s,\tau)d\tau$.
Wlog $\xi=\xi_1$.
We see that $\|\xi\|_{L^2}\le c\epsilon\|xi\|_{L^2}$.

Suppose $(M,\omega)$ is symplectically aspherical, $H\colon S^1\times M\to\R$,
$J_t$ ($t\in S^1$) is a family of almost complex structures.
Then $\HF_*(H,J)$ is independent of $(H,J)$ and $\HF_*(H,J)=\HF_{*+n}(M,\Z/(2))$.

Take $H\colon M\to\R$ independent of~$t$, $H\to\epsilon H$, $\epsilon>0$ small,
$J$ independent of~$t$ corresponding to~$g$.
Then $\CF_i(H,J)=C^M_{i+n}(f,g)\otimes\Z/(2)$.
Now $t$-independent $J$-holomorphic curve corresponds to gradient flow line.
Transversality as a holomorphic cylinder corresponds to transversality
as a gradient flow line.

Last step: If $\epsilon$ is sufficiently small, then every solution to
the equation $\partial_su+J(\partial_tu-X_H)=0$~$(*)$,
where $\lim_{s\to\pm\infty}u(s,t)=x_\pm$ and $\ind(x_+)-\ind(x_-)=1$ is $t$-independent.
Morally $S^1$ acts on the space of solution by rotating~$t$.  If $J$ is regular then
all solutions are $S^1$-independent, otherwise the dimension of moduli space is too big.
If $J$ is not regular, then ``localization'' works.
Direct argument in this case: Suppose that for any $\epsilon>0$ there is a $t$-dependent
solution.  Start with $\epsilon_0>0$ sufficiently small that all previous steps work.
For every positive integer~$n$ there is a $t$-dependent solution to the equation
for~$\epsilon_0/n$: $\partial_su_n+J(\partial_tu_n-X_H/n)=0$.
Define $v_n(s,t)=u_n(ns,nt)$.  Then $\partial_sv_n+J(\partial_tv_n-X_H)=0$
and $v_n(s,t+1/n)=v_n(s,t)$.
By Gromov compactness, a subsequence of~$v_n$ converges to a solution of~$(*)$.
By previous equation, $v_\infty$ is $S^1$-invariant.
Since $v_\infty$ is transverse, $v_n=v_\infty$ for large~$n$ up to $\R$-translation.

\proclaim Definition.  A symplectic manifold $(M,\omega)$ is called monotone
if there is a $\lambda>0$ such that $\langle c_1(TM),A\rangle=\lambda\langle\omega,A\rangle$ for
all~$A\in\pi_2(M)$.
Definition of $\HF_*(H,J)$ in the monotone case: Again, $\CF_*(H,J)$ is the free $\Z/(2)$-module
generated by contractible loops $\gamma\colon S^1\to M$ such that $\gamma'(t)=X_{H_t}$.

Grading is only defined in $\Z/N$, where $N=2\min\{\langle c_1(TM),A\rangle\mid A\in\pi_2(M)\land
\langle c_1(TM),A\rangle>0\}$.
If $\gamma\colon S^1\to M$ is a generator, let $u\colon D^2\to M$ be a map such that it restriction
to the boundary is~$\gamma$.  Let $\tau$ be a trivialization of $\gamma^*TM$ that extends
to a trivialization of $u^*TM$ with respect to~$\tau$,
$d\phi_t\colon TM_{\gamma(0)}\to TM_{\gamma(t)}$ is a symplectic matrix~$\psi_t$.
Define grading $\mu(\gamma)=\CZ\{\psi_t\mid t\in[0,1]\}$.
If $u'$ is another disk, then $\CZ\{\psi_t\}=\CZ\{\psi'_t\}=\pm2\langle c_1(TM),u-u'\rangle$.
Differential: $\partial\gamma=\sum_{\mu(\gamma)-\mu(\gamma')=1}\#m_1(\gamma,\gamma')\gamma'$,
where $m_1(\gamma,\gamma')=\{u\in m(\gamma,\gamma')\mid\ind(D_u)=1\}$.
Claim: $\partial$ is well-defined, $\partial^2=0$.
Compactness argument.  Suppose $u_n\in m(\gamma,\gamma')$, $\ind(D_{u_*})\in\{1,2\}$.
Subsequence converges to a cylinder with bubbles.
We have $\ind(u_n)=\sum_i\ind(v_i)+\sum_j2\langle c_1(TM),[S_j]\rangle$,
hence there are no bubbles and previous argument applies.
As before, $\HF_*(H,J)=\oplus_{i\equiv *+n\pmod{N}}H_i(M,\Z/(2))$.

Bad case: there are holomorphic spheres~$S$ with $\langle c_1(TM),[S]\rangle<0$.
Multiple covers of~$S$ have very negative~$c_1$.
Not so bad case: $\langle c_1(TM),A\rangle=0$ for all $A\in\pi_2(M)$.  Novikov rings.

