1: contact manifolds and Reeb flow

definition 1.0.1

Let \(M\) be a closed \(2n+1\)-real dimensional manifold. A contact form on \(M\) is one form \(\alpha\in \Omega^1(M)\) so that \[\alpha \wedge (d\alpha)^n \] is a volume form on \(M\). We call the pair \((M, \alpha)\) a contact manifold.

example 1.0.2

A key set of examples of contact manifolds come as hypersurfaces of symplectic manifolds. Let \((X, \omega)\) be a symplectic manifold. Suppose that there is an expanding vector field \(Z\) on \(X\), that is, a vector field so that \[\mathcal L_Z \omega = \omega.\] The symplectic manifold \(X\) is exact, with primitive given by \(\lambda=\iota_Z \omega\). Let \(i:M\into X\) be a hypersurface which is transverse to \(Z\). Then the restriction \(\alpha:=\lambda|_M\) is an example of a contact form. We see that the form \begin{align*} \alpha \wedge d\alpha^{n-1} =& i^* (\iota_Z \omega \wedge \omega^{n-1})\\ \end{align*} is nonvanishing, as \(\omega^n\) is a volume form and \(Z\) is transverse to \(M\). The simplest example to consider come from \(\CC^n= \RR^{2n}\), where the radial vector field \(Z=\frac{1}{2}\sum_i \left(x_i \partial_{x_i} + y_i\partial_{y_i}\right)\) provides an example of an expanding vector field. The associated primitive for the symplectic form is \[\iota_Z \omega =\sum_{i=1}^n x_i dy_i-y_idx_i \] This radial vector field plays especially nicely with respect to the moment map, \begin{align*} p: \RR^{2n}\to& (\RR_{\geq 0})^n\\ (x_i, y_i)\mapsto&\frac{1}{2} (x_i^2+y_i^2) \end{align*} We give the base of the moment polytope \((\RR_{\geq 0})^n\) coordinates \((p_1, \ldots, p_n\)). At every point \((x_i, y_i)\in \RR^{2n}\), we can project the Liouville vector field to \[p_*Z_{(x_i, y_i)}= \sum_{i=1}^n (x_i^2+y_i^2) \partial_{p_i} =\frac{1}{2}\sum_{i=1}^n p_i \partial_{p_i}.\] In particular, if we have a hypersurface \(N\subset (\RR_{\geq 0})^n\) which is transverse to the radial vector field \(\sum_{i=1}^n p_i \partial_{p_i}\) and whose preimage \(M:=p^{-1}(N)\subset \RR^{2n}\) is a smooth hypersurface, then \(M\) is a contact manifold.

figure 1.0.3:The Liouville structure on \(\CC^2\) as viewed from the moment map

definition 1.0.4

Let \((M, \alpha)\) be a contact manifold. The Reeb vector field is the unique vector field \(R_\alpha\) characterized by \begin{align*} R_\alpha\in \ker(d\alpha) && \alpha(R_\alpha)=1.\end{align*}

example 1.0.5

We return to example 1.0.2 of hypersurfaces in \(\RR^{2n}\) given by \(M=p^{-1}(N)\). Consider the hypersurface \(N\) defined by the equation \(p_1+\cdots p_n=1\). Then \(M=S^{2n-1}\subset \RR^n\). We give \(\CC^n\) the polar coordinates \((r_i, \theta_i)\). Let \(f=\sum_{i=1}^n |z_i|^2.\) The tangent space to \(M\) is the orthogonal complement to \(\grad(f).\) Consider the vector field \[V:=\sum_{i=1}^n x_i \partial_{y_i}- y_i \partial_{x_i}.\] First, observe that \[V\cdot \grad(f)=\sum_{i=1}^n( x_i y_i - y_i x_i )= 0 \] so \(V\) restricts to a vector field on \(M\). Let \(v=\sum_{i}a_i \partial_{x_i}+b_i\partial_{y_i}\) be any vector in \(TM\). Then the pairing \begin{align*} \omega(v, V)=&\sum_{i=1}^n a_ix_i+b_iy_i\\ =&v\cdot \grad(f)=0 \end{align*} From this, we conclude that \(V\in \ker(d\alpha)\). Finally, we have that \(\alpha=\iota_Z\omega\), so \[\omega(Z, V)=\sum_{i=1}^n (x^2+y^2)=1\] From which we conclude that \(V\) is the Reeb vector field for \((M, \alpha)\). A Reeb orbit is a map \(\gamma: \RR/\ell\ZZ\to M\) with \(\frac{d}{dt}\gamma = R\). Unlike the setting of Hamiltonian Floer theory, we do not restrict ourselves to time-one orbits. We have a contact-version of the Arnold conjecture. Every contact manifold \((M, \alpha)\) has at least one Reeb orbit. Since we do not restrict ourselves to time one-orbits, if there is an orbit, there are an infinite number of orbits (by considering their multiple covers). Even when one considers orbits up to multiplicity, an infinite number of orbits can occur.

example 1.0.6

We return to example 1.0.5 of the Reeb vector field on \((S^{n-1},\alpha)\), where \(S^{n-1}\) is considered as a hypersurface of \(\CC^n\). Recall we have a map \(p:S^{2n-1}\to N\), where \(N\subset \RR_{\geq 0}^n\) is the simplex defined by \(\sum_{i=1}^n p_i^2=1\). The fibers of \(p\) are \(n\)-dimensional tori in \(S^{2n-1}\) which are parallel to the Reeb vector field \(V_\alpha\). In fact, Reeb vector field \(V_\alpha\) acts on the fibers \(p^{-1}(p_1, \ldots, p_n)\) by translation in the \((\sqrt p_1, \ldots, \sqrt p_n)\) direction. We therefore identify two types of fibers of \(p\):

If \((\sqrt p_1, \ldots, \sqrt p_n)\) has integral slope (that is, there exists a scalar \(r\) so that \(r\cdot(\sqrt p_1, \ldots, \sqrt p_n) \in \ZZ^n\)) then every point on the fiber belongs to a closed orbit.
Otherwise, no point on the fiber belongs to a closed orbit.

