Michael Hutchings. Floer theory.

Notes by Dmitri Pavlov, pavlov@math.

Morse theory. Suppose X is a finite-dimensional smooth manifold and f: X → 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 ∑_{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: LX → R: f(γ) = ∫_{S^{1}} |γ′(t)|^{2}dt. 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.

Theorem. (Arnold Conjecture.) If M is a closed symplectic manifold and ϕ is a Hamiltonian symplectomorphism with nondegenerate fixed points, then the number of fixed points is at least ∑_{i} dim H_{i}(M).

Much weaker Lefschetz theorem says that the number of fixed points is at least |∑_{i}(−1)^{i} dim H_{i}(M)|. 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 λ such that λ ∧ dλ ≠ 0. The kernel of λ is called a contact structure. Associated to λ is the Reeb vector field R defined by dλ(R, x) = 0 and λ(R) = 1.

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: X → R is a smooth function. A critical point of f is a point p ∈ 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): T_{p}X ⊗ T_{p}X → R as follows: if ∇ is a any connection on cotangent bundle, then H(f, p) = ∇(df)_{p}: T_{p}X → T_{p}^{∗}X. Equivalently, we can regard df as a section df: X → T^{∗}X. Then H(f, p) is the composition T (df)_{p}: T_{p}X → T_{(p,0)}(T^{∗}X) = T_{p}X ⊕ T_{p}^{∗}X → 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.

Proposition. The set of Morse functions is open and dense in the space of smooth functions X → R in the smooth topology.

Example. Height function on torus.

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.

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}(−∞, a]. This is a smooth manifold with boundary ∂X_{a} = f^{−1}(a).

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} × D^{n−i} along S^{i−1} × 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} + ⋯ + (−1)^{i}c_{0} ≥ b_{i} − b_{i−1} + ⋯ + (−1)^{i}b_{0}.

Morse complex and Morse homology.

Suppose that X is a closed smooth manifold and f: X → R is a Morse function. Choose a generic Riemannian metric on X. Hence we have ∇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 −∇f between critical points. This complex leads to Morse homology H_{∗}^{M}(X, f, g).

Fundamental theorem. H_{∗}^{M}(X, f, g) = H_{∗}^{s}(X).

Exercise. If C is a chain complex, c_{i} = dim C_{i} and b_{i} = dim H_{i}, then c_{i} − ⋯ + (−1)^{i}c_{0} ≥ b_{i} − ⋯ + (−1)^{i}b_{0} for all i.

Let ξ = −∇f, let ϕ_{s}: X → X for s ∈ R denote the flow generated by ξ. If p is a critical point, define the descending manifold of p as D(p) = {x ∈ X | lim_{s→−∞} ϕ_{s}(x) = p} and the ascending manifold of p as A(p) = {x ∈ X | lim_{s→∞} ϕ_{s}(x) = p}.

Proposition. Let p be an index i critical point. Then D(p) is an embedded open disc of dimension i. Also T_{p}D(p) is the negative eigenspace of H(f, p): T_{p}X → T_{p}^{∗}X → T_{p}X. Likewise, A(p) is an embedded open disc of dimension dim(X) − i and T_{p}A(p) is the positive eigenspace.

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).

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 ξ = −∇f from p to q is a path γ: R → X such that lim_{s→−∞} γ(s) = p and lim_{s→∞} γ(s) = q and γ′ = ξ. Flow lines are not unique.

Note: Morse cohomology is obtained by replacing −∇f with ∇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 γ → γ(0) identifies the set of all flow lines from p to q with D(p) ∩ A(q). The Morse-Smale condition implies that if p ≠ 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 [γ] ∈ m(p, q) let x = γ(0) ∈ D(p)∩A(q). We have T_{x}D(p) = T_{x}(D(p)∩A(q))⊕(T_{x}X/T_{x}A(q)) (follows from transversality). The last space is isomorphic to T_{[γ]}m(p, q) ⊕ T_{x}ℑ(γ) ⊕ T_{q}D(q). Orient T_{[γ]}m(p, q) so that this isomorphism is orientation preserving.

Definition of Morse complex C_{∗}(X, f, g). Denote by C_{i} the free abelian group generated by index i critical points. A differential ∂_{i}: C_{i} → C_{i−1} is defined as follows: ∂p = ∑_{q} q · #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).

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 m̄(p, q) whose codimension k stratum is m̄(p, q)_{k} = ∪_{r_{i}∈Crit(f)}m(p, r_{1}) × m(r_{1}, r_{2}) × ⋯ m(r_{k−1}, r_{k}) × m(r_{k}, q). Here p, r_{1}, …, r_{k}, and q must be all different.

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.

Corollary. If ind(p) − ind(q) = 2, then m̄(p, q) is a compact oriented manifold with boundary

⋃_{r:ind(p)−ind(r)=1} m(p, r) × m(r, q).

Now we shall prove that ∂^{2} = 0. Suppose that ind(p) − ind(q) = 2. Then m̄(p, q) is a compact oriented manifold with boundary ∪_{r:ind(p)−ind(r)=1}m(p, r)×m(r, q). Then ⟨∂^{2}p, q⟩ = ∑_{r:ind(p)−ind(r)=1}⟨∂p, r⟩·⟨∂r, q⟩ = #∂m̄(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}) | t ∈ [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 β: [0, 1] → R such that β^{−1}(0) = {0, 1}, β′(0) > 0 and β′(1) < 0.

Define a vector field V on [0, 1] × X by V = β(t)∂_{t} + ζ_{t}, where ζ_{t} is the negative of g_{t}-gradient of f_{t}. Note that Crit(V) = {0} × Crit(f_{0}) ∪ {1} × 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 ∈ Crit_{i}(f_{0}) define Φ(p) = ∑_{q∈Crit_{i}(f_{1})} #m^{V} ((0, p), (1, q))q.

Lemma. Φ is a well-defined chain map.

Proof. Usual arguments show that if p ∈ Crit_{i}(f_{0}) and q ∈ Crit_{i}(f_{1}), then m^{V} ((0, p), (1, q)) is compact (finite), hence Φ is well-defined. If p ∈ Crit_{i}(f_{0}) and q ∈ Crit_{i−1}(f_{1}), then m^{V} ((0, p), (1, q)) has a compactification to a 1-manifold with boundary ∂\overline{m^{V} ((0, p), (1, q))} = ∪_{r∈Crit_{i−1}(f_{0})}m^{V} ((0, p), (0, r))×m^{V} ((0, r), (1, q))∪ ∪_{r∈Crit_{i}(f_{1})}m^{V} ((0, p), (1, r)) × 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}) ⊕ C_{i}(X, f_{1}, g_{1}) and ∂ = ( \begin{matrix}−∂_{0} & 0 \\ Φ & ∂_{1} \end{matrix} ), therefore

0 = ∂^{2} = ( \begin{matrix}∂_{0}^{2} & 0 \\ −Φ∂_{0} + ∂_{1}Φ & ∂_{1}^{2} \end{matrix} ).

Example. If (f_{t}, g_{t}) does not depend on t, then Φ = id.

Proof. Since V = β(t)∂_{t} + ζ, any flow line of V on [0, 1] × X projects to a flow line of ζ on X. If p, q ∈ Crit_{i}(f), then m_{ζ}(p, q) = ∅ when p ≠ q and one-element set consisting of constant map if p = q. It follows that Φ = ± id and it is easy to check that Φ = id.

Lemma. Suppose we have two generic paths between a pair of Morse-Smale pairs. Then associated continuation maps Φ_{0} and Φ_{1} are chain homotopic.

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 Ṽ on [0, 1]^{2} × X by Ṽ = β(t)∂_{t} + ζ_{s,t}. Define K: C_{∗}^{M}(X, f_{0}, g_{0}) → C_{∗}^{M}(X, f_{1}, g_{1}) by counting flow lines of Ṽ with appropriate signs. If p ∈ Crit_{i}(f_{0}), then K(p) = ∑_{q∈Crit_{i+1}(f_{1})} # ∪_{s∈[0,1]} mṼ (((s, 0), p), ((s, 1), q))q. Trivial computations show that K is a chain homotopy.

Corollary. The map Φ_{∗}: H_{∗}^{M}(X, f_{0}, g_{0}) → 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.

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

Lemma. Φ 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.)

Theorem. Given (f_{0}, g_{0}) and (f_{1}, g_{1}), let Γ be a generic path from (f_{0}, g_{0}) to (f_{1}, g_{1}) and let Δ be the reverse path. Obviously Φ_{Δ}Φ_{G}amma ∼ id_{C_{∗}^{M} (X,f_{0},g_{0})}. Likewise for Φ_{Γ}Φ_{Δ}. Hence Φ_{Γ} 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 β by ϵβ for sufficiently small ϵ, then they become equal. Continuation maps for different β are chain homotopic.

Bifurcations. For a generic family (f_{t}, g_{t}) there are times t_{1} < ⋯ < 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′ ∈ Crit_{i} and q ≠ q′. A flow line from q′ to r ∈ Crit_{i−1} gets glued to a new flow line from q to r either before or after the bifurcation. Thus, ∂_{+}q = ∂_{−}q ± ∂q′. Also ∂_{+}p = ∂_{−}p ∓ ⟨∂_{−}p, q⟩q′ for p ∈ Crit_{i+1}. Define an isomorphism Φ: C_{−} → C_{+} as Φ(q) = q ± q′ and Φ(s) = s for all other critical points s. (3) Birth or death: two critical points p ∈ Crit_{i} and q ∈ Crit_{i−1} cancel. Algebraically: ∂_{+} = ∂_{−}p′ ± ⟨∂_{−}p′, q⟩∂_{−}p for all p′ ≠ p. Define chain homotopy Φ: C_{−} → C_{+} as Φ(p) = 0, Φ(q) = ±∂_{−}p ∓ ⟨∂_{−}p, q⟩q and Φ(r) = r for all r ≠ p, q.

Exercise. With corrections as necessary, Φ induces an isomorphism on homology.

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

Conjecture. For a given family (f_{t}, g_{t}) if we replace β by ϵβ for sufficiently small ϵ, then Φ′ = Φ.

Morse homology versus singular homology.

Idea: Define a map C_{∗}^{M} → C_{∗} by sending a critical point to its descending manifold.

Proposition. For each critical point p the descending manifold D(p) has a compactification to a manifold D̄(p) with corners such that codimension k stratum is a union of m(p, q_{1})×m(q_{1}, q_{2})×⋯×m(q_{k−1}, q_{k})× D(q_{k}) over all distinct q_{i} ∈ Crit(f) distinct from p. The endpoint map to D(q_{k}) is continuous.

Claim. In general D̄(p) is homeomorphic to a closed ball with dimension ind(p). Hence the compactified manifolds D̄(p) together with the maps e: D̄(p) → 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.

Other approach. Define chain map C_{∗}^{M}(X, f, g) → C_{∗}(X). Roughly p → D̄(p).

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

We shall not take these approaches.

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

Proof of the Proposition. We have

∂D̄(p) = ⋃_{q∈Crit(f)\{p}}(−1)^{ind(p)+ind(q)+1}m(p, q) × D(q)

and

∂m̄(p, q) = ⋃_{p≠q≠r≠p}(−1)^{ind(p)+ind(r)+1}m(p, r) × m(r, q).

Lemma. For each p ≠ q there is a cubical singular chain m_{p,q} ∈ C_{|p|−|q|−1}^{s}(m̄(p, q)) such that m_{p,q} represents the relative fundamental chain in H_{|p|−|q|−1}(m̄(p, q), ∂m̄(p, q)) and at the chain level ∂m_{p,q} = ∑_{p≠q≠r≠p}(−1)^{|p|+|r|+1}m_{p,r} × m_{r,q}. Here × is the cross product on cubical chains.

Proof. By induction on |p| − |q|. Given p and q the right hand side represents the fundamental class of ∂m̄(p, q). It is easy to see that ∂ of right hand side equals zero. So right hand side is a cycle, hence it clearly represents the fundamental class of ∂.

Lemma. For each critical point p we can choose d_{p} ∈ C_{|p|}^{s}(D̄(p)) such that d_{p} represents the relative fundamental class in H_{∗}(D̄(p), ∂D̄(p)) and ∂d_{p} = ∑_{p≠q}(−1)^{|p|+|q|+1}m_{p,q} × d_{q}.

Proof. User previous lemma and induction.

Rest of the proof of the Proposition. Define a map D: C_{∗}^{M}(X, f, g) → C_{∗}^{s}(X) by D(p) = e_{#}d_{p}. Now D is a chain map: ∂^{s}D = D∂^{M}. Proof: Let p ∈ Crit_{i}(f). We have ∂^{s}d_{p} = ∑_{q∈Crit(f)\{p}}(−1)^{i+|q|+1}e_{#}(m_{p,q}×d_{q}). If ind(q) ≥ i, then m(p, q) = ∅, if ind(q) ≤ i − 2, then the corresponding cube is degenerate. Hence we have ∑_{q∈Crit_{i−1}(f)} #m(p, q)e_{#}(dq) = D(∂^{M}p).

Hence we have a map H_{∗}^{M}(X, f, g) → 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 σ: [−1, 1]^{i} → 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 σ and p ∈ Crit(f), define m(σ, p) = {(t, γ) | t ∈ dom σ ∧ γ: [0, ∞) → X ∧ γ(0) = σ(t) ∧ γ′(s) = ξ(γ(s)) ∧ lim_{s→∞} γ(s) = q}. Now T_{t}[−1, 1]^{i} is isomorphic to T_{γ}m(σ, p) ⊕ T_{p}D(p). We have dim m(σ, p) = i − ind(p).