Recall that a Lagrangian in $(M^{2n},\omega)$ is a closed submanifold
$L^n\subset M$ such that $\omega$ restricted to~$L$ is~0.
Let $L_1$~and~$L_2$ be two Lagrangians intersecting transversally.
Idea: define $\HF_*(L_1,L_2)$ and $\CF_*(L_1,L_2)$ generated by intersection points.
Choose $J_t$, an $\omega$-tame almost complex structure for $t\in[0,1]$.
Differential $\langle\partial x_+,x_-\rangle$ counts $u\colon\R\times[0,1]\to M$ such that
$u(s,0)\in L_1$, $u(s,1)\in L_2$, $\lim_{s\to\pm\infty}u(s,t)=x_\pm$ and
$\partial_su+J_t\partial_tu=0$.  Under favorable circumstances, $\partial$ is well defined,
$\partial^2=0$, $\HF_*(L_1,L_2)$ is invariant under appropriate isotopy of $L_1$~and~$L_2$.
Bad stuff: bubbling of holomorphic spheres and bubbling of holomorphic discs with boundary
on $L_1$~and~$L_2$.
Let $L$ be the manifold of all Lagrangian liner subspaces of~$(\R^{2n},\omega)$.
We claim that $\pi_1(L)=\Z$.
The generator comes from $\{\exp(i\pi t)\R\subset\C\mid t\in[0,1]\}$.
Relative grading: If $x_+$~and~$x_-$ belong to~$L_1\cap L_2$ and there is~$u$ satisfying
conditions above, define $\mu(x_+)-\mu(x_-)\in\Z$.
Given~$u$, trivialize $u^*TM$ such that $TL_1=\R^n\oplus\{0\}\subset\R^{2n}$ over~$\R\times\{0\}$
and $TL_2=\{0\}\oplus\R^n$ over $\{0\}\times[0,1]$.  Along $\R\times\{1\}$, $TL_2$ defines
a loop of Lagrangians starting and ending at~$\{0\}\oplus\R^n$.  Then $\mu(x_+)-\mu(x_-)$ is the
integer corresponding to the given element of the fundamental group.

Last time we learned Lagrangian Floer homology.  If $L_0$~and~$L_1$ are two
Lagrangian submanifolds of~$(M,\omega)$ intersecting transversally
and $\CF_*(L_0,L_1)$ is a free $\Z/(2)$-module generated by intersection points
with relative grading given by Maslov index.  Relative grading lies inside~$\Z/(n)$.
If $L_0$~and~$L_1$ are oriented, then we have absolute $\Z/(2)$ grading by intersection sign.
Differential counts holomorphic strips $u\colon\R\times[0,1]\to M$
such that $u(s,0)\in L_0$, $u(s,1)\in L_1$, $\lim_{s\to\pm\infty}u(s,t)=x_\pm$,
$\partial_su+J_t\partial_tu=0$.
In good cases, $\HF_*(L_0,L_1)$ is well-defined and invariant under Hamiltonian isotopy of
$L_0$~or~$L_1$, e.g., if two noncontractible circles on a surface are Hamiltonian isotopic
then they must intersect.

Why does it matter that $L_0$~and~$L_1$ are Lagrangian?  If two strips are homotopic as such,
then the corresponding integrals coincide.  Why is $\HF_*(L_0,L_1)$ invariant only under
Hamiltonian and not symplectic isotopy?

\proclaim Example.  Suppose $f\colon(M,\omega)\to(M,\omega)$ is a symplectomorphism.
Its graph~$\Gamma$ is Lagrangian.  The diagonal~$\Delta$ is also Lagrangian.
The intersection $\Gamma\cap\Delta$ is the set of fixed points of~$f$.

\proclaim Theorem.  $\HF_*(\Gamma,\Delta)=\HF_*(f)$.
To define $\HF_*(f)$, choose~$J_t$, $t\in\R$, $J_{t+1}=f_*J_tf_*^{-1}$.

\proclaim Floer homology for symplectomorphisms of surfaces.  Suppose we have a surface
of genus greater than~1.  Let $f\colon\Sigma$

Nielsen-Thurston classification of surfaces diffeomorphism.
Properties.  Every diffeomorphism~$f$ is isotopic to one of the following:
finite order: ($f^n=f$ for some~$m$);
reducible: there is an essential arc $\gamma\subset\Sigma$ such that $f(\gamma)=\gamma$;
Pseudo-Anosov: two transverse singular foliations $f_1$~and~$f_2$ on~$\Sigma$.

Mapping torus: $Y_f=[0,1]\times\Sigma/(1,x)\tilde(0,f(x))$.  $\omega$ on~$\Sigma$
induces a closed form~$\omega$ on~$Y_f\to[\omega]+H^2(\Upsilon,\R)$.

\proclaim Definition.  $f$ is monotone if $[\omega]=\lambda c_1(E)$ in $H^2(Y,\R)$
for some~$\lambda\in\R$.  If $f$ is monotone, define $\HF_*(f)$ as follows.
$\CF_*(f)=\Z/(2)\Fix(f)$.
For $J_t$ on $\Sigma$ we have $J_{t+1}=F_*J_tF_*^{-1}$.
We obtain a $\Z/(2)$-grading by sign of fixed points.
Now $\langle\partial x_+,x_-\rangle=\#m_1(x_+,x_-)/\R$.
For compactness part of proof that $\partial$ is well-defined, $\partial^2=0$,
need that $u_n$ is a sequence in $m_1(x_+,x_-)$ then $\int_{u_n}\omega<c$.
If $u$~and~$u'$ are in~$m_1(x_+,x_-)$, then $\langle c_1(E),u-u'\rangle=0$,
therefore $\langle[\omega],u-u'\rangle=0$.