The Reeb orbits of \((S^{n-1}, \alpha)\) are in bijection with \(\bigoplus_{\vec v\in \NN^n\setminus \{0\}} T^n/\vec v\). In example 1.0.6 we have an uncountable number of orbits, but the set of periods of these orbits is only countable. Given a Reeb orbit \(\gamma\), let \(\ell(\gamma)\) denote its period.

proposition 1.0.7

Let \(M\) be a contact manifold. Let \[\Gamma:=\{\gamma: \RR/\ell\RR\to M \st\gamma'=R\}\] denote the set of Reeb orbits. The image of \(\ell(\Gamma)\subset \RR_{\geq 0}\) is countable and closed. Give a Reeb orbit, the reparameterization of the orbit satisfying \(\dot \gamma = \ell V_\alpha\) is a critical point of the action \begin{align*} A:W^{1,2}(S^1, M)\to& \RR\\ \gamma\mapsto& \int_{\gamma}\alpha \end{align*} where \(W^{1,2}(S^1, M)\) is the space of maps of Sobelov class \(1,2\). On a Reeb orbit, observe that \[\int_{\gamma}\alpha= \int_{S^1} \alpha(\cdot \gamma) = \ell\] the action computes the period of \(\gamma\). Therefore the set of periods is a subset of the critical values of \(A\). By applying the Sard-Smale theorem, the critical values of a functional are countable and closed in \(\RR\). While the Weinstein conjecture is stated in terms of contact geometry, there is a standard method to translate questions in contact geometry to questions in symplectic geometry.

definition 1.0.8

Let \((M, \alpha)\) be a contact manifold. The symplectization of \((M, \alpha)\) is the symplectic manifold \[(\RR\times M, d(\exp(r)\cdot \alpha)).\] To see that this is a symplectic manifold, observe that \begin{align*} (d(\exp(r)\cdot \alpha)^n=( \exp(r) (dr\wedge\alpha + d\alpha))^n = \exp(nr) ( dr\wedge \alpha\wedge d\alpha^{n-1}) \end{align*} which is non-vanishing everywhere as \(\alpha\) is a contact form.

definition 1.0.9

Let \(M\) be a contact manifold, and let \(\RR\times M\) be its symplectization. A linear Hamiltonian of slope \(m\) is the Hamiltonian \(H^m(r, x)= m\cdot \exp(r)\). Observe that the level sets of a linear Hamiltonian are \(\{r\}\times M\). The Hamiltonian vector field therefore can be restricted to a vector field on the contact manifold \(M\). It is not a surprise that the \(m V_\alpha\) and Hamiltonian vector field \(H_V\) agree: \begin{align*} \iota_{i_* m V_\alpha}\omega=& d(\exp(r) \alpha )(m V_\alpha)\\ =&(\exp(r) dr \wedge d\alpha - \exp(r) d\alpha) (m V_\alpha)\\ =&- m \cdot \exp(r)dr = dH^m \end{align*} Therefore, the time one Hamiltonian orbits of \(H^m\) correspond to the time \(m\) Reeb orbits of \(V_\alpha\); by studying Hamiltonian Floer theory on the symplectization \(\RR\times M\) we obtain some understanding of the Reeb dynamics of \((M, \alpha)\). A useful observation is the following: suppose that \(m\) is not a period of some Reeb orbit. Then \(\RR\times M\) has no time one orbits for the linear Hamiltonian of slope \(m\). If we take \(h: \RR_{\geq 0}\to \RR\) a function with slope increasing to infinity the Hamiltonian \(H:=h(\exp(r)): X\times \RR\to \RR\) our Hamiltonian vector field is related to the linear Hamiltonian vector fields by \[V_H = h'(\exp(r)) V_{H^1}\] Therefore, the time-one Hamiltonian orbits contained in the level set \(\exp(r)\times M\) will compute the \(h'(\exp(r))\)-period Reeb orbits of \(V_\alpha\).

2: Liouville domains

definition 2.0.1

A Liouville domain is a pair \((X,\lambda)\), where

\(X\) is a \(2n\)-manifold with boundary \(\partial X\) and,
\(\lambda\in \Omega^1(X, \RR)\) is a one form on \(\Omega^1(X, \RR)\) so that \(\omega=d\lambda\) is a symplectic form for \(X\).

To this data we can associate a Liouville vector field \(Z\) defined by the property \(\iota_Z\omega= \lambda\). We require that this vector field transversely points outward along \(\partial X\). In some ways, Liouville domains are the ``simplest'' symplectic manifolds to study. While the presence of boundary on \(X\) may seem problematic, this is more than compensated by the fact that \(X\) is exact. This is an honest trade-off: by Stoke's theorem, there are no closed exact symplectic manifolds. By example 1.0.2, \(\alpha:= \lambda|_{\partial X}\) is a contact form on the boundary of a Liouville domain \(X\). A Liouville domain can be extended to a Liouville manifold by attaching the symplectization of \(\partial X\) to the boundary of \(X\). The