Define A: C_{∗}^{a}(X) → C_{∗}^{M}(X, f, g) as A(σ) = ∑_{p∈Crit_{i}(f)} #m(σ, p)p. We can compactify m(σ, p). The boundary consists of the boundary of the cube times the broken lines. We have ∂m̄(σ, p) = m(∂σ, p) ∪ ⋃_{q∈Crit(f)\{p}}(−1)^{i+ind(q)}m(σ, q)×m(q, p). Lemma: A is a chain map: A∂^{s} = ∂^{M}A. Proof: If p ∈ Crit_{i−1}(f), then m̄(σ, p) is a compact 1-manifold with boundary. Obvious computation completes the proof of the lemma. Exercise: The induced map A_{∗}: H_{∗}^{s}(X) → 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 ∈ Crit_{i} and q ∈ Crit_{i}, then D(p) ∩ A(q) is empty if p ≠ q and {p} if p = q. Chain homotopy K: C_{∗}^{a}(X) → C_{∗+1}^{a}(X) is defined by flowing a cube down. More precisely, define the forward orbit of σ to be the space F (σ) = [−1, 1]^{i} × [0, ∞) with the map e: F (σ) → X: e(t, s) = ϕ_{s}(σ(t)) where ϕ_{s} is the gradient flow. Compactify to e: F̄(σ) → X, choose appropriate singular chains on F̄(σ).

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

Poincaré duality. If X^{n} is oriented, then H_{i}^{M}(X) = H_{M}^{n−i}(X).

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

Proof of Poincaré 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_{p}X = T_{p}D(p, f) ⊕ T_{p}A(p, f) and A(p, f) = D(p, −f).

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 (σ, g) where σ is a cube (or a simplex) in X and g ∈ G_{σ(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 \ {x}) (an orientation of X at x is a generator of O_{x}).

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 ⊗ 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_{i}^{M}(X, f, g, L) = ⊕_{p∈Crit_{i}(f)}L_{p} and ∂(p, l) = ∑_{q∈Crit_{i−1}(f)} ∑_{γ∈m(p,q)} ϵ(γ)(q, ϕ_{γ}(l)).

Proof. We have ∂^{2}(p, l) = ∑_{r∈Crit_{i−2}}(r, ∑_{q∈Crit_{i−2}} ∑_{{γ_{1}∈m(p,q) \atop γ_{2}∈m(q,r)}} ϵ(γ_{1})ϵ(γ_{2})ϕ_{γ_{2}}(ϕ_{γ_{1}}(l))) = 0, because two cor-

responding paths are homotopic.

Poincaré duality. C_{i}^{M}(X, f, g, L) = C_{M}^{n−i}(X, −f, g, L ⊗ O) is an isomorphism of chain complexes.

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 ∗: C_{M}^{i}(X, f_{1}, g_{1})⊗C_{M}^{j}(X, f_{2}, g_{2}) → C_{M}^{i+j}(X, f_{3}, g_{3}) as follows: ⟨p∗q, r⟩ 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 δ(p ∗ q) = (δp) ∗ q + p ∗ (δq). Trivial configuration counting show that this is true.

We can also do this for general coefficient systems: H_{M}^{i}(X, G_{1}) ⊗ H_{M}^{j}(X, G_{2}) → H_{M}^{i+j}(X, G_{1} ⊗ G_{2}).

Theorem. Under the canonical isomorphism H_{M}^{∗} = H_{s}^{∗} the product ∗ agrees with the cup product.

Proof. Idea: ⟨p ∗ q, r⟩ = #(A(p) ∩ A(q) ∩ D(r)).

Spectral sequences in Morse theorem.

Let F → E → B be a smooth fiber bundle where F, E, and B are closed smooth manifolds. Define E_{i,j}^{2} = H_{i}(B, H_{j}(E_{b})). Here H_{j}(E_{b}) is a local coefficient system. For k ≥ 2 we have maps ∂_{k}: E_{i,j}^{k} → E_{i−k,j+k−1}^{k} such that ∂_{k}^{2} = 0 and E^{k+1} = H(E^{k}). We define E_{i,j}^{∞} = E_{i,j}^{k} for k > i and k > j.

H_{∗}(E) has a filtration: α ∈ F_{i}H_{∗}(E) iff α 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_{i}H_{∗}(E) = F_{i}H_{∗}(E)/F_{i−1}H_{∗}(E). Note that G_{m}H_{n}(E) = E_{m,n−m}^{∞}. Example: SU(2) → SU(3) is a fibration with base S^{5} and fiber S^{3}. We have E^{2} = E^{∞}. Hence

H_{∗}(SU(3)) = H_{∗}(S^{5} × 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}) | b ∈ B} where f_{b}: E_{b} → 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 ∈ Crit(f^{B}). Choose a connection ∇ on E. For b ∈ B denote by ξ_{b} the negative gradient of f_{b} with respect to g_{b}. Denote by ξ^{B} the negative gradient of f^{B} with respect to g^{B}. Define a vector field V on E by V (b, e) = ξ_{b} + H(ξ^{B}), where H(ξ^{B}) is the horizontal lift with respect to ∇. Now Crit(V) = ∪_{b∈Crit(f^{B})} Crit(f_{b}). Define a chain complex C_{∗} = ⊕_{i+j=∗} ⊕_{b∈Crit_{i}(f^{B})} Crit_{j}(f_{b}). Define ∂: C_{∗} → C_{∗−1} by counting flow lines of V in the usual manner. Usual arguments show that ∂ is well defined, ∂^{2} = 0, and H_{∗}(C_{∗}, ∂) = H_{∗}(E).

In general, this chain complex has a filtration defined by F_{i}C_{∗} = ⊕_{{b∈Crit(f^{B}) \atop ind(b)≤i}} Z\{Crit_{∗−ind(b)\}(f_{b})}.

This filtered chain complex gives us a spectral sequence. We have E_{i,j}^{0} = G_{i}C_{i+j} = ⊕_{b∈Crit_{i}(f^{B})} Crit_{j}(f_{b}). The differential ∂_{0}: E_{i,j}^{0} → E_{i,j−1}^{0} is induced by ∂. We have E_{i,j}^{1} = ⊕_{b∈Crit_{i}(f^{B})}H_{j}^{M}(E_{b}, f_{b}, g_{b}).

It is easy to see that E_{i,j}^{2} = H_{i}^{M}(B, f^{B}, g^{B}, {H_{j}^{M}(E_{b}, f_{b}, g_{b})}).

Theorem. For k ≥ 2 Morse theory spectral sequence agrees with Leray-Serre spectral sequence.

Idea of proof. Leray-Serre spectral sequence comes from filtration on C_{∗}^{s}(E) defined by F_{i}C_{∗}(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: C_{∗}^{M} → 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.

Definition. A 1-form ω on X is Morse if it is closed and is locally a differential of Morse function.

Let p and q be critical points of ω. Let γ_{n} be a sequence of flow lines from p to q. Let r_{0}, …, r_{k+1} be critical points such that r_{0} = p and r_{k+1} = q. Let η_{i} be a flow line from r_{i} to r_{i+1}. Say that γ_{n} converges to (η_{0}, …, η_{k}) if there are real numbers s_{n,0} < ⋯ < s_{n,k} such that γ_{n} restricted to corresponding interval converges to η_{i} in C^{∞} on compact sets as n → ∞. Also we require that homology class of γ_{n} − ∑_{i} η_{i} is zero for sufficiently large n. Define the energy of a flow line γ by E(γ) = ∫ |γ^{′2}| = ∫ γ^{∗}ω.

Proposition. If γ_{n} is a sequence of flow lines with E(γ_{n}) < C, then there is a subsequence converging to a broken flow line.

Lemma. Let γ: R → X such that γ′(s) is dual to −ω(γ(s)) (a) E(γ) ≥ 0 and equality holds iff γ is a constant map to a critical point. (b) If E(γ) < ∞, then there are critical points p and q such that γ is a flow line from p to q: lim_{s→−∞} γ(s) = p and lim_{s→∞} γ(s) = q. (c) There is δ > 0 such that if γ is nonconstant then E(γ) > δ.

Proof of Proposition. Suppose γ_{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_{n} E(γ_{n}) = D. Choose ϵ > 0 such that ϵ-balls around critical points are disjoint. Define s_{n,0} = inf{s ∈ R | d(γ_{n}(s), ω^{−1}(0)) > ϵ}. Pass to a subsequence so that γ_{n} restricted to [s_{n,0}, s_{n+1,0}] converges to η_{9} in C^{∞} on compact sets. Must have E(η_{0}) ≤ C_{0}. If E(η_{0}) = C_{0}, then γ_{n} → (η_{0}) in the defined sense. If E(η_{0}) < C_{0}, do the following. Let t_{0} = sup{t | d(η_{0}(t), ω^{−1}(0)) > ϵ}. Define s_{n,1} = inf{s | s > s_{n,0} + t_{0} + 1 ∧ d(γ_{n}(s)ω^{−1}(0)) > ϵ}. Pass to a subsequence so that γ_{n} restricted to [s_{n,1}, s_{n+1,1}] converges to η_{1}. And so on. Note that γ_{n} is homotopic rel endpoints to η_{0} … η_{k}.

Morse theory for f: X → S^{1}. (Simplest version.) Assume f is Morse, choose generic metric. Let Σ ⊂ X be a level set of f not containing any critical points. Define C_{∗} over Λ = {∑_{n≥n_{0}} a_{n}t^{n}} for a_{n} ∈ Z. Let C_{i} be a free abelian group generated by index i critical points. If p ∈ Crit_{i}(f), then ∂p = ∑_{q∈Crit_{i−1}(f)} ∑_{n≥0} t^{n}z_{n}, where z_{n} is the number of flow lines from p to q that cross Σ n times. Usual argument shows that ∂^{2} = 0, homology as a Λ-module is a topological invariant of X and the homotopy class of f: X → S^{1} in H^{1}(X, Z).

Let X be a closed smooth manifold, ω be a Morse 1-form (dω = 0 and locally ω is a differential of a Morse function), g be a generic metric on X. Choose K ⊂ H_{1}(X) such that ∫ ω = 0 for all α ∈ K. (Usual choices: K = {0} or K = ker(∫).) Let H = H_{1}(X)/K.

Idea: Classify flow lines modulo the following equivalence relation: If γ and γ′ are elements of m(p, q), then γ ∼ γ′ iff [γ − γ′] ∈ K. Note: if γ ∼ γ′, then E(γ) = E(γ′). 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: H → 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 ∈ H with nonzero coefficient and N(h) in the neighborhood. Multiplication is defined as in group ring. There is an obvious injection Z[H] → Nov(H, N). If N = 0, then is an iso.

Definition of Novikov complex. Fix a base point x_{0} ∈ X. An anchored critical point is a pair (p, η) where p ∈ ω^{−1}(0) and η is a path from p to x_{0}, modulo ∼. Define the energy E(p, η) = ∫_{η} −ω. (Alternatively: Let π: (X̃, x̃_{0}) → (X, x_{0}) be the covering space determined by K. The pair (p, η) corresponds to a zero of π^{∗}ω on X̃. We have π^{∗}ω = df̃, where f̃(x̃_{0}) = 0 and E(p, η) = f̃(p̃).

Let C_{i} be a submodule of Novikov ring generated by all points p̃ with index i such that only finite number of terms have nonzero coefficients and energy lying outside given neighborhood of 0.

Exercise. C_{i} is a module over Λ = Nov(H, −ω) where H acts by h(p, η) = (p, η + h). Moreover this is a free module with one generator for each index i critical point of ω in X.

Define ∂: C_{i} → C_{i−1} by ∂(p, η) = ∑_{q∈Crit_{i−1}(ω)} ∑_{γ∈m(p,q)} ϵ(γ)(q, η − γ).

Exercise. ∂ is well-defined.

Theorem. H_{∗}^{N}(X, x_{0}, ω, g, K) depends only on X, [ω], and K. (The isomorphisms are canonical only up to multiplication by elements of H.)

Usually we cannot vary [ω] except by positive scaling.

Interpretation in classical topology: Chose a cell decomposition of X, lift it to a cell decomposition of X̃. Then C_{∗}^{c}(X̃) is a module over Z[H].

Theorem. (Novikov.) H_{∗}^{N} = H_{∗}(C_{∗}^{c}(X̃) ⊗_{Z[H]} Λ) = H_{∗}(C_{∗}^{∞/2}(X̃)). Here C_{∗}^{∞/2}(X̃) is the half-infinite singular chain complex of X̃ (formal sums of simplices such that for any real c only finitely many simplices intersect {x̃ ∈ X̃ | f̃(x̃) > c}.

Often H_{∗}^{N} is torsion. Suppose H has no torsion so that the quotient ring of Λ is a field. Suppose H_{∗}^{N} ⊗ Q(Λ) = 0.

Theorem. The Reidemeister torsion of C_{∗}^{c}(X̃) ⊗_{Z[H]} Q(Λ) is equal to the Reidemeister torsion of C_{∗}^{N} ⊗_{Λ} Q(Λ)) times a certain count of closed orbits of the gradient flow.

Pseudoholomorphic curves in symplectic manifolds. Note: we shall call them holomorphic curves.

Definition. A symplectic manifold is a smooth manifold with a closed nondegenerate 2-form (nondegenerate means that it defines an isomorphism T M → T^{∗}M, equivalently ω^{n} ≠ 0 everywhere, where 2n is the dimension of the manifold, which must be even).

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 symplectomorphism.

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.

Note. Every symplectomorphism preserves volume form ω^{n}, hence it preserves volume.

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

Answer. For n = 1 these notions are the same.

Gromov nonsqueezing theorem. Suppose there is a symplectic embedding ϕ: B^{2n}(r) → B^{2}(R)×R^{2n−2}. Then r ≤ R.