\proclaim Example.  If $f$ is isotopic to identity then $f$ is monotonic iff $f$ is Hamiltonian
isotopic to identity.  More generally, suppose $\{f_t\mid t\in[0,1]\}$ is a symplectic
isotopy.  This induces a diffeomorphism $\phi\colon Y_{f_0}\to Y_{f_1}$,
$[\omega_0]\ne\phi^*[\omega_1]$ and $c_1(E_0)=\phi^*c_1(E_1)$.

$[\omega_0]$ relates to $\phi^*[\omega_1]$ as follows:
$0\to\Z\to H_2(Y_f)\to\ker(1-f_*)\to0$.

\proclaim Conclusion.  $\{f_t\}$ preserves monotonicity iff flow $\{f_t\}$ iff $(\Sigma,\R)$ annihilates ker. %???
Also, any $f$ is isotopic to a monotonic symplectomorphism.

Suppose $\Sigma$ is a closed surface, $\omega$ is an area form on~$\Sigma$,
$\phi\colon(\Sigma,\omega)\to(\Sigma,\omega)$ is a symplectomorphism.
The mapping torus of~$\phi$ is $Y_\phi=[0,1]\times\Sigma/(1,x)\sim(0,\phi(x))$.
Suppose that $\omega$ extends to a closed 2-form on~$Y$
and $\R^2\to E\to Y$ is the vertical tangent bundle of $\Sigma\to Y\to S^1$.
$\phi$ is monotone if $[\omega]=\lambda c_1(E)$ in $H^2(Y,\R)$.

\proclaim Fact.  (Seidel, Pacific Journal of Mathematics.)
This map from monotone symplectomorphisms to ori\-en\-ta\-tion-preserving diffeomorphisms of~$\Sigma$
is a homotopy equivalence.

Assume $\phi$ is monotone and has nondegenerate fixed points.  Define $\HF_*(\phi)$ as follows:
$C_*$ is the $\Z/(2)$-module generated by fixed points graded by
Lefschetz sign.  Choose $J_t$ on~$\Sigma$ as usual.
Now $\partial p=\sum_qq\cdot\#m_1(p,q)/\R$.
Monotonicity implies that $\partial$ is well-defined.

\proclaim Lemma.  If $J$ is generic then for any $p,q\in\Fix(\phi)$ the set
$m_1(p,q)/\R$ is finite.

\proclaim Proof.  Key is that if $c,c'\in m_1(p,q)$ then $\int_c\omega=\int_{c'}\omega$.
Because if $\ind(c)=\ind(c')$ then $\lambda\langle c_1(E),[c-c']\rangle=0=\langle[\omega],[c-c']\rangle=\int_c\omega=\int_{c'}\omega$.

Gromov compactness: if $\{c_n\}$ is a sequence in $m(p,q)$ with $\int_{c_n}\omega<R$ then
there is a subsequence converging to a ``broken trajectory'' from~$p$ to~$q$.  No bubbling
because $\pi_2(\Sigma)=0$.

Suppose $m_1(p,q)/\R$ is infinite.  Let $\{c_n\}$ be a sequence of distinct elements
in $m_1(p,q)/\R$.  Then a subsequence converges to a broken trajectory
$(\hat c_0,\ldots,\hat c_k)$.  If $k=0$ then $\hat c_0\in m_1(p,q)/\R$ is not isolated,
contradicting transversality.  If $k>0$ then $\sum_{0\le i\le k}\ind(c_i)=1$.
So some $\hat c_i$ has $\ind(\hat c_i)\le0$, again contradicting transversality.
Similarly, $\HF_*(\phi)$ depends only on mapping class of~$\phi$.