Recognition of R^{4}. Suppose M is a symplectic manifold such that H̃_{∗}(M) = 0 and M is asymptotically symplectomorphic to R^{4}, more precisely, there are compact sets K_{1} ⊂ M and K_{2} ⊂ R^{4} and a symplectomorphism ϕ: M \ K_{1} → R^{4} \ K_{2}. Then M is symplectomorphic to R^{4}, by a symplectomorphism agreeing with ϕ outside a compact set.

Symplectomorphisms of S^{2} × S^{2}. Let ω_{1} and ω_{2} be two symplectic forms on S^{2} with ∫ ω_{1} = ∫ ω_{2}. Then Symp(S^{2} × S^{2}, ω_{1} ⊕ ω_{2}) has two connected components. The identity component is homotopy equivalent to SO(3) × SO(3).

Definition. An almost complex structure on M is a bundle map J: T M → T M such that J^{2} = −1. Example: M is a complex manifold, and J is multiplication by i. In such a case J is called integrable.

Definition. An almost complex structure is ω-tame if ω(v, Jv) > 0 whenever v ≠ 0. It is ω-compatible if it is ω-tame and ω(Jv, Jw) = ω(v, w).

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

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

Proposition. There is no obstruction to finding or extending tame or compatible almost complex structures. More precisely, let Ω 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 Ω with contractible fibers.

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

Exercise. Let ω be a linear symplectic form and g a linear metric on R^{4}. Then g comes from ω and ω-compatible J iff ω is self-dual with respect to g and |ω| = \sqrt{2}.

Definition. A Riemann surface is a closed surface σ is a closed surface Σ with an almost complex structure J (necessarily integrable). m_{g,n} is the space of genus g Riemann surfaces with n marked points.

Fact. m_{g,n} is a smooth orbifold with dimension 3g − 3 + n for g ≥ 2 or g = 1 and n ≥ 1 or g = 0 and n ≥ 3. For g = 1 and n = 0 the dimension is 1 and for g = 0 and n ≤ 2.

Definition. Let M be an almost complex manifold. A (pseudo)holomorphic curve in M is a smooth map u: (Σ, j) → (M, J) such that J ∘ du = du ∘ j. (This means that du respects almost complex structure.) Two curves are equivalent if there is a biholomorphic map from (Σ, j) to (Σ′, j′) such that the obvious diagram commutes.

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 Σ such that J maps T Σ to itself.

Proposition. If u: Σ → M is a holomorphic curve, then its area is equal to ∫ u^{∗}ω = ⟨[ω], u_{∗}[Σ]⟩.

Proof. To prove the first part it is enough to show that Area(v, Jv) = ω(v, Jv) for all v ∈ T M. This follows from the fact that Area(v, Jv) = ω(v, Jv). To prove the second part it is enough to show that for all v and w in T_{p}M we have Area(v, w) ≥ ω(v, w) with equality iff w is in the span of v and Jv. Since J is ω-compatible, we can find a basis e_{i} and f_{i} for T_{p}M such that at p we have ω = ∑_{i} e_{i}^{∗} ∧ f_{i}^{∗} and J(e_{i}) = f_{i}. Wlog v = e_{1}. Write w = ∑_{i} a_{i}e_{i} + b_{i}f_{i}. Now Area(v, w) = (∑_{i}(a_{i}^{2} + b_{i}^{2}) − a_{1}^{2})^{1/2} ≥ b_{1}. We have equality iff a_{i} = b_{i} = 0 for i ≠ 1.

Trivial example of holomorphic curves.

1. The zero set of a homogeneous polynomial over C in 3 variables is a holomorphic curve in CP^{3}.

2. If u: (Σ, j) → (M, J) is J-holomorphic (we assume that J is ω-tame) and u_{∗}[Σ] = 0 ∈ H_{2}(M), then u is constant. (Proof: Area(u) = 0.)

3. Suppose M = Σ × V, where Σ is a Riemann surface and V is a symplectic manifold. Take ω and J be product symplectic and almost-complex structure. For any p ∈ V the map Σ → Σ × V sending x → (x, p) is holomorphic. Claim: These are they only holomorphic curves in the homology class [Σ] × [·]. Proof: Let u′: (Σ′, j′) → (Σ × V, J) be another holomorphic curve in this homology class. The projection Σ′ → Σ × V → 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 Σ × {p} for some p ∈ V. The projection Σ′ → Σ × V → Σ is also holomorphic. A non-constant holomorphic map Σ′ → Σ is a branched cover of degree at least 1. Since homology class is [Σ] the degree of this map is 1 and it is an isomorphism.

Gromov Non-Squeezing Theorem. If ϕ: B^{2n}(r) → B^{2}(R) × R^{2n−2} is symplectomorphism, then r ≤ R.

Proof. (Modulo some stuff.) Choose c > 0 such that ℑ(ϕ) ⊂ B^{2}(R)×[−c, c]^{2n−2}. We get a map ϕ: B^{2n}(r) → S^{2}(R + ϵ) × T^{2n−2}. Choose an ω-tame almost complex structure on the last manifold in such a way that on ℑ(ϕ) it agrees with ϕ-pushforward of standard complex structure on R^{2n}. The key is to show the existence of J-holomorphic sphere u: S^{2} → S^{2}(R + ϵ) × T^{2n−2} such that ℑ(u) ∋ ϕ(0) and u_{∗}[S^{2}] = [S^{2}(R + ϵ)] × [·]. Let Σ denote ϕ^{−1}(ℑ(u)) ⊂ B^{2n}(r). We have 0 ∈ Σ. If t < r and if we replace Σ ∩ B^{2n}(t) with another surface with the same boundary, area does not decrease. Monotonicity lemma for minimal surfaces: Above conditions on Σ imply Area(ϵ) ≥ πr^{2}. Proof: Define Σ_{t} = Σ ∩ B^{2n}(t), θ(t) = Area(Σ_{t})/(πt^{2}), and l(t) as the length of Σ ∩ S^{2n}(t). We have lim_{t→0} θ(t) = 1 and θ′(t) ≥ 0. Also Area′(Σ_{t}) ≥ l(t) and Area(Σ_{t}) ≤ tl(t)/2. We have θ(r) ≥ θ(0) = 1. (Monotonicity.) Now πr^{2} ≤ Area(σ) ≤ Area(u) = π(R + ϵ)^{2}. This is true for any ϵ > 0, hence r ≤ 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, ω) be a symplectic manifold, J an ω-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}: m_{g,n}(X, A) → 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, ω) be a compact symplectic manifold with π_{2}(V) = 0. Let X = (S^{2} × V, ω_{0} ⊕ ω). Let A = [S^{2}] × [·] ∈ H_{2}(X).

Let J be any ω-tame almost complex structure on X. Then ev_{1}: m_{0,1}(X, A) → 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} | t ∈ [0, 1]} is a generic homotopy then ⋃_{t}{t} × 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.

Genericity and transversality.

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).

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⟨c_{1}(T X), A⟩ + 2k.

Sard-Smale theorem. Let f: X → 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 ≥ 1 and k ≥ l + 1. Then a generic y ∈ Y is a regular value of f so that f^{−1}(y) is a manifold of dimension l.

Theorem. Let Z be a smooth finite-dimensional manifold. Let k ≥ 2. Then a generic C^{k} function f: Z → R is Morse.

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_{z}^{∗}Z. Define a section ψ: Y × Z → E by ψ(f, z) = df_{z}. Then ψ^{−1}(0) = ⋃_{f}{f} × Crit(f). Claim: For (f, z) ∈ ψ^{−1}(0) we have dψ: T_{f,z}Y × Z → T_{f,z,0}E is surjective, therefore ψ^{−1}(0) is a Banach manifold. Actually we show that T_{f,z}Y × Z → T_{f,z,0}E → E_{f,z} = T_{z}^{∗}Z is surjective. If f_{1} ∈ C^{k}(Z, R) and v ∈ T_{z}Z, then ∇ψ(f_{1}, v) = df_{1}(z) + ∇_{v}(df) can be anything. Claim: The projection ψ^{−1}(0) → Y has dπ_{f,z} Fredholm so that Sard-Smale applies. We have ker(dπ_{f,z}) = ker(∇ψ: T_{z}Z → E_{f,z}. There is an automorphism ∇ψ: coker(dπ: T_{f,z}ψ^{−1}(0) → T_{f}Y) → coker(∇ψ: T_{z}Z → E_{f,z}). It follows that a generic f ∈ C^{k}(Z, R) is a regular value of π: ψ^{−1}(0) → C^{k}(Z, R). If f is a regular value of π then for every z ∈ Crit(f) we have dπ: T_{f,z}ψ^{−1}(0) → T_{f}C^{k}(Z, R) is surjective.

We say that E → X is a Banach vector bundle if E and X are Banach manifolds, π^{−1}(x) is a Banach space for any x ∈ X, and X has an open cover such that the corresponding restrictions are trivial bundles.

Theorem. If ∇ψ: T_{x}X → E_{x} is surjective for all x ∈ ψ^{−1}(0), then ψ^{−1}(0) is a Banach submanifold of X.

Proposition. Let Y and Z be separable Banach manifolds, π: E → Y × Z a Banach vector bundle, ψ: Y × Z → E a smooth section. Suppose that for all (y, z) ∈ ψ^{−1}(0) the following holds: (1) ∇ψ: T_{y,z}(Y ×Z) → E_{y,z} is surjective. (2) ∇ψ: T_{z}Z → E_{y,z} is Fredholm of index l. Then for generic y ∈ Y the set {z ∈ Z | ψ(y, z) = 0} is an l-dimensional submanifold of Z (and moreover, at each point in this set ∇ψ is surjective on tangent space to Z).

Proof. It follows from (a) that ψ^{−1}(0) ⊂ Y × Z is a Banach manifold (by implicit function theorem). Let π: ψ^{−1}(0) → Y denote the projection. By Sard-Smale theorem, it is enough to show that for all (y, z) ∈ ψ^{−1} the differential of π, D = dπ: T_{y,z}ψ^{−1}(0) → T_{y}Y is Fredholm. By (b) it is sufficient to prove these two statements: (1) ker(D) = ker(∇ψ: T_{z}Z → E_{y,z}) and (2) The map ∇ψ: T_{y}Y → E_{y,z} induces an isomorphism coker(D) → coker(∇ψ: T_{z}Z → E_{y,z}). Proof of (1): ker(D) = {(0, ż) | (0, ż) ∈ T_{y,z}ψ^{−1}(0)} = ker(∇ψ: T_{z}Z → E_{y,z})}. Proof of (2): Check that ∇ψ: T_{y}Y → E_{y,z} sends im(D) → im(∇ψ: T_{z}Z → E_{y,z}). Let (ẏ, ż) ∈ T_{y,z}ψ^{−1}(0). Need to show ∇ψ(D(ẏ, ż)) ∈ im(∇ψ: T_{z}Z → E_{y,z}). Check ∇ψ: coker(D) → coker(∇ψ: T_{z}Z → E_{y,z}) is injective. Have ∇ψ(ẏ, 0) = ∇ψ(0, ż) for some ż ∈ T_{z}Z. Then (ẏ, −ż) ∈ T_{y,z}ψ^{−1}(0) and ẏ = D(ẏ, −ż). Check ∇ψ: coker(D) → coker(∇ψ: T_{z}Z → E_{y,z}) is surjective. Let e ∈ E_{y,z}. By hypothesis (a), e = ∇ψ(ẏ, ż) for some (ẏ, ż) ∈ T_{y,z}Y × Z. We have e = ∇ψ(ẏ, 0) + ∇ψ(0, ż) = 0 in coker(∇ψ: T_{z}Z → E_{y,z}).

Example. Suppose Z is a closed smooth manifold, Y = C^{k}(Z, R), E_{f,z} = T_{z}^{∗}Z, and ψ(f, z) = df_{z}. It follows that for generic f ∈ C^{k}(R) and for any z ∈ df^{−1}(0) = Crit(f) we have ∇ψ: T_{z}Z → E_{f,z} is surjective.

Spectral flow. Reference: Robbin and Salaman, Spectral flow and the Maslov index. Let H be a Hilbert space and let A_{s}: H → H be a continuous family of unbounded operators parametrized by s ∈ R. Assume there are invertible self-adjoint operators A_{+} and A_{−} such that lim_{s→∞} A_{s} = A_{+} and lim_{s→−∞} 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 −∞ to ∞. If H is finite-dimensional, then SF(A) is the difference between the number of positive eigenvalues of A_{+} and A_{−}.

Theorem. Under certain assumptions ∂_{s} + A_{s}: L_{1}^{2}(R, H) → L^{2}(R, H) is Fredholm and its index is the spectral flow of A.

Proof. Assume H is finite-dimensional. For each h ∈ H by fundamental theorem of ODE’s there is a function f: R → H such that (∂_{s} + A_{s})f_{h}(s) and f_{h}(0) = h. Define H^{+} = {h ∈ H | lim_{s→∞} f_{h}(s) = 0} and H^{−} = {h ∈ H | lim_{s→−∞} f_{h}(s) = 0}. Then ker(∂_{s} + A_{s}) is isomorphic to H^{+} ∩ H^{−}. Claim: H^{+} is the negative eigenspace of A^{+} and H^{−} is the positive eigenspace of A^{−}. Claim: ∂_{s}+A_{s} is bounded and has closed image. Then coker(∂_{s} +A_{s}) = ker(−∂_{s} +A_{s}^{∗}) = ker(∂_{s} −A_{s}^{∗}). Similarly to above, ker(∂_{s} −A_{s}^{∗}) = H^{+⊥} ∩H^{−⊥}. We have ind(∂_{s} +A_{s}) = dim(H^{+} ∩H^{−})−dim(H^{+⊥} −H^{−⊥}) = dim(H^{+} ∩H^{−})+dim(H^{+} +H^{−})−dim(H) = dim H^{+} + dim H^{−} − dim H = SF(A).