\proclaim Examples.  (1)~$\phi=\id$, $Y=S^1\times\Sigma$.  $\HF_*(\id)=H_*(\Sigma)$.
(2)~$\phi$ is finite order ($\phi^n=1$).
All fixed points have the same $\Z/(2)$-grading, hence $\HF_*(\phi)=\oplus_{p\in\Fix(\phi)}\Z/(2)$.
(3)~$\phi$ is a Dehn twist.  Let $\gamma\subset\Sigma$ be an embedded circle,
$N$ be a neighborhood of~$\gamma$, $N\cong[0,1]\times S^1\supset N'\cong[\epsilon,1-\epsilon]\times S^1$.
$\phi=\id$ on~$\Sigma\setminus N$.  On $N$, $\psi(x,y)=(x,y-x)$.
We have $\HF_*(\phi)=H_*(\Sigma\setminus\gamma)$.  Proof: Do Hamiltonian isotopy so that~$\phi$
has the following form.
On~$N'$, $\phi(x,y)=(x,y-x)$.  On $\Sigma\setminus N'$, $\phi$ is the time~1 flow of~$X_H$,
where $H\colon\Sigma\setminus N'\to\R$ is a Morse-Smale.
$\Fix(\phi)=\Crit(H)$.  Choose $J$ as usual.  Lemma: If $c\in m(p,q)$, then $c$ does not
intersect~$N'$.  This lemma implies that $\HF_*(\phi)=H^M_*(H)=H_*(\Sigma\setminus N')
=H_*(\Sigma\setminus\gamma)$.  Proof: Let $c\in m_1(p,q)$.  Let $x\in[\epsilon,1-\epsilon]$.
Let $T$ be the mapping tows of $\phi$ restricted to $\{x\}\times S^1$.
Want to show that $C\cap T=\phi$, $\R\times T=\R_s\times S^1_t\times S^1_y$.
$J(\partial/\partial s)=\partial/\partial t-x\partial/\partial y$.
Let $F$ be the foliation of~$T$ generated by~$\partial/\partial t-x\partial/\partial y$.
Then $\R\times F$ is a holomorphic foliation of~$\R\times T$.
Wlog $x$ is rational and $c$ is transverse to~$T$.
$[c\cap(\R\times T)]=(a,b)\in H_1(\R\times T)$.  Since $c$ has positive intersection
with the holomorphic cylinders in~$\R\times F$, we have $ax-b\ge0$, equality only
if~$c\cap(\R\times T)=\emptyset$.
In particular, if $c\cap(\R\times T)\ne\emptyset$, then $[c\cap(\R\times T)]\ne0$.
To prove lemma, show $[c\cap(\R\times T)]=0$.
Write $[c]=z_0+z$, where $z_0$ is the real homology class of Morse cylinder from~$p$ to~$q$
and $z\in H_2(Y)$.  Need to show $z\cap(\R\times T)=0$.
$\ind(c)=\ind(H,p)-\ind(H,q)+2\langle c_1(E),z\rangle$.
$H_2(Y)=(S^1\otimes\{\alpha\in\Sigma\mid\alpha\cdot\gamma=0\})\oplus H_2(\Sigma)$.
Write $z=(z_1,z_2)$.  $1=\ind(c)=\ind(H,p)-\ind(H,q)+2(2-2g)z_2$, therefore $z_2=0$.

\proclaim Contact geometry.  Let $Y$ be a closed oriented 3-manifold.
A {\it contact form\/} on~$Y$ is a 1-form~$\lambda$ such that $\lambda\wedge d\lambda>0$.
Let $\xi=\ker(\lambda)$.  This is an oriented 2-plane field on~$Y$.
$\xi$ is called a {\it contact structure}.
Note: $\xi$ is totally nonintegrable.  (The kernel of~$\lambda$ is a foliation iff $\lambda\wedge
d\lambda=0$.)
$\lambda$~and~$\lambda'$ determine the same~$\xi$ iff $\lambda'=f\lambda$ for some
$f\colon Y\to\R_{>0}$.
(If $\dim(Y)=2n-1$, we require $\lambda\wedge(d\lambda)^{n-1}>0$.)
Example: Standard contact form on~$\R^3$: $\lambda=dz-ydx$.
Darboux-type theorem: Any contact structure is locally isomorphic to this one.
Gray stability theorem: If $\xi_t$ is a family of contact structures for~$t\in[0,1]$,
then there is a family of diffeos $\phi_t\colon Y\to Y$ such that $\phi_0=\id$
and $\phi_{t*}\xi_0=\xi_t$.

Reference: John Etnyre, Introductory Lectures in Contact Geometry.

An overtwisted contact form on~$\R^3$: $\lambda=\cos(r)dz+\sin(r)d\theta$.
This contact structure is not diffeomorphic to the previous one.

Why do we care?  Contact manifolds are natural odd-dimensional counterparts of symplectic
manifolds.  Information from contact geometry can give topological invariants of 3-manifolds.

If $Y$ is a 3-manifold with a contact form~$\lambda$, define the {\it symplectization\/}
$(\R_s\times Y,d(\exp(s)\lambda))$.  Check symplectic: $d(\exp(s)\lambda)
=\exp(s)(ds\wedge\lambda+d\lambda)$.

\proclaim Definition.  Let $(Y_+,\xi_+)$ and $(Y_-,\xi_-)$ be contact 3-manifolds.
A {\it symplectic cobordism\/} from $(Y_+,\xi_+)$ to $(Y_-,\xi_-)$ is a compact symplectic
4-manifold $(X,\omega)$ such that $\partial X=Y_+\sqcup-Y_-$ and there are contact
forms $\lambda_\pm$ with $\xi_\pm=\ker(\lambda_\pm)$ such that $\omega|_{Y_+}=d\lambda_+$
and $\omega|_{Y_-}=d\lambda_-$.

\proclaim Example.  $(\R^4,\omega)\supset(U,\omega|_U)$.
Under appropriate convexity conditions, $\partial U$ has a contact
form~$\lambda$ with $\omega|_{\partial U}=d\lambda$.

\proclaim Definition. $(Y,\xi)$ is symplectically fillable if there exists a symplectic
cobordism from~$(Y,\xi)$ to~$\phi$.

\proclaim Example.  There is a ``functor'' from differential topology to contact geometry.
Let $M$ be any smooth manifold.  Choose a metric on~$M$.
Let $ST^*M$ be the unit cotangent bundle of~$M$.  This has an obvious canonical contact form,
which is obtained by tautological mapping of tangent bundle of~$ST^*M$ into~$T^*M$.
This contact manifold does not depend on a metric on~$M$.

Let $(Y^{2n-1},\xi)$ be a contact manifold.  Let $\xi=\ker(\lambda)$ and $\lambda\wedge(d\lambda)^{n-1}>0$.
A {\it Legendrian submanifold\/} of~$(Y,\xi)$ is a submanifold~$L^{n-1}\subset Y$ such that
$TL\subset\xi|_L$.  ($\R\times L$ is Lagrangian in~$\R\times Y$.)

\proclaim Example.  Legendrian knots in~$\R^3$.  Tangent vector cannot be vertical.
They are uniquely determined by the topological type of their projection to xy-plane
and the areas of all parts of the plane obtained by projection.
They have two invariants: rotation number (if we orient the knot)
and Thurston-Bennequin invariant.
Note: contact homology distinguishes Legendrian knots which are isotopic
as smooth knots and have the same rotation number and Thurston-Bennequin invariants.

If $M$ is a smooth manifold and $N$ is a submanifold,
then $(ST^*M,\xi)$ contains the conormal bundle of~$N$ ($\{(x,y)\}\mid x\in N\land y\in ST^*_xM\land y|_{T_xN}=0\}$.
Smooth isotopy of~$N$ gives a Legendrian isotopy of~$L(N)$.
For example, smooth knot in~$\R^3$ turns into Legendrian submanifold of~$\R^3\times S^2$,
then contact homology gives us an invariant that distinguished the unknot.

\proclaim Definition.
Let $(Y^3,\xi)$ be a contact 3-manifold.  $\xi$ is overtwisted if there is an embedded
disk $D\in Y$ such that $\xi|_{\partial D}=TD|_{\partial D}$.

Example: Standard contact structure on~$\R^3$ is tight.

\proclaim Theorem.  (Eliashberg.)  For any closed oriented~$Y$, overtwisted contact
structures are homotopy equivalent to oriented 2-plane fields.

Classification of tight contact structures is much more subtle.  Some 3-manifolds have none.

\proclaim Theorem.  (Eliashberg and Gromov.)  Symplectically fillable implies tight.

\proclaim Reeb vector field.  This is a vector field~$R$ such that $d\lambda(R,\cdot)=0$
and $\lambda(R)=1$.  It depends on~$\lambda$, not just~$\xi$.
A Reeb orbit is a map $\gamma\colon\R/\T\to Y$ such that $\gamma'(t)=R(\gamma(t))$.
We mod out by reparametrization.
The $k$-fold iterate of~$Y$ is the pullback to~$\R/KT$, where $K$ is a positive integer.
$\gamma$ is embedded iff $\gamma$ is not the $K$-fold iterate of some $\gamma'$ where $k>1$.

\proclaim Weinstein Conjecture.  For any contact form on any closed 3-manifold
there is a Reeb orbit.

\proclaim Strategy.  Define Floer homology generated by Reeb orbits whose differential counts
holomorphic curves.  Show Floer homology is a topological invariant.
Compute invariant, show its nontriviality.

\proclaim Definition.  An almost complex structure $J$ on $\R_s\times Y$ is admissible
if $J$ acts compatibly with~$d\lambda$, $J(\partial_s)=R$, and $J$ is $\R$-invariant.

Look at holomorphic curves in~$\R\times Y$.  Let $\gamma\colon\R/\T\to Y$ be a Reeb orbit.
The Reeb flow preserves~$\lambda$.  $L_R\lambda=di_R\lambda+i_Rd\lambda=d(1)+0=0$.
Linearization of the Reeb flow on the contact planes along~$\gamma$.
$P_\gamma\colon\xi_{\gamma(0)}\to\xi_{\gamma(0)}$ is symplectic with respect to~$d\lambda$.
$\gamma$ is nondegenerate if $1\notin P_\gamma$.  Assume all Reeb orbits are nondegenerate.

Let $\alpha_1$, \dots, $\alpha_k$, $\beta_1$, \dots, $\beta_l$ be Reeb orbits, $g\ge0$.
Define $m_g(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,b_l)$ as the set of all $J$-holomorphic
curves $u\colon\Sigma\to\R\times Y$ where u has positive ends at $\alpha_1$, \dots, $\alpha_k$,
negative ends at $\beta_1$, \dots, $\beta_l$ and no other ends.
Here $\Sigma$ is a genus~$g$ surface with $k+l$ punctures.

If $\sum_i[\alpha_i]=\sum_j[\beta_j]$ in~$H_1(Y)$, define $H_2(Y,\alpha_1,\ldots,\alpha_k,
\beta_1,\ldots,\beta_l)$ to be the set of relative homology classes of 2-chains~$z$
with $\partial z=\sum_i\alpha_i-\sum_j\beta_j$.  This is an affine space over~$H_2(Y)$.
We have $m_g(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_l,z)=\{u\colon\Sigma\to\R\times Y\in m_g\mid u_*[\Sigma]=z\}$.
Choose a trivialization~$\tau$ of~$\xi$ over the $\alpha_i$~and~$\beta_j$.
Expected dimension of the moduli space $m_g(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_l,z)
=(n-3)\chi(\Sigma)+2c_1(u^*\xi,\tau)+\sum_i\CZ_\tau(\alpha_i)-\sum_j\CZ_\tau(\beta_j)$.
This is the actual dimension if $J$ is generic and $u\colon\Sigma\to\R\times Y$ is not multiply
covered, then moduli space is a manifold near~$u$ of this dimension.
Multiple covers prevent this theory from being complete (work in progress by Hofer-Wysocki-Zehnder).