Proposition. If X is a closed smooth manifold, f: X → R is a Morse function. then for a generic C^{k}- metric g on X, the pair (f, g) is Morse-Smale.

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_{1}^{2} maps γ: R → X such that lim_{s→−∞} γ(s) = p and lim_{s→∞} γ(s) = q, and γ(−∞, −R] and γ[R, ∞) are L_{1}^{2} for sufficiently large R.

Proposition. Let Y and Z be C^{k} separable Banach manifolds, E → Y × Z a Banach space bundle, ψ: Y × Z → E a C^{k} section such that k ≥ 1 and k ≥ l + 1. Suppose for all (y, z) ∈ ψ^{−1}(0) we have (a) ∇ψ: T_{(y,z)}(Y × Z) → E_{(y,z)} is surjective and (b) restriction ∇ψ: T_{z}Z → E_{(y,z)} is Fredholm of index l. Then for generic y ∈ Y the set {z ∈ Z | ψ(y, z) = 0} is an l-dimensional C^{k} submanifold of Z (and on this set ∇ψ is surjective onto tangent space to Z).

Proposition. Let X be a closed smooth manifold, f: X → 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.

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 γ: R → X such that lim_{s→−∞} γ(s) = p, for R sufficiently small so that γ(−∞, R] ⊂ U_{p}, where U_{p} is a coordinate chart around p, we have γ: (−∞, R] → R^{n} is L_{1}^{2}, Also we must have lim_{s→∞} = q and an analogous statement for U_{q}. Exercise: Z is a C^{∞} Banach manifold, and T_{γ}Z = L_{1}^{2}(γ^{∗}T X). Let E_{g,γ} = L^{2}(γ^{∗}T X). This is a C^{∞} Banach space bundle. Let ψ be a section of E such that ψ(g, γ)(s) = γ′(s) − ξ(γ(s)). Thus ψ(g, γ) = 0 iff γ 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 T X → X. If ψ(g, γ) = 0 then ∇ψ(ġ, γ̇) = ∇_{γ′} γ̇ − ∇γ̇ξ − ξ̇. Given (g, γ), choose a trivialization of γ^{∗}T X, which is parallel with respect to connection on T X. We have ∇ψ(ġ, γ̇) = ∂_{s}γ̇ − A_{s}γ̇ − ξ̇, A_{s} = ∇ξ: T_{γ(s)}X → T_{γ(s)}X. We also have lim_{s→−∞} A_{s} = H(f, p) and lim_{s→∞} A_{s} = H(f, q). (a) We check ∇ψ is surjective. Can show ∇ψ has closed range. (Skip.) Enough to show if ω ∈ L^{2}(γ^{∗}T X) is orthogonal to im(∇ψ) then ω = 0. Such an ω satisfies ∫_{R}⟨ξ̇, ω⟩ds = 0 for every ġ. At any given point in the image of γ can find a ġ such that ξ̇ = ω there. Choosing ġ supported near that point gives ω = 0 there. (b) L_{1}^{2}(γ^{∗}T X) → L^{2}(γ^{∗}T X) sends γ̇ to ∂_{s}γ̇ − A_{s}γ̇ 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 γ ∈ m(p, q), operators ∂_{s} − A_{s} is surjective. Observe H_{+} = T_{γ(0)}D(p) and H_{−} = T_{γ(0)}A(q). Recall coker(∂_{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, ω) be a symplectic manifold, J be an ω-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).

Theorem. For generic J the manifold m_{g,n}^{∗}(X, A) is oriented. Its dimension is

(n − 3)(2 − 2g) + 2⟨c_{1}(T X), A⟩ + 2k.

Theorem. Fix a complex structure j on Σ_{g}. Let Σ = (Σ_{g}, j), m(Σ, X, A) be the space of J-holomorphic maps (Σ_{g}, j) → (X, J) in class A, m^{∗}(Σ, X, A) the space of simple curves in m(Σ, X, A). For generic J the manifold m^{∗}(Σ, X, A) is oriented and has dimension n(2 − 2g) + 2⟨c_{1}(T X), A⟩.

Proof Sketch. Ignore the issue of Banach space completions. Let Y be the space ω-tame almost complex structures J on X, Z be the space of smooth maps u: Σ → X representing the class A. Let E_{J,u} = T (T^{0,1}Σ⊗_{C} u^{∗}T X) is the space of C-antilinear bundle maps T Σ → u^{∗}T X. Let ψ(J, u) be the map T Σ → u^{∗}T X defined by ψ(J, u) = du + J ∘ du ∘ j. We have ψ(J, u) = 0 iff J ∘ du = du ∘ j, i.e., u is holomorphic. We say that u is somewhere injective whenever there is a z ∈ Σ such that u^{−1}(u(z)) = {z} and du: T_{z}Σ → 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) ∈ ψ^{−1}(0): (a) ∇ψ: T_{J,u}Y × Z → E_{J,u} is simple; (b) ∇ψ: T_{u}Z → E_{J,u} is Fredholm of index n(2−2g)+2⟨c_{1}(T X), A⟩. We see that T_{J}Y is the space of bundle maps J̇: T X → T X such that J̇J +JJ̇ = 0. and T_{u}Z = T (u^{∗}T X). We have ψ(J, u) = du + J ∘ du ∘ j. (b) We have ∇ψ(0, u̇) = (∂_{s}u + J∂_{t}u)(ds − idt) plus zero order term. We can deform the zeroth order term so that this is ¯∂ operator on holomorphic vector bundle u^{∗}T X over Σ. Riemann-Roch theorem implies that the index is n(2 − 2g) + 2c_{1}(u^{∗}T X). To complete the proof, we use somewhere injective to prove (a).

Suppose we have a symplectic manifold (M, ω), J is ω-tame almost complex structure. Fix (Σ, j) and u: Σ → M. We have ¯∂(u) = du+J ∘du∘j ∈ Ω^{0,1}(Σ, u^{∗}T M) = Γ(T^{0,1}Σ⊗_{C}u^{∗}T M). The map u is holomorphic iff ¯∂(u) = 0. If u is holomorphic, derivative of ¯∂ defines an operator D_{u}: T (u^{∗}T M) → Γ(T^{0,1}Σ ⊗_{C} u^{∗}T M).

Definition. u is transverse if D_{u} is surjective (on C^{∞} 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(Σ, M) is a manifold near u, and T_{u}m(Σ, M) = ker(D_{u}). Also dim = ind(D_{u}) = n(2 − 2g)+2⟨c_{1}(T M), A⟩. 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_{J}g ⊕ Γ(u^{∗}T M) → Γ(T^{0,1}Σ ⊗ u^{∗}T M), (J̇, ξ) → D_{u}ξ + J̇ ∘ du ∘ j. Can show closed range. More precisely, suppose η is perpendicular to image. Since u is somewhere injective, find z ∈ Σ such that u embeds a neighborhood of z into M, disjoint from the rest of u(Σ). It follows that η = 0 in a neighborhood of z. So we have ⟨D(J̇, 0), η⟩ = ∫_{Σ}⟨J̇ ∘du∘j, η⟩ = ∫_{u^{−1}(U)}⟨J̇ ∘du∘j, η⟩ = ∫_{N}⟨J̇ ∘du∘j, η⟩ > 0. Since η ⊥ im D_{u}⟩ we have D_{u}^{∗}η = 0. Unique continuation theorem shows that if D_{u}^{∗}η = 0 and η vanishes to infinite order at a point, then η = 0.

Back to the Gromov nonsqueezing theorem. Take M = S^{2} × V, ω = ω_{S^{2}} ⊕ ω_{V}, J = J_{S^{2}} ⊕ J_{V}. Claim: The holomorphic spheres S^{2} × {v} are transverse.

Proof. Let u be the map S^{2} → S^{2} × {v}. We have the operator D_{u}: Γ(u^{∗}T M) → Γ(T^{0,1}S^{2} ⊗ u^{∗}T M). We have splitting u^{∗}T M = T S^{2} ⊕ T_{v}V = T S^{2} ⊕ C^{n−1}. D_{u} respects this splitting. D_{u} is a sum of operators of the form Γ(L) → Γ(T^{0,1}S^{2} ⊗ 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 η ∈ Ω^{0,1}(S^{2}, L) satisfies such an equation. It follows that any zero of η has negative multiplicity. From Carleman it follows that deg(T^{0,1}S^{2} ⊗ L) ≤ 0.

Remark. Consider M = Σ_{g} × V. Are the curves Σ × {v} transverse? No, if g > 0.

Also: moduli spaces are canonically oriented.

Gromov Compactness Theorem. (Simplest version.) M, ω, J as usual. Let A ∈ H_{2}(M). Suppose there is no B ∈ π_{2}(M) such that 0 < ∫_{B} ω < ∫_{A} ω. Then m(S^{2}, M, A) is compact. Moreover, if J_{t} is a family of ω-tame almost complex structures, then ∪_{t}{t} × m_{J_{t}}(S^{2}, M, A) is compact.

Idea of proof. Consider a sequence of maps u_{k}: S^{2} → M. The energy E(u_{k}) = ∫_{S^{2}} |du_{k}|^{2} = ∫_{A} Cω. 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 ∫_{A} ω.

Intersection property. Let (M, ω) 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.

Adjunction formula. Let C be a simple J-holomorphic curve in X. Define c_{1}(C) = c_{1}(T X), where T X is restricted to C. Let χ(C) be the Euler characteristic of domain of C. Then c_{1}(C) = χ(C) + C · C − 2 ∑ δ(p), where the sum is taken over all singular points of C, and δ(p) is a positive integer, δ(p) = 1 iff C has a transverse self-intersection at p.

Theorem. (Recognition of R^{4}.) Let (M, ω) be (noncompact) symplectic manifold, H̃_{∗}(M) = 0. Suppose there is a compact subset K of M such that (M \ K, ω) is symplectomorphic to (R^{4} \ B, ω_{s}). Then (M, ω) is symplectomorphic to R^{4} with standard symplectic structure.

Proof. We see that M̃ \K is symplectomorphic to S^{2} ×S^{2} \B. Also H_{∗}(M̃) = H_{∗}(S^{2} ×S^{2}). Choose ω-tame J on M̃ which is product structure outside of K. Take A = (1, 0) ∈ H_{∗}(M̃) and B = (0, 1) ∈ H_{∗}(M̃). Look at ev: (M̃, A) → M̃. Similarly to proof of nonsqueezing, this evaluation map has degree 1. Consider (p, q) as shown. There is a holomorphic sphere S^{2} × {q} ⊂ m_{0,1}. We claim that S^{2} × {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 · c′ > 0. But (1, 0) · (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 m̃. There is a unique such sphere through each point. Any such sphere is embedded: 2 = c_{1}(C) = χ(C) + C · C − 2δ(C) = 2 + 0 − 0. Different spheres don’t intersect.

Now we know that every point in M̃ is in a unique embedded holomorphic sphere in class (1, 0) = A or (0, 1) = B respectively. Define ϕ: S^{2} × S^{2} → M̃ as follows. Given (x, y) ∈ S^{2} × 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 ϕ(x, y) to be the intersection of C_{1} with C_{2}. (Unique by intersection positivity.) Claim: ϕ is a diffeomorphism. Surjective because every point in M̃ 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 ϕ is the identity at infinity. Claim: ϕ is isotopic to a symplectomorphism that is still identity at infinity. Proof: Let ω denote symplectic forms. Claim: ω ∧ ϕ^{∗}ω > 0. We have (ω ∧ ϕ^{∗}ω)(∂_{x_{1}}, ∂_{y_{1}}, ∂_{x_{2}}, ∂_{y_{2}}) = ω(∂_{x_{1}}, ∂_{y_{1}})ϕ^{∗}ω(∂_{x_{2}}, ∂_{y_{2}}) + ω(∂_{x_{2}}, ∂_{y_{2}})ϕ^{∗}ω(∂_{x_{1}}, ∂_{y_{1}}). Let ω_{t} = tω + (1 − t)ϕ^{∗}ω on S^{2} × S^{2} for t ∈ [0, 1]. Since ω ∧ ϕ^{∗}ω > 0 it follows that ω_{t} ∧ ω_{u} > 0 for all t and u. Hence all ω_{t}’s are all in the same cohomology class. Choose 1-form β_{t} with dβ_{t} = (d/dt)ω_{t} and β_{t} = 0 at infinity. Look for ϕ_{t}: S^{2} × S^{2} → M̃ such that ϕ_{t}^{∗}ω = ω_{t} and ϕ_{0} = ϕ and ϕ_{t} depends smoothly on t. Then ϕ_{1}^{∗}ω = ω_{1}. To get ϕ_{t}^{∗}ω = ω_{t} for all t it is enough to get (d/dt)ϕ_{t}^{∗}ω = (d/dt)ω_{t}. We have (d/dt)ϕ_{t}^{∗}ω = L_{X_{t}}ω = (d/dt)ω_{t} = dβ_{t}. X_{t} is a vector field on S^{2} × S^{2} determining the isotopy. Since ω is nondegenerate there is X_{t} with I_{X_{t}} = β_{t′} X_{t} = 0 at infinity so ϕ_{1} = ϕ_{0} at infinity. Conclusion: we have a symplectic diffeomorphism between S^{2} × S^{2} and M̃, which is identity at infinity.

Theorem. The group of symplectic diffeomorphisms of S^{2} × S^{2} is SO(3) × SO(3).