CZ index in 3-dimensional case: $P_\gamma\colon\xi_{\gamma(0)}\to\xi_{\gamma(0)}$.
Elliptic if eigenvalues are $\exp(\pm2\pi i\theta)$, positive hyperbolic if eigenvalues are
$\lambda>0$~and~$\lambda^{-1}$, negative hyperbolic if eigenvalues are $\lambda<0$~and~$\lambda^{-1}$.
Elliptic case: linearized flow rotates by angle $2\pi\theta$ for some $\theta\in\R\setminus\Z$.
We have $\CZ_\tau(\gamma)=2\lfloor\theta\rfloor+1$.
Hyperbolic case: linearized flow rotates eigenspaces by angle $\pi n$ for some $n\in\Z$,
$\CZ_\tau(\gamma)=n$.

Note: if $m_g(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_l)$ is nonempty then
$\sum_i\int_{\alpha_i}\lambda\ge\sum_j\int_{\beta_j}\lambda$, equality only if
$\{\alpha_1,\ldots,\alpha_k\}=\{\beta_1,\ldots,\beta_l\}$.  Here $\int\lambda$ is ``symplectic
action''.

Proof: if $u\colon\Sigma\to\R\times Y$ is in $m_g$, then $u^*d\lambda\ge0$ on all of~$\Sigma$,
with equality only where $\pi\circ du=0$.
$u^*d\lambda(v_1,v_2)=d\lambda(\pi du(v_1),\pi du(v_2))$.
$u^*d\lambda(v,jv)=d\lambda(\pi du(v),\pi du(jv))=d\lambda(\pi du(v),J\pi du(v))\ge0$, equality
iff $\pi du(v)=0$.  Apply Stokes theorem on~$\Sigma$.
In particular every holomorphic curve in~$\R\times Y$ has at least one positive end.
But it is possible to have no negative ends.

\proclaim Compactness.  (Bargeois-Eliashberg-Hofer-Wysocki-Zehnder.)  Any sequence
in~$m_g$ has a subsequence which converges to a ``broken'' curve.

Key point: for any $u\colon\Sigma\to\R\times Y$ in $m_g$ we have $\int_\Sigma u^*d\lambda=
\sum_i\int_{\alpha_i}\lambda-\sum_j\int_{\beta_j}\lambda$.

\proclaim Cylindrical contact homology.  Chain complex generated by ``good'' Reeb orbits over~$\Q$.
Differential counts holomorphic cylinders in~$\R\times Y$.
Trouble with coverings.  Either assume $\lambda$ is ``nice'' so that there are no bad holomorphic
discs or add correction term (augmentation) to deal with bad discs.

Reference: Introduction to Symplectic Field Theory by Eliashberg-Givental-Hofer.

\section Cylindrical contact homology (in 3 dimensions)

Let $Y$ be a closed oriented 3-manifold, $\lambda$ be a contact form, $\lambda\wedge d\lambda>0$,
$R$ be the corresponding Reeb vector field.

\proclaim Example.  $Y=S^3$, $\lambda=(x_1dy_1-y_1dx_1+x_2dy_2-y_2dx_2)/2$.

\proclaim Exercise.  $R$ is tangent to the Hopf circles.

\proclaim Example.  $Y=T^3=(\R/2\pi\Z)^3$.  $\lambda_n=\cos(nz)dx+\sin(nz)dy$,
$R_n=\cos(nz)\partial/\partial x+\sin(nz)\partial/\partial y$.

To define CCH, assume all Reeb orbits are nondegenerate.
(Above examples are ``Morse-Bott''.)
Choose $J$ on $\R_s\times Y$.

$m_g(\alpha_1,\ldots,\beta_1,\ldots,z)$ is the set of all $J$-holomorphic curves
$u\colon\Sigma\to\R\times Y$ such that domain~$\Sigma$ is a genus~$g$ surface
with $k+l$ punctures, $u$ has positive ends at $\alpha_i$, negative ends at $\beta_j$.
If $u$ has a positive or negative end at the $k$-fold iterate of an embedded Reeb orbit~$\gamma$,
there are $k$ possible asymptotic markings.

Pretend that all of these moduli spaces are manifolds of the expected dimension.
(Requires abstract perturbation of Cauchy-Riemann equation: Hofer-Wysocki-Zehnder.)
A Reeb orbit is ``bad'' if it is the $k$-fold iterate of~$\gamma$ where $k$ is even
and $\gamma$ is negative hyperbolic.  Otherwise it is ``good''.
Let $C_*$ be the free $\Q$-module generated by good Reeb orbits.
If $\alpha$ is a good Reeb orbit, then $\partial\alpha=\sum_\beta k^{-1}\#m_0(\alpha,\beta,z)/\R\cdot\beta$, where $\alpha$ is the $k$-fold iterate of an embedded orbit.