Proof. Key point is that for any ω-tame J on S^{2} × S^{2} we have two foliations by A-spheres and B-spheres. Note: The claim is false for S^{2} ×S^{2} with symplectic structure ω_{1} ⊕ω_{λ}, where ∫_{S^{2}} ω_{1} = 1, ∫_{S^{2}} ω_{λ} = λ ∈ (1, 2]. Trouble is that (0, 1) = (1, 0) + (−1, 1). So in constructing B foliation the compactness argument fails.

Suppose (X, ω) is a closed symplectic manifold. A symplectic isotopy is a smooth family of symplectomorphisms of X. We say that isotopy ϕ is generated by vector fields X_{t} if (d/dt)ϕ_{t}(x) = X_{t}(ϕ(x)). Differentiating ϕ_{t}^{∗}ω = ω we get 0 = L_{X_{t}}ω = di_{X_{t}}ω + i_{X_{t}}dω. Conclusion: X_{t} generates a symplectic isotopy iff ω(X_{t}, ·) is a closed 1-form.

Definition. The isotopy ϕ_{t} is Hamiltonian if ω(X_{t}, ·) is exact, i.e., ω(X_{t}, ·) = dH_{t}, where H_{t}: X → R.

Degenerate Arnold Conjecture. If ϕ: (X, ω) → (X, ω) is Hamiltonian isotopic to id_{X} then | Fix(ϕ)| ≥ min{| Crit(f)| | f: X → R}. It is still open!

Nondegenerate Arnold Conjecture. If ϕ: (X, ω) → (X, ω) is Hamiltonian isotopic to id_{X} then | Fix(ϕ)| ≥ ∑_{i} dim(H_{i}(X, Q)). Proved using Floer homology.

A fixed point p of f is nondegenerate if 1 − df_{p}: T_{p}X → T_{p}X is invertible. Equivalently, graph T (f) = {(x, f(x))} ⊂ X × X is transverse to the diagonal Δ = {(x, x)} ⊂ X × X at (p, p).

Lefschetz Fixed Point Theorem. If X is a closed smooth manifold and f: X → X is a smooth map with nondegenerate fixed points then ∑_{p∈Fix(f)} sign det(1 − df_{p}) = ∑_{i}(−1)^{i} tr(f_{∗}: H_{i}(X, Q) → H_{i}(X, Q)). It follows that if f ∼ id_{X} then | Fix(f)| ≥ |∑_{i}(−1)^{i} dim H_{i}(X, Q)| = |χ(X)|.

Note. Arnold conjecture is false if we only require ϕ to be symplectically isotopic to id_{X}.

Counterexample. The map ϕ: (T^{2}, ω) → (T^{2}, ω) such that ϕ(x, y) = (x + 1/2, y) is symplectically isotopic to identity via the map ϕ_{t}(x, y) = (x + t/2, y). This symplectic isotopy is not Hamiltonian because it is generated by X_{t} = ∂_{x}/2 and i_{X_{t}}ω = dy/2, which is not exact. In fact, there is no Hamiltonian isotopy from ϕ to id. If ϕ is a symplectic isotopy, define the flux of ϕ to be an element of H^{1}(X, R) such that its value on γ: S^{1} → X is ∫_{[0,1]×S^{1}} f^{∗}ω, where f: [0, 1] × S^{1} → X is the map such that f(t, 0) = ϕ_{t}(γ(0)).

Fact. This gives a well-defined element of H^{1}(X, R) and ϕ is homotopic relative to endpoints to a Hamiltonian isotopy iff flux of ϕ is zero.

If ϕ is symplectically isotopic to identity, then the flux of ϕ is defined and is zero iff ϕ is Hamiltonian isotopic to identity. If X = T^{2}, the flux of ϕ in the previous example is (0, 1/2) ≠ 0.

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

Symplectic action functional. Let L be the space of contractible smooth loops in X. Let L̃ be the space of pairs (γ, [u]), where γ ∈ L and [u] is homotopy class of u: D^{2} → X relative boundary such that γ is the restriction of u to the boundary. Consider the map A: L̃ → R such that A(γ, [u]) = ∫_{[0,1]} H_{t}(γ(t))dt+∫_{D^{2}} u^{∗}ω.

Lemma. (γ, [u]) ∈ Crit(A) iff γ′(t) = X_{H_{t}}(γ(t)) iff γ(t) = ϕ_{t}(γ(0)). Hence Crit(A) = Fix(ϕ).

Proof. Let ξ ∈ T_{(γ,[u])}L̃ = Γ(γ^{∗}T X). Now

\begin{matrix}dA_{γ}(ξ) = (d/ds) (∫_{[0,1]} H_{t}(γ_{s}(t))dt + ∫_{u_{s}} ω) = ∫_{[0,1]} dH_{t}(ξ(t))dt + ∫_{[0,1]} ω(ξ, γ′(t))dt \\ = ∫_{[0,1]}(ω(X_{H_{t}}, ξ(t)) + ω(ξ(t), γ′(t))dt = ∫_{[0,1]} ω(ξ(t), γ′(t) − X_{H_{t}})dt. \end{matrix}

(γ, [u]) ∈ Crit(A) iff the above integral is zero for all ξ iff γ′(t) − X_{H_{t}} = 0 for all t.

Key example. Suppose H_{t} = H: X → R is a Morse function, X_{H_{t}} = X and ω(X, ·) = dH. Now Crit(H) ⊂ Fix(ϕ). Arnold conjecture is trivial in this case.

Let J_{t}, where t ∈ S^{1} be a family of ω-compatible almost complex structures on X. Each J_{t} defines a metric g_{t} on X by g_{t}(v, w) = ω(v, J_{t}w). This defines a metric on L̃ as follows: If (γ, [u]) ∈ L̃ and ξ_{1}, ξ_{2} ∈ T_{(γ,[u])}, then ⟨ξ_{1}, ξ_{2}⟩ = ∫_{S^{1}} g_{t}(ξ_{1}(t), ξ_{2}(t))dt. Consider a path ũ: R → L̃ such that ũ(s) = (u(s, ·), [D_{s}]). Lemma: ũ is an upward gradient flow line of A iff ∂_{s}u + J_{t}(∂_{t}u − X_{H_{t}}) = 0. Note: This is almost the equation for u to define a pseudoholomorphic map R × S^{1} → X. (Only almost because J depends on t and there is X_{H_{t}} term.) Proof: Need to check that (d/ds)ũ(s) = ∇A(u(s)), i.e., fix s, let γ(t) = u(s, t), want for any ξ ∈ T_{γ}L = Γ(γ^{∗}T X) the following: dA_{γ}(ξ) = ⟨(d/ds)ũ(s), ξ⟩, i.e., ∫_{[0,1]} ω(ξ, γ′(t) − X_{H_{t}}) = ⟨J_{t}(−γ′(t) + X_{H_{t}}), ξ⟩, i.e., ∫_{[0,1]} ω(ξ, γ′(t) − X_{H_{t}}) = ∫_{[0,1]} ω(J_{t}(−γ′(t) + X_{H_{t}}), J_{t}ξ)dt. (Tame ok.)

Key example. H_{t} = H: X → R is a Morse function, J_{t} = J is an ω-compatible almost complex structure, therefore g_{t} = g, g(v, w) = ω(v, Jw). Suppose η: R → X is an upward gradient flow line of H, i.e., η′(s) = ∇H(η(s)). Define u(s, t) = η(t). This satisfies ∂_{s}η = JX_{H} because JX_{H} = ∇H because ⟨JX_{H}, v⟩ = dH(v) because ⟨JX_{H}, v⟩ = ω(JX_{H}, Jv) = ω(X_{H}, v) = dH(v).

Conclusion. When H_{t} and J_{t} do not depend on t we have Crit(H) ⊂ Crit(A)/π_{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, ω) is a closed symplectic manifold, H is a 1-periodic Hamiltonian: H: S^{1} × M → R, X_{H_{t}} is the corresponding vector field, and ϕ_{t} is a family of symplectomorphisms such that ϕ_{0} = id_{M}, (d/dt)ϕ_{t}(x) = X_{H_{t}}(ϕ_{t}(x)). Let ϕ = ϕ_{1}. We say that ϕ_{t} is a Hamiltonian isotopy from id_{M} to ϕ.

Arnold Conjecture: ϕ has at least as many fixed points as the minimum number of critical points of a function M → R. Nondegenerate version: If fixed points of ϕ are nondegenerate then the number of fixed points is at least ∑_{i} dim H_{i}(M, Q).

Now let L be the space of contractible loops in M. Define a symplectic action functional A: L → R (if ⟨ω, π_{2}(M)⟩ = 0) A(γ) = ∫_{S^{1}} H_{t}(γ(t))dt + ∫_{D} ω, where D is a disc in M with boundary γ. We have Crit(A) = Fix(ϕ) = {γ: S^{1} → M | γ′(t) = X_{H_{t}}(γ(t)). If J_{t} is an ω-compatible almost complex structure for t ∈ S^{1} then we have a metric on L.

An upward gradient flow line of A from γ_{−} to γ_{+} is a map u: R_{s} → S_{t}^{1} → M satisfying ∂_{s}u + J_{t}(∂_{t}u − X_{H_{t}}) = 0. We have lim_{s→−∞} u(s, t) = γ_{−}(t) and lim_{s→∞} u(s, t) = γ_{+}(t).

Floer homology: define Morse homology for A generated by fixed points of ϕ 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 (∂^{2} = 0), orientations (counting with signs), removing symplectically aspherical assumption.

Let ϕ: (M, ω) → (M, ω) be any symplectomorphism (not necessarily Hamiltonian isotopic to identity). Define the mapping torus Y_{ϕ} = [0, 1] × M/(1, x) ∼ (0, ϕ(x)). Y_{ϕ} fibers over S^{1} with fiber M. There is a vector field ∂_{t} on Y_{ϕ}. Fixed points of ϕ correspond to circles in Y that are tangent to ∂_{t} and go once around the S^{1} direction. R × Y_{ϕ} has a symplectic form Ω = ω + ds ∧ dt. Choose an Ω-tame almost complex structure J on R×Y such that J: T M → T M, J(∂_{s}) = ∂_{t} and J does not depend on s. Equivalently, choose ω-tame almost complex structure J_{t} on M for each t ∈ R such that J_{t+1} = ϕ_{∗} ∘ J_{t} ∘ ϕ_{∗}^{−1}. If X_{+} and X_{−} are fixed points of ϕ corresponding to circles γ_{+} and γ_{−} in Y_{ϕ}, define a flow line from X_{+} to X_{−} to be a J-holomorphic cylinder C ⊂ R × Y_{ϕ} such that C is asymptotic to R × Y_{t} as s → ∞.

This is more general version of Floer homology for any ϕ ∈ Symp(M, ω). Chain complex is generated by Fix(ϕ). Differential counts holomorphic cylinders in R × Y_{ϕ}. Any flow line as above is a section of R × Y_{ϕ} → R × S^{1}.

Why this generalizes previous setup? Suppose ϕ = ϕ_{1} comes from H: S^{1} × M → R. Y_{ϕ} ↔ S^{1} × M, (t, x) ↔ (t, ϕ_{t}(x)), γ: S^{1} → Y_{ϕ} ↔ γ: S^{1} → M.

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: R × R → M such that u(s, t + 1) = ϕ^{−1}(u(s, t)), ∂_{s}u + J_{t}∂_{t}u = 0 and lim_{s→±∞} u(s, t) = γ_{±}(t). What is the dimension of m(x_{+}, x_{−}) near u? Assume everything is transverse. u corresponds to a cylinder C ⊂ R × Y.

Deformation operator D: L_{1}^{2}(C, T M) → L^{2}(C, T^{0,1}C ⊗_{C} T M). If we choose some trivialization of T M over C, then D has the form D: L_{1}^{2}(R × S^{1}, R^{2n}) → L^{2}(R × S^{1}, R^{2n}) such that Dξ = ∂_{s}ξ + J_{0}∂_{t}ξ + A(s, t)ξ). 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_{±}(t) = lim_{s→±∞} A(s, t) is a symmetric matrix. We have trivialized T M over γ_{±}. Linearization of the flow ∂_{t} along γ_{±} from t = 0 to a given t defines a pair of symplectic linear maps ψ_{t}^{±}: T_{x}M → T_{x}M such that ψ_{0}^{±} = id_{T_{x}M} and ψ_{1}^{±} = dϕ: T_{x}M → T_{x}M. With respect to trivialization we get two paths of symplectic matrices ψ_{t}^{±}. Assume fixed points are nondegenerate, so 1 is not an eigenvalue of ψ_{1}^{±}. A_{±}(t) is given by (d/dt)ψ_{t}^{±} = J_{0}A_{t}ψ_{t}^{±}.

Theorem. Let A(s, t) be matrices parametrized by (s, t) ∈ R × S^{1} such that lim_{s→±∞} A(s, t) = A_{±}(t) is symmetric and 1 is not an eigenvalue of ψ^{p}m_{1}. Then the operator D: L_{1}^{2}(R × S^{1}, R^{2n}) → L^{2}(R × S^{1}, R^{2n}) defined by Dξ = ∂_{s}ξ + J_{0}∂_{t}ξ + A(s, t)ξ is Fredholm and ind(D) = CZ(ψ_{t}^{+}) − CZ(ψ_{t}^{−}). CZ is the integer Conley-Zehnder index associated to a path of symplectic matrices.