\proclaim ``Theorem''.  Suppose there is no contractible Reeb orbit~$\gamma$ bounding a disk~$D$
such that $-1+2c_1(\xi|_D,\tau)+\CZ_\tau(\gamma)=1$.
Then $\partial^2=0$.

\proclaim ``Proof''.  Let $\alpha$, $\gamma$ be generators, $z\in H_2(Y,\alpha,\gamma)$,
$\CZ_\tau(\alpha)-\CZ_\tau(\gamma)+2c_1(\xi|_z,\tau)=2$.
Look at $m_0(\alpha,\gamma,z)$.  Compactify and look at the boundary.
We have $\#\partial(m_0(\alpha,\beta,z)/\R)=a\langle\partial^2\alpha,\gamma\rangle$.
For $\Gamma\in H_1(Y)$, let $\CH^\Gamma_*$ be the part corresponding to Reeb orbits in homology
class~$\Gamma$.

\proclaim ``Theorem''.
Let $\lambda_1$, $\lambda_2$ be two different contact forms and $J_1$, $J_2$ two complex
structures.
Suppose that both forms are nice and correspond to some contact structure.
Then $\CH^\Gamma_*(\lambda_1,J_1)=\CH^\Gamma_*(\lambda_2,J_2)$.

\proclaim Definition.  A contact form~$\lambda$ on~$Y$ is called nice, if
all Reeb orbits are nondegenerate and there are no contractible Reeb orbits~$\gamma$
bounding a disk~$D$ with $-1+2c_1(\xi|_D,\tau)+\CZ_\tau(\gamma)\in\{1,0,-1\}$.

\proclaim ``Proof''.  Usual argument with continuation maps and chain homotopies.
Niceness assumption implies that boundaries of relevant moduli spaces of cylinders consist of
broken curves involving only cylinders.

\proclaim Corollary.  Let $Y$ be a closed oriented manifold with a contact structure~$\xi$.
If there is a nice contact form for~$\xi$, then $\CH^\Gamma_*(Y,\xi)$ is well defined.

\proclaim Corollary.  If $\xi$ has a nice contact form and $\CH^\Gamma_*(Y,\xi)\ne0$,
then Weinstein conjecture holds for any contact form for~$\xi$.

\proclaim Examples.  $T^3$.  After perturbation each circle of Reeb orbits splits into
two Reeb orbits, one elliptic and one positive hyperbolic.
Therefore for each $z$ with $\tan(z)\in\Q\cup\{\infty\}$ we have generators
$e^k_z$, $h^k_z$ for $k\ge1$ integer.
Claim: $\partial=0$.
Idea: There are two index~1 cylinders from~$e_z$ to~$h_z$, opposite sign.  Like $H^M_*(S^1)$.
No other cylinders because of symplectic action.
Conclusion: Let $T=(a,b,c)\in H_1(T^3)$.  If $c\ne0$, then $\CH^\Gamma_*=0$.
If $c=0$ and $(a,b)\ne(0,0)$, then $\CH^\Gamma_*=\oplus_n H_*(S^1,\Q)$.
Corollary: Weinstein conjecture holds for these contact structures.  The contact structures
determined by $\lambda_n$ for different~$n$ are not isomorphic.

Let $(Y,\xi)$ be a contact manifold ($\dim Y=3$, can be generalized to other dimensions).
Choose (1)~contact form~$\lambda$ with~$\xi=\ker\lambda$ and nondegenerate Reeb orbits;
(2)~almost complex structure on~$\R_s\times Y$ such that $J(\xi)=\xi$,
$J$ is compatible with~$d\lambda$, $J(\partial_s)=R$, $J$ is $\R$-invariant;
(3)~abstract perturbations to make moduli spaces of holomorphic curves transverse.
Then we have contact homology algebra~$A$ over~$\Q$.  The generators are good Reeb orbits.
Relations: if $\alpha$~and~$\beta$ are good Reeb orbits
then $\alpha\beta=(-1)^{|\alpha|\cdot|\beta|}\beta\alpha$,
where $|\alpha|=\CZ_\tau(\alpha)-1\pmod2$.
If $\alpha$ is a good Reeb orbit, then $$\partial\alpha=\sum_{k\ge0}\sum_\beta\hbox{(combinatorial factor)}\sum_{z\in H_2(Y,\alpha,\beta_i)} k-1+2c_1(\xi|_z,\tau)+\CZ_\tau(\alpha)-\sum_i\CZ_\tau(\beta_i)=1,$$ where $\beta_i$ are good Reeb orbits. %???
Extend $\partial$ to~$A$ by the Leibniz rule.
Theorem: $\partial^2=0$ implies that $\HC_*(Y,\xi)$ depends only on~$Y$ and~$\xi$ and not on the
other choice.  Also a symplectic cobordism~$X$ from~$Y_+$ to~$Y_-$ induces a DGA morphism
$A_+\to A_-$.

\proclaim Example.  $(S^3,\xi)$, $\xi$ is the standard contact structure.  Reeb orbits
are Hopf circles.  Perturbation gives two Reeb orbits (plus very long Reeb orbits).
$A_*$ is generated by $a_k$~and~$b_k$ for $k\ge1$.
Also $\partial=0$ and $|a_k|=|b_k|=0\pmod2$.
Since $H_1(S^3)=H_2(S^3)=0$, $A$ has a $\Z$-grading.
Conclusion: $\HC_*(S^3,\xi)=\Q[z_2,z_4,\ldots]$.  $\deg(z_{2k}=2k$.