Recall that we have a symplectic manifold (M, ω) with an automorphism ϕ. Consider the mapping torus of ϕ: Y_{ϕ} = [0, 1] × /(1, x) ∼ (0, ϕ(x) = R × M/(t + 1, x) ∼ (t, ϕ(x)). Fixed points of ϕ correspond to parallel sections. Choose an ω-tame almost complex structure on E, or equivalently J_{t} on M such that J_{t+1} = ϕ_{∗}J_{t}ϕ_{∗}^{−1}.

Floer homology of ϕ: Chain complex generators are fixed points of ϕ, regarded as circles in Y_{ϕ}, differential counts holomorphic sections of R × Y_{ϕ}: If γ_{+} and γ_{−} are circles in Y_{ϕ} corresponding to x_{+} and x_{−} in Fix(ϕ), then ⟨∂x_{+}, x_{−}⟩ counts maps u: R × R → M such that u(s, t + 1) = ϕ^{−1}(u(s, t)), ∂_{s}u + J_{t}∂_{t}u = 0 and lim_{s→±∞} u(s, t) = γ_{±}(t) = x_{±}.

The difference between gradings of x_{+} and x_{−} equals expected dimension of m(x_{+}, x_{−}). Compute this. Given u ∈ m(x_{+}, x_{−}) compute ind(D_{u}). We have D_{u}: L_{1}^{2}(R × S^{1}, u^{∗}E) → L^{2}(R × S^{1}, u^{∗}E). and D_{u}ξ = ∂_{s}ξ + J_{t}∂_{t}ξ + A(s, t)ξ. Choose Hermitean trivialization τ of u^{∗}E over R × S^{1}. We have u^{∗}E = R × S^{1} × R^{2n} and D_{u}ξ = ∂_{s}ξ + J_{0}∂_{s}ξ + A(s, t)ξ, where J_{0} is the standard complex structure on R^{2n}. τ restricts to a trivialization τ_{±} of E over γ_{±}. For |s| ≫ 0 we have ∂_{s}ξ + J_{0}∂_{t}ξ + A(s, t)ξ ≈ ∂_{s}ξ + J_{0}∇_{t}ξ. With respect to τ_{±} parallel transport along γ_{±} from 0 to t defines a symplectic map ψ_{t}^{±} ∈ Symp(R^{2n}, ω_{0}). The path ψ_{t}^{±} of symplectic matrices is equivalent to a path A_{t}^{±} of symmetric matrices via (d/dt)ψ_{t}^{±} = J_{0}A_{t}^{±}ψ_{t}. Conclusion: D_{u}ξ = ∂_{s}ξ + J_{0}∂_{t}ξ + A(s, t)ξ where lim_{s→±∞} A(s, t) = A_{t}^{±}. Theorem: If 1 ∉ Spec(ψ_{1}^{±}) then D is Fredholm and ind(D) = CZ(ψ_{t}^{+}) − CZ(ψ_{t}^{−}).

Let {ψ_{t} | t ∈ [0, 1]} be a path of symplectic matrices on R^{2n} with ψ_{0} = id and 1 ∉ Spec(ψ_{1}). Define the Conley-Zehnder index CZ(ψ_{t}) ∈ Z as follows. Define the Maslov cycle M = {A ∈ Sp_{2n}(R) | 1 ∈ Spec(A)}. Roughly speaking, CZ(ψ_{t}) is the signed count of ψ_{t} with M. Note: U(n) = Sp(2n) ∩ O(n) is a maximal compact subgroup of Sp(2n). H^{1}(Sp(2n), Z) = Z is generated by a continuous extension of det: U(n) → S^{1}. M is a co-oriented codimension 1 subvariety of Sp(2n), Poincaré dual to the generator of H^{1}(Sp(2n), Z). So if {ψ_{t} | t ∈ S^{1}} is an arbitrary loop in Sp(2n) then {ψ_{t}} ∩ M ∈ Z is defined. To define CZ index, declare

CZ (t → ⨁_{n} ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} )) = 0. To defined CZ(ψ_{t}), let γ be a path from ⨁_{n} ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} ) to ψ_{1}, which is

transverse to M such that η + γ − {ψ_{t}} = 0 ∈ H_{1}(Sp(2n)). Define CZ(ψ_{t}) = #(γ ∩ M).

General properties: Naturality: If ϕ: [0, 1] → Sp(2n) is an arbitrary path, then CZ(ϕψϕ^{−1}) = CZ(ψ). Inverses: CZ(ψ^{−1}) = − CZ(ψ). Change of trivialization: If ϕ: [0, 1] → Sp(2n) is a path such that ϕ(0) = ϕ(1) = id, then CZ(ϕψ) = CZ(ψ) + 2 deg ϕ. Signature: If A is a symmetric matrix with det(A) ≠ 0 and ∥A∥ < 2π then CZ(exp(J_{0}At)) = σ(A)/2.

Why is ind(∂_{s} + J_{0}∂_{t} + A(s, t)) = CZ(ψ_{t}^{+}) − CZ(ψ_{t}^{−})? We omit the proof of Fredholm property. To compute index we assume that A(s, t) is symmetric. For each s ∈ R take a family of symplectic matrices ψ_{s,t} defined by ψ_{s,0} = id, (d/dt)ψ_{s,t} = J_{0}A_{s,t}ψ_{s,t}. Claim: For a given s we have 0 ∈ Spec(J_{0}∂_{t} + A(s, t)) iff 1 ∈ Spec(ψ_{s,1}). CZ(ψ_{t}^{+}) − #(ψ_{s,1}) ∩ M − CZ(ψ_{t}^{−}) = 0.

Theorem. If the almost complex structures J_{t} on M are generic, then for any m ∈ m(x_{+}, x_{−}), the operator D_{u} is surjective, so m(x_{+}, x_{−}) is a manifold near u of dimension ind(u).

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_{∅} is somewhere injective.

Index theorem for Cauchy-Riemann operators on Riemann surfaces with cylindrical ends. Assumptions: C is a Riemann surface with ends identified with [0, ∞)×S^{1}, E is a rank n complex vector bundle on C (with Hermitean metric), D: L_{1}^{2}(E) → L^{2}(T^{0,1}C ⊗ E) such that in local coordinates and trivialization D = ∂_{s} + i∂_{t} + zeroth order term, on each end, for some trivialization of E, D = ∂_{s} + i∂_{t} + A(s, t) where lim_{s→∞} A(s, t) = A(t) symmetric, and if ψ_{0} = id and (d/dt)ψ_{t} = J_{0}A(t)ψ_{t}, then 1 ∉ Spec(ψ).

Theorem. D is Fredholm and ind(D) = nχ(C) + 2c_{1}(E, τ) + ∑_{ends} CZ_{τ}.

To compute c_{1}(E, τ): take a generic section s of Λ^{n}E such that on each end, s is nonvanishing and constant with respect to trivialization τ. Then c_{1}(E, τ) = #s^{−1}(0). This depends only on E and homotopy class of τ. CZ_{τ} is the CZ index of the path ψ_{t} obtained as above. (Only depends on homotopy class of τ.) We can identify some ends with (−∞, 0] × 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 τ; 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 nχ(C)+2c_{1}(E, τ)+∑_{ends} CZ_{τ} independent of τ? For any given end, the set of homotopy classes of trivializations is an affine space over π_{1}U(n) = Z. If you shift the trivialization by 1, then c_{1} changes by ±1. If s is any generic section of E which is nonvanishing on ends, then c_{1}(E, τ) = #s^{−1}(0) − ∑. If you shift trivialization by 1, CZ_{τ} shifts by ∓2. 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.

Back to Floer homology. H: S^{1} × M → R generates ϕ: (M, ω) → (M, ω) and Hamiltonian isotopy ϕ_{t} from 1 to ϕ. Assume M is symplectically aspherical: ⟨c_{1}(T M), π_{2}(M)⟩ = ⟨ω, π_{2}(M)⟩ = 0. CF_{∗}(H) is the free Z-module generated by contractible loops γ: S^{1} → M with γ′(t) = X_{H_{t}}. (These are some of the fixed points of ϕ.) Grading: Let γ: S^{1} → M be a generator. Define μ(γ) ∈ Z as follows. Choose a map η: D^{2} → M such that η restricted to ∂D^{2} is γ. Trivialize η^{∗}T M use this to trivialize γ^{∗}T M. Homotopy class of trivialization of γ^{∗}T M does not depend on η because ⟨c_{1}(T M), η − η′⟩ = 0. Linearization of equation, γ′(t) = X_{H_{t}} defines a family of symplectic matrices ψ(t): T M_{γ(0)} → T M_{γ(t)}.

Floer homology. We use Z/(2) coefficients. Assume ϕ has nondegenerate fixed points. Choose a generic family of ω-tame almost complex structures J_{t} on M for t ∈ S^{1}. Define (CF_{∗}(H, J), ∂) as follows. Fix(ϕ) = {γ: S^{1} → M | γ′(t) = X_{H_{t}}(γ(t))}. CF_{∗} is the free Z/(2)-module generated by fixed points corresponding to contractible γ, with Z-grading. Given a generator γ, let u: D^{2} → M be a map such that its restriction to S^{1} is γ. This determines a homotopy class of trivialization of γ^{∗}T M, independent of u because ⟨c_{1}(T M), u − u′⟩ = 0. With this trivialization, {dϕ_{t}: T_{γ(0)}M → T_{γ(0)}M}_{t∈[0,1]} is a path of symplectic matrices from 1 to μ(x) = CZ.

Given fixed points x_{+} and x_{−} corresponding to γ_{+} and γ_{−} let m(x_{+}, x_{−}) = {u: R_{s} × S^{1} → M | ∂_{s}u + J_{t}(∂_{t}u − X_{H_{t}}) = 0 ∧ lim_{s→±∞} u(s, ·) = γ_{±}}. By previous theorem, if J is generic, dim m(x_{+}, x_{−}) = μ(x_{+}) − μ(x_{−}). R acts on m(x_{+}, x_{−}) by translating s.

Definition. ∂: CF_{∗} → CF_{∗−1}: ∂x_{+} = ∑_{x_{−}:μ(x_{+})−μ(x_{−})=1} #(m(x_{+}, x_{−})/R)x_{−}.

Lemma. ∂ is well defined, i.e., m(x_{+}, x_{−})/R is finite when μ(x_{+}) − μ(x_{−}).

Gromov compactness. For any closed (M, ω) 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, ω) is symplectically aspherical, then we do not have any bubbles because any holomorphic sphere has ∫ ω > 0.

Theorem. ∂^{2} = 0.

Theorem. HF_{∗}(H, J) does not depend on (H, J).

Proof idea. Consider generic family {(H_{s}, J_{s}) | s ∈ R}. Assume (H_{s}, J_{s}) = (H_{+}, J_{+}) for large positive s, (H_{s}, J_{s}) = (H_{−}, J_{−}) for large negative s. Define Φ: HF_{∗}(H_{+}, J_{+}) → HF_{∗}(H_{−}, J_{−}). Choose x_{+} and x_{−} in Fix(ϕ_{±}) corresponding to γ_{±}: S^{1} → M. Let Φ(x_{+}) = ∑_{x_{−}:μ(x_{+})=μ(x_{−})} #m(x_{+}, x_{−})x_{−}. Similarly to Morse theory case, Φ is a chain map, induces an isomorphism on homology depending only on the homotopy class of the path from (H_{+}, J_{+}) to (H_{−}, J_{−}).

Remark. HF_{∗}(H, J) does depend on ϕ in the sense that if (H_{+}, J_{+}) and (H_{−}, J_{−}) has ϕ_{+} and ϕ_{−} the map Φ: HF_{∗}(H_{+}, J_{+}) → HF_{∗}(H_{−}, J_{−}) might be nontrivial.

Theorem. HF_{∗}(H, J) = H_{∗+n}(M, Z/(2)).

Proof. Take H_{t}: M → R independent of t, Morse function H: M → R. (May have to replace H by ϵH, ϵ > 0 small.) Take J_{t} = J independent of t, let g be the corresponding metric. Can arrange that (H, g) is Morse-Smale.

Claim. If we replace H by ϵH for ϵ > 0 sufficiently small, then (CF_{∗}(H, J), ∂) is well-defined and equal to (C_{∗}^{M}(H, g), ∂). Need: Generators are the same. Gradings. Floer differential is defined. Differentials agree.

(1) Crit(H) ⊂ Fix(ϕ). This is an equality if ϵ is sufficiently small. (2) Recall CZ{exp(J_{0}At) | t ∈ [0, 1]} = σ(A)/2. If x ∈ Crit(H), then dϕ_{t}: T_{x}M → T_{x}M is exp(t∇X_{H}) = exp(−tJ · Hess(H, x)), therefore μ(x) = −σ(Hess)/2 = −n + ind(f, x). (3) and (4): If μ(x_{+}) − μ(x_{−}) = 1, then every u ∈ m(x_{+}, x_{−}) is independent of t. If u ∈ m(x_{+}, x_{−}) is independent of t, then D_{u} is surjective. D_{u}: L_{1}^{2}(R × S^{1}, R^{2n}) → L^{2}(R × S^{1}, R^{2n}). D_{u}ξ = ∂_{s}ξ + J_{0}∂_{t}ξ + A(s)ξ, where ∂_{s} + A(s) is the Morse theory deformation operator. Show: If ϵ is small enough, then any ξ ∈ 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 ξ ∈ ker(D_{u}). Write ξ = ξ_{0} + ξ_{1}, where ξ_{0}(s, t) = ∫_{τ∈S^{1}} ξ(s, τ)dτ. Wlog ξ = ξ_{1}. We see that ∥ξ∥_{L^{2}} ≤ cϵ∥xi∥_{L^{2}}.