``Theorem.''  If $\xi$ is an overtwisted contact structure on~$Y$, then $\HC_*(Y,\xi)=0$.
``Proof.'' (Mei-Lin Yau, Eliashberg.)  Can find $\lambda$~and~$J$ such that there is a Reeb
orbit~$\gamma$ such that $\gamma$ bounds a unique index~1 holomorphic disc in~$\R\times Y$
and $\gamma$ is shorter than all other Reeb orbits.
Now $\partial\gamma=1$.  If $\partial\alpha=0$ then $\partial(\gamma\alpha)=(\partial\gamma)\alpha
\pm\gamma(\partial\alpha)=1\alpha\pm\gamma\cdot0=\alpha$.

\proclaim Corollary.  If $Y$ is symplectically fillable, then $\xi$ is tight.

\proclaim Proof.  Suppose $X^4$ is a symplectic cobordism from~$(Y^3,\xi)$ to~$\phi$.
Make choices to define~$A$ for~$Y$.  Then $X$ induces a DGA morphism $\Phi\colon A\to\Q$.
Suppose $\xi$ is overtwisted.  Then there is an $\alpha$ such that $\partial\alpha=1$
and $0=\Phi(\partial\alpha)=\Phi(1)=1$.  Symplectically fillable implies $1\ne0$ in~$\HC_*$.
Overtwisted implies $1=0$ in $\HC_*$.
Possible: tight and $1=0$.  Examples: tight but not fillable.

\proclaim Morse-Bott theory.
Model case: $X$ is a closed smooth manifold.  A smooth function $f\colon X\to\R$ is Morse-Bott
if $\Crit(f)$ is a union of closed submanifolds of~$X$, and for each $p\in\Crit(f,p)$ the
map $H\colon T_pX\otimes T_pX\to\R$ is nondegenerate
on the orthogonal complement of $T_pS$, where $S$ is the critical submanifold containing~$p$.

\proclaim Example.  Height function on a torus lying on its side.

\proclaim Example.  If $F\to E\to B\to\R$ and $E\to\R$ with $E\to\R$ Morse-Bott,
then $\Crit(\pi^*f)=\pi^{-1}\Crit(f)$, where $\pi=E\to B$.

The index of a critical submanifold can be regarded as an interval $[i_-(s),i_+(s)]$,
where $i_-(s)$ is the number of negative eigenvalues of Hessian and $i_+(s)=i_-(s)+\dim S$.
Perturb $f$ to $f+\sum_s\epsilon_s\pi^*f_s$ where $f_s\colon s\to\R$ is Morse, 
$\pi\colon N\to S$ is a tubular neighborhood,
$\epsilon_s$ is a small function which is positive near~$s$ and 0 elsewhere.
$\Crit(\hat f)=\cup_s\Crit_j(f_s)$.  Near $s$: $d\hat f=df+\epsilon_s\pi^*df_s$.
Pick a generic metric~$g$ on~$X$.  What are the gradient flow lines of~$\hat f$,
in terms of~$f$?  Answer: ``Cascades''.
Bourgeois (contact homology).  Frauenfelder (Morse theory).
Claim: flow lines of~$\hat f$ correspond bijectively to cascades.

\proclaim Exercise.  Expected dimensions agree.

\proclaim Exercise.  Find an example where cascades with $k>1$ contribute.

Example: torus lying on its side.

\proclaim Floer homology.  Let $(X^{2n},\omega)$ be a closed symplectic manifold (assume monotone)
and $\phi\colon(X,\omega)\to(X,\omega)$ be a Hamiltonian symplectomorphism.
Assume fixed point come in nondegenerate manifolds, i.e., $\Fix(\phi)$ is a union of
closed submanifolds of~$X$ and for any $p\in\Fix(\phi)$ we have
$\ker(1-d\phi_p\colon T_pX\to T_pX)=T_pS$ where $S$ is the corresponding manifold of fixed points.
Choose Morse functions $f_s\colon s\to\R$.
Define $\CF^{MB}_*(\phi)=\oplus C_{*-\ind_-(s)}(f_s)$.
Differential counts cascades.

``Theorem.''  Can extend differential of Floer homology, continuation maps, etc. to this
setting.  Corollary.  $\HF_*(\id_X)=H_{*-n}(X)$.  Proof.  $k=0$ is reduced to Morse homology.
$k>0$ cascades: holomorphic spheres in~$X$ ruled out by monotonicity.

\proclaim Example.  Cylindrical contact homology of $(T^3,\lambda_n=\cos(nz)dx+\sin(nz)dy)$.
Claim: $$\CH_*(T^3,\lambda_n,(a,b,0))=\bigoplus_n H_*(S^1).$$
Proof: Just need to show there are now cascades with $k\ge1$, i.e.,
no holomorphic cylinders between Reeb orbits in different critical submanifolds.
But all orbits~$\gamma$ with $[\gamma]=(a,b,0)$ have the same action.
A simple calculation completes the proof.
Can also show that differential in contact homology algebra vanishes.  (Lots of
holomorphic curves with 2 positive ends.)

\bye