Suppose (M, ω) is symplectically aspherical, H: S^{1} × M → R, J_{t} (t ∈ 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: M → R independent of t, H → ϵH, ϵ > 0 small, J independent of t corresponding to g. Then CF_{i}(H, J) = C_{i+n}^{M}(f, g) ⊗ 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 ϵ is sufficiently small, then every solution to the equation ∂_{s}u+J(∂_{t}u−X_{H}) = 0 (∗), where lim_{s→±∞} u(s, t) = x_{±} 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 ϵ > 0 there is a t-dependent solution. Start with ϵ_{0} > 0 sufficiently small that all previous steps work. For every positive integer n there is a t-dependent solution to the equation for ϵ_{0}/n: ∂_{s}u_{n} +J(∂_{t}u_{n} −X_{H}/n) = 0. Define v_{n}(s, t) = u_{n}(ns, nt). Then ∂_{s}v_{n} + J(∂_{t}v_{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_{∞} is S^{1}-invariant. Since v_{∞} is transverse, v_{n} = v_{∞} for large n up to R-translation.

Definition. A symplectic manifold (M, ω) is called monotone if there is a λ > 0 such that ⟨c_{1}(T M), A⟩ = λ⟨ω, A⟩ for all A ∈ π_{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 γ: S^{1} → M such that γ′(t) = X_{H_{t}}.

Grading is only defined in Z/N, where N = 2 min{⟨c_{1}(T M), A⟩ | A ∈ π_{2}(M) ∧ ⟨c_{1}(T M), A⟩ > 0}. If γ: S^{1} → M is a generator, let u: D^{2} → M be a map such that it restriction to the boundary is γ. Let τ be a trivialization of γ^{∗}T M that extends to a trivialization of u^{∗}T M with respect to τ, dϕ_{t}: T M_{γ(0)} → T M_{γ(t)} is a symplectic matrix ψ_{t}. Define grading μ(γ) = CZ{ψ_{t} | t ∈ [0, 1]}. If u′ is another disk, then CZ{ψ_{t}} = CZ{ψ′_{t}} = ±2⟨c_{1}(T M), u − u′⟩. Differential: ∂γ = ∑_{μ(γ)−μ(γ′)=1} #m_{1}(γ, γ′)γ′, where m_{1}(γ, γ′) = {u ∈ m(γ, γ′) | ind(D_{u}) = 1}. Claim: ∂ is well-defined, ∂^{2} = 0. Compactness argument. Suppose u_{n} ∈ m(γ, γ′), ind(D_{u_{∗}}) ∈ {1, 2}. Subsequence converges to a cylinder with bubbles. We have ind(u_{n}) = ∑_{i} ind(v_{i}) + ∑_{j} 2⟨c_{1}(T M), [S_{j}]⟩, hence there are no bubbles and previous argument applies. As before, HF_{∗}(H, J) = ⊕_{i≡∗+n (mod N)}H_{i}(M, Z/(2)).

Bad case: there are holomorphic spheres S with ⟨c_{1}(T M), [S]⟩ < 0. Multiple covers of S have very negative c_{1}. Not so bad case: ⟨c_{1}(T M), A⟩ = 0 for all A ∈ π_{2}(M). Novikov rings.

Recall that a Lagrangian in (M^{2n}, ω) is a closed submanifold L^{n} ⊂ M such that ω 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 ω-tame almost complex structure for t ∈ [0, 1]. Differential ⟨∂x_{+}, x_{−}⟩ counts u: R × [0, 1] → M such that u(s, 0) ∈ L_{1}, u(s, 1) ∈ L_{2}, lim_{s→±∞} u(s, t) = x_{±} and ∂_{s}u + J_{t}∂_{t}u = 0. Under favorable circumstances, ∂ is well defined, ∂^{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}, ω). We claim that π_{1}(L) = Z. The generator comes from {exp(iπt)R ⊂ C | t ∈ [0, 1]}. Relative grading: If x_{+} and x_{−} belong to L_{1} ∩ L_{2} and there is u satisfying conditions above, define μ(x_{+}) − μ(x_{−}) ∈ Z. Given u, trivialize u^{∗}T M such that T L_{1} = R^{n} ⊕ {0} ⊂ R^{2n} over R × {0} and T L_{2} = {0} ⊕ R^{n} over {0} × [0, 1]. Along R × {1}, T L_{2} defines a loop of Lagrangians starting and ending at {0} ⊕ R^{n}. Then μ(x_{+}) − μ(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, ω) 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: R × [0, 1] → M such that u(s, 0) ∈ L_{0}, u(s, 1) ∈ L_{1}, lim_{s→±∞} u(s, t) = x_{±}, ∂_{s}u + J_{t}∂_{t}u = 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?

Example. Suppose f: (M, ω) → (M, ω) is a symplectomorphism. Its graph Γ is Lagrangian. The diagonal Δ is also Lagrangian. The intersection Γ ∩ Δ is the set of fixed points of f.

Theorem. HF_{∗}(Γ, Δ) = HF_{∗}(f). To define HF_{∗}(f), choose J_{t}, t ∈ R, J_{t+1} = f_{∗}J_{t}f_{∗}^{−1}.

Floer homology for symplectomorphisms of surfaces. Suppose we have a surface of genus greater than 1. Let f: Σ

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 γ ⊂ Σ such that f(γ) = γ; Pseudo-Anosov: two transverse singular foliations f_{1} and f_{2} on Σ.

Mapping torus: Y_{f} = [0, 1] × Σ/(1, x)˜(0, f(x)). ω on Σ induces a closed form ω on Y_{f} → [ω] + H^{2}(Υ, R).

Definition. f is monotone if [ω] = λc_{1}(E) in H^{2}(Y, R) for some λ ∈ R. If f is monotone, define HF_{∗}(f) as follows. CF_{∗}(f) = Z/(2) Fix(f). For J_{t} on Σ we have J_{t+1} = F_{∗}J_{t}F_{∗}^{−1}. We obtain a Z/(2)-grading by sign of fixed points. Now ⟨∂x_{+}, x_{−}⟩ = #m_{1}(x_{+}, x_{−})/R. For compactness part of proof that ∂ is well-defined, ∂^{2} = 0, need that u_{n} is a sequence in m_{1}(x_{+}, x_{−}) then ∫_{u_{n}} ω < c. If u and u′ are in m_{1}(x_{+}, x_{−}), then ⟨c_{1}(E), u − u′⟩ = 0, therefore ⟨[ω], u − u′⟩ = 0.

Example. If f is isotopic to identity then f is monotonic iff f is Hamiltonian isotopic to identity. More generally, suppose {f_{t} | t ∈ [0, 1]} is a symplectic isotopy. This induces a diffeomorphism ϕ: Y_{f_{0}} → Y_{f_{1}}, [ω_{0}] ≠ ϕ^{∗}[ω_{1}] and c_{1}(E_{0}) = ϕ^{∗}c_{1}(E_{1}).

[ω_{0}] relates to ϕ^{∗}[ω_{1}] as follows: 0 → Z → H_{2}(Y_{f}) → ker(1 − f_{∗}) → 0.

Conclusion. {f_{t}} preserves monotonicity iff flow {f_{t}} iff (Σ, R) annihilates ker. Also, any f is isotopic to a monotonic symplectomorphism.

Suppose Σ is a closed surface, ω is an area form on Σ, ϕ: (Σ, ω) → (Σ, ω) is a symplectomorphism. The mapping torus of ϕ is Y_{ϕ} = [0, 1] × Σ/(1, x) ∼ (0, ϕ(x)). Suppose that ω extends to a closed 2-form on Y and R^{2} → E → Y is the vertical tangent bundle of Σ → Y → S^{1}. ϕ is monotone if [ω] = λc_{1}(E) in H^{2}(Y, R).

Fact. (Seidel, Pacific Journal of Mathematics.) This map from monotone symplectomorphisms to orientation-preserving diffeomorphisms of Σ is a homotopy equivalence.

Assume ϕ is monotone and has nondegenerate fixed points. Define HF_{∗}(ϕ) as follows: C_{∗} is the Z/(2)- module generated by fixed points graded by Lefschetz sign. Choose J_{t} on Σ as usual. Now ∂p = ∑_{q} q · #m_{1}(p, q)/R. Monotonicity implies that ∂ is well-defined.

Lemma. If J is generic then for any p, q ∈ Fix(ϕ) the set m_{1}(p, q)/R is finite.

Proof. Key is that if c, c′ ∈ m_{1}(p, q) then ∫_{c} ω = ∫_{c′} ω. Because if ind(c) = ind(c′) then λ⟨c_{1}(E), [c − c′]⟩ = 0 = ⟨[ω], [c − c′]⟩ = ∫_{c} ω = ∫_{c′} ω.

Gromov compactness: if {c_{n}} is a sequence in m(p, q) with ∫_{c_{n}} ω < R then there is a subsequence converging to a “broken trajectory” from p to q. No bubbling because π_{2}(Σ) = 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 (ĉ_{0}, …, ĉ_{k}). If k = 0 then ĉ_{0} ∈ m_{1}(p, q)/R is not isolated, contradicting transversality. If k > 0 then ∑_{0≤i≤k} ind(c_{i}) = 1. So some ĉ_{i} has ind(ĉ_{i}) ≤ 0, again contradicting transversality. Similarly, HF_{∗}(ϕ) depends only on mapping class of ϕ.

Examples. (1) ϕ = id, Y = S^{1} × Σ. HF_{∗}(id) = H_{∗}(Σ). (2) ϕ is finite order (ϕ^{n} = 1). All fixed points have the same Z/(2)-grading, hence HF_{∗}(ϕ) = ⊕_{p∈Fix(ϕ)}Z/(2). (3) ϕ is a Dehn twist. Let γ ⊂ Σ be an embedded circle, N be a neighborhood of γ, N ≅ [0, 1] × S^{1} ⊃ N′ ≅ [ϵ, 1 − ϵ] × S^{1}. ϕ = id on Σ \ N. On N, ψ(x, y) = (x, y − x). We have HF_{∗}(ϕ) = H_{∗}(Σ \ γ). Proof: Do Hamiltonian isotopy so that ϕ has the following form. On N′, ϕ(x, y) = (x, y − x). On Σ \ N′, ϕ is the time 1 flow of X_{H}, where H: Σ \ N′ → R is a Morse-Smale. Fix(ϕ) = Crit(H). Choose J as usual. Lemma: If c ∈ m(p, q), then c does not intersect N′. This lemma implies that HF_{∗}(ϕ) = H_{∗}^{M}(H) = H_{∗}(Σ \ N′) = H_{∗}(Σ \ γ). Proof: Let c ∈ m_{1}(p, q). Let x ∈ [ϵ, 1 − ϵ]. Let T be the mapping tows of ϕ restricted to {x} × S^{1}. Want to show that C ∩ T = ϕ, R × T = R_{s} × S_{t}^{1} × S_{y}^{1}. J(∂/∂s) = ∂/∂t − x∂/∂y. Let F be the foliation of T generated by ∂/∂t − x∂/∂y. Then R × F is a holomorphic foliation of R × T. Wlog x is rational and c is transverse to T. [c ∩ (R × T)] = (a, b) ∈ H_{1}(R × T). Since c has positive intersection with the holomorphic cylinders in R × F, we have ax − b ≥ 0, equality only if c ∩ (R × T) = ∅. In particular, if c ∩ (R × T) ≠ ∅, then [c ∩ (R × T)] ≠ 0. To prove lemma, show [c ∩ (R × T)] = 0. Write [c] = z_{0} + z, where z_{0} is the real homology class of Morse cylinder from p to q and z ∈ H_{2}(Y). Need to show z ∩ (R × T) = 0. ind(c) = ind(H, p) − ind(H, q) + 2⟨c_{1}(E), z⟩. H_{2}(Y) = (S^{1} ⊗ {α ∈ Σ | α · γ = 0}) ⊕ H_{2}(Σ). Write z = (z_{1}, z_{2}). 1 = ind(c) = ind(H, p) − ind(H, q) + 2(2 − 2g)z_{2}, therefore z_{2} = 0.

Contact geometry. Let Y be a closed oriented 3-manifold. A contact form on Y is a 1-form λ such that λ ∧ dλ > 0. Let ξ = ker(λ). This is an oriented 2-plane field on Y. ξ is called a contact structure. Note: ξ is totally nonintegrable. (The kernel of λ is a foliation iff λ ∧ dλ = 0.) λ and λ′ determine the same ξ iff λ′ = fλ for some f: Y → R_{>0}. (If dim(Y) = 2n − 1, we require λ ∧ (dλ)^{n−1} > 0.) Example: Standard contact form on R^{3}: λ = dz − ydx. Darboux-type theorem: Any contact structure is locally isomorphic to this one. Gray stability theorem: If ξ_{t} is a family of contact structures for t ∈ [0, 1], then there is a family of diffeos ϕ_{t}: Y → Y such that ϕ_{0} = id and ϕ_{t∗}ξ_{0} = ξ_{t}.

Reference: John Etnyre, Introductory Lectures in Contact Geometry.

An overtwisted contact form on R^{3}: λ = cos(r)dz+sin(r)dθ. 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 λ, define the symplectization (R_{s} × Y, d(exp(s)λ)). Check symplectic: d(exp(s)λ) = exp(s)(ds ∧ λ + dλ).

Definition. Let (Y_{+}, ξ_{+}) and (Y_{−}, ξ_{−}) be contact 3-manifolds. A symplectic cobordism from (Y_{+}, ξ_{+}) to (Y_{−}, ξ_{−}) is a compact symplectic 4-manifold (X, ω) such that ∂X = Y_{+} ⊔ −Y_{−} and there are contact forms λ_{±} with ξ_{±} = ker(λ_{±}) such that ω|_{Y_{+}} = dλ_{+} and ω|_{Y_{−}} = dλ_{−}.

Example. (R^{4}, ω) ⊃ (U, ω|_{U}). Under appropriate convexity conditions, ∂U has a contact form λ with ω|_{∂U} = dλ.

Definition. (Y, ξ) is symplectically fillable if there exists a symplectic cobordism from (Y, ξ) to ϕ.

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}, ξ) be a contact manifold. Let ξ = ker(λ) and λ ∧ (dλ)^{n−1} > 0. A Legendrian submanifold of (Y, ξ) is a submanifold L^{n−1} ⊂ Y such that T L ⊂ ξ|_{L}. (R × L is Lagrangian in R × Y.)

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, ξ) contains the conormal bundle of N ({(x, y)} | x ∈ N ∧ y ∈ ST_{x}^{∗}M ∧ y|_{T_{x}N} = 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} × S^{2}, then contact homology gives us an invariant that distinguished the unknot.

Definition. Let (Y^{3}, ξ) be a contact 3-manifold. ξ is overtwisted if there is an embedded disk D ∈ Y such that ξ|_{∂D} = T D|_{∂D}.

Example: Standard contact structure on R^{3} is tight.

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.

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

Reeb vector field. This is a vector field R such that dλ(R, ·) = 0 and λ(R) = 1. It depends on λ, not just ξ. A Reeb orbit is a map γ: R/T → Y such that γ′(t) = R(γ(t)). We mod out by reparametrization. The k-fold iterate of Y is the pullback to R/KT, where K is a positive integer. γ is embedded iff γ is not the K-fold iterate of some γ′ where k > 1.

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

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.

Definition. An almost complex structure J on R_{s}×Y is admissible if J acts compatibly with dλ, J(∂_{s}) = R, and J is R-invariant.

Look at holomorphic curves in R × Y. Let γ: R/T → Y be a Reeb orbit. The Reeb flow preserves λ. L_{R}λ = di_{R}λ + i_{R}dλ = d(1) + 0 = 0. Linearization of the Reeb flow on the contact planes along γ. P_{γ}: ξ_{γ(0)} → ξ_{γ(0)} is symplectic with respect to dλ. γ is nondegenerate if 1 ∉ P_{γ}. Assume all Reeb orbits are nondegenerate.

Let α_{1}, …, α_{k}, β_{1}, …, β_{l} be Reeb orbits, g ≥ 0. Define m_{g}(α_{1}, …, α_{k}, β_{1}, …, b_{l}) as the set of all J-holomorphic curves u: Σ → R × Y where u has positive ends at α_{1}, …, α_{k}, negative ends at β_{1}, …, β_{l} and no other ends. Here Σ is a genus g surface with k + l punctures.

If ∑_{i}[α_{i}] = ∑_{j}[β_{j}] in H_{1}(Y), define H_{2}(Y, α_{1}, …, α_{k}, β_{1}, …, β_{l}) to be the set of relative homology classes of 2-chains z with ∂z = ∑_{i} α_{i} − ∑_{j} β_{j}. This is an affine space over H_{2}(Y). We have m_{g}(α_{1}, …, α_{k}, β_{1}, …, β_{l}, z) = {u: Σ → R × Y ∈ m_{g} | u_{∗}[Σ] = z}. Choose a trivialization τ of ξ over the α_{i} and β_{j}. Expected dimension of the moduli space m_{g}(α_{1}, …, α_{k}, β_{1}, …, β_{l}, z) = (n − 3)χ(Σ) + 2c_{1}(u^{∗}ξ, τ) + ∑_{i} CZ_{τ} (α_{i}) − ∑_{j} CZ_{τ} (β_{j}). This is the actual dimension if J is generic and u: Σ → R × 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_{γ}: ξ_{γ(0)} → ξ_{γ(0)}. Elliptic if eigenvalues are exp(±2πiθ), positive hyperbolic if eigenvalues are λ > 0 and λ^{−1}, negative hyperbolic if eigenvalues are λ < 0 and λ^{−1}. Elliptic case: linearized flow rotates by angle 2πθ for some θ ∈ R \ Z. We have CZ_{τ} (γ) = 2⌊θ⌋ + 1. Hyperbolic case: linearized flow rotates eigenspaces by angle πn for some n ∈ Z, CZ_{τ} (γ) = n.

Note: if m_{g}(α_{1}, …, α_{k}, β_{1}, …, β_{l}) is nonempty then ∑_{i} ∫_{α_{i}} λ ≥ ∑_{j} ∫_{β_{j}} λ, equality only if {α_{1}, …, α_{k}} = {β_{1}, …, β_{l}}. Here ∫ λ is “symplectic action”.

Proof: if u: Σ → R × Y is in m_{g}, then u^{∗}dλ ≥ 0 on all of Σ, with equality only where π ∘ du = 0. u^{∗}dλ(v_{1}, v_{2}) = dλ(πdu(v_{1}), πdu(v_{2})). u^{∗}dλ(v, jv) = dλ(πdu(v), πdu(jv)) = dλ(πdu(v), Jπdu(v)) ≥ 0, equality iff πdu(v) = 0. Apply Stokes theorem on Σ. In particular every holomorphic curve in R × Y has at least one positive end. But it is possible to have no negative ends.

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: Σ → R × Y in m_{g} we have ∫_{Σ} u^{∗}dλ = ∑_{i} ∫_{α_{i}} λ − ∑_{j} ∫_{β_{j}} λ.

Cylindrical contact homology. Chain complex generated by “good” Reeb orbits over Q. Differential counts holomorphic cylinders in R × Y. Trouble with coverings. Either assume λ 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.

Cylindrical contact homology (in 3 dimensions) Let Y be a closed oriented 3-manifold, λ be a contact form, λ ∧ dλ > 0, R be the corresponding Reeb vector field.

Example. Y = S^{3}, λ = (x_{1}dy_{1} − y_{1}dx_{1} + x_{2}dy_{2} − y_{2}dx_{2})/2.

Exercise. R is tangent to the Hopf circles.

Example. Y = T^{3} = (R/2πZ)^{3}. λ_{n} = cos(nz)dx + sin(nz)dy, R_{n} = cos(nz)∂/∂x + sin(nz)∂/∂y.

To define CCH, assume all Reeb orbits are nondegenerate. (Above examples are “Morse-Bott”.) Choose J on R_{s} × Y.

m_{g}(α_{1}, …, β_{1}, …, z) is the set of all J-holomorphic curves u: Σ → R × Y such that domain Σ is a genus g surface with k + l punctures, u has positive ends at α_{i}, negative ends at β_{j}. If u has a positive or negative end at the k-fold iterate of an embedded Reeb orbit γ, 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 γ where k is even and γ is negative hyperbolic. Otherwise it is “good”. Let C_{∗} be the free Q-module generated by good Reeb orbits. If α is a good Reeb orbit, then ∂α = ∑_{β} k^{−1}#m_{0}(α, β, z)/R · β, where α is the k-fold iterate of an embedded orbit.

“Theorem”. Suppose there is no contractible Reeb orbit γ bounding a disk D such that −1+2c_{1}(ξ|_{D}, τ)+ CZ_{τ} (γ) = 1. Then ∂^{2} = 0.

“Proof”. Let α, γ be generators, z ∈ H_{2}(Y, α, γ), CZ_{τ} (α) − CZ_{τ} (γ) + 2c_{1}(ξ|_{z}, τ) = 2. Look at m_{0}(α, γ, z). Compactify and look at the boundary. We have #∂(m_{0}(α, β, z)/R) = a⟨∂^{2}α, γ⟩. For Γ ∈ H_{1}(Y), let CH_{∗}^{Γ} be the part corresponding to Reeb orbits in homology class Γ.

“Theorem”. Let λ_{1}, λ_{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_{∗}^{Γ}(λ_{1}, J_{1}) = CH_{∗}^{Γ}(λ_{2}, J_{2}).

Definition. A contact form λ on Y is called nice, if all Reeb orbits are nondegenerate and there are no contractible Reeb orbits γ bounding a disk D with −1 + 2c_{1}(ξ|_{D}, τ) + CZ_{τ} (γ) ∈ {1, 0, −1}.

“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.

Corollary. Let Y be a closed oriented manifold with a contact structure ξ. If there is a nice contact form for ξ, then CH_{∗}^{Γ}(Y, ξ) is well defined.

Corollary. If ξ has a nice contact form and CH_{∗}^{Γ}(Y, ξ) ≠ 0, then Weinstein conjecture holds for any contact form for ξ.

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) ∈ Q ∪ {∞} we have generators e_{z}^{k}, h_{z}^{k} for k ≥ 1 integer. Claim: ∂ = 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) ∈ H_{1}(T^{3}). If c ≠ 0, then CH_{∗}^{Γ} = 0. If c = 0 and (a, b) ≠ (0, 0), then CH_{∗}^{Γ} = ⊕_{n}H_{∗}(S^{1}, Q). Corollary: Weinstein conjecture holds for these contact structures. The contact structures determined by λ_{n} for different n are not isomorphic.

Let (Y, ξ) be a contact manifold (dim Y = 3, can be generalized to other dimensions). Choose (1) contact form λ with ξ = ker λ and nondegenerate Reeb orbits; (2) almost complex structure on R_{s} × Y such that J(ξ) = ξ, J is compatible with dλ, J(∂_{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 α and β are good Reeb orbits then αβ = (−1)^{|α|·|β|}βα, where |α| = CZ_{τ} (α) − 1 (mod 2). If α is a good Reeb orbit, then

∂α = ∑_{k≥0} ∑_{β} (combinatorial factor) ∑_{z∈H_{2}(Y,α,β_{i})} k − 1 + 2c_{1}(ξ|_{z}, τ) + CZ_{τ}(α) − ∑_{i} CZ_{τ}(β_{i}) = 1,

where β_{i} are good Reeb orbits. Extend ∂ to A by the Leibniz rule. Theorem: ∂^{2} = 0 implies that HC_{∗}(Y, ξ) depends only on Y and ξ and not on the other choice. Also a symplectic cobordism X from Y_{+} to Y_{−} induces a DGA morphism A_{+} → A_{−}.

Example. (S^{3}, ξ), ξ 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 ≥ 1. Also ∂ = 0 and |a_{k}| = |b_{k}| = 0 (mod 2). Since H_{1}(S^{3}) = H_{2}(S^{3}) = 0, A has a Z-grading. Conclusion: HC_{∗}(S^{3}, ξ) = Q[z_{2}, z_{4}, …]. deg(z_{2k} = 2k.

“Theorem.” If ξ is an overtwisted contact structure on Y, then HC_{∗}(Y, ξ) = 0. “Proof.” (Mei-Lin Yau, Eliashberg.) Can find λ and J such that there is a Reeb orbit γ such that γ bounds a unique index 1 holomorphic disc in R × Y and γ is shorter than all other Reeb orbits. Now ∂γ = 1. If ∂α = 0 then ∂(γα) = (∂γ)α ± γ(∂α) = 1α ± γ · 0 = α.

Corollary. If Y is symplectically fillable, then ξ is tight.

Proof. Suppose X^{4} is a symplectic cobordism from (Y^{3}, ξ) to ϕ. Make choices to define A for Y. Then X induces a DGA morphism Φ: A → Q. Suppose ξ is overtwisted. Then there is an α such that ∂α = 1 and 0 = Φ(∂α) = Φ(1) = 1. Symplectically fillable implies 1 ≠ 0 in HC_{∗}. Overtwisted implies 1 = 0 in HC_{∗}. Possible: tight and 1 = 0. Examples: tight but not fillable.

Morse-Bott theory. Model case: X is a closed smooth manifold. A smooth function f: X → R is Morse-Bott if Crit(f) is a union of closed submanifolds of X, and for each p ∈ Crit(f, p) the map H: T_{p}X⊗T_{p}X → R is nondegenerate on the orthogonal complement of T_{p}S, where S is the critical submanifold containing p.

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

Example. If F → E → B → R and E → R with E → R Morse-Bott, then Crit(π^{∗}f) = π^{−1} Crit(f), where π = E → 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 + ∑_{s} ϵ_{s}π^{∗}f_{s} where f_{s}: s → R is Morse, π: N → S is a tubular neighborhood, ϵ_{s} is a small function which is positive near s and 0 elsewhere. Crit(f̂) = ∪_{s} Crit_{j}(f_{s}). Near s: df̂ = df + ϵ_{s}π^{∗}df_{s}. Pick a generic metric g on X. What are the gradient flow lines of f̂, in terms of f? Answer: “Cascades”. Bourgeois (contact homology). Frauenfelder (Morse theory). Claim: flow lines of f̂ correspond bijectively to cascades.

Exercise. Expected dimensions agree.

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

Example: torus lying on its side.

Floer homology. Let (X^{2n}, ω) be a closed symplectic manifold (assume monotone) and ϕ: (X, ω) → (X, ω) be a Hamiltonian symplectomorphism. Assume fixed point come in nondegenerate manifolds, i.e., Fix(ϕ) is a union of closed submanifolds of X and for any p ∈ Fix(ϕ) we have ker(1 − dϕ_{p}: T_{p}X → T_{p}X) = T_{p}S where S is the corresponding manifold of fixed points. Choose Morse functions f_{s}: s → R. Define CF_{∗}^{MB}(ϕ) = ⊕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.

Example. Cylindrical contact homology of (T^{3}, λ_{n} = cos(nz)dx + sin(nz)dy). Claim:

CH_{∗}(T^{3}, λ_{n}, (a, b, 0)) = ⨁_{n} H_{∗}(S^{1}).

Proof: Just need to show there are now cascades with k ≥ 1, i.e., no holomorphic cylinders between Reeb orbits in different critical submanifolds. But all orbits γ with [γ] = (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.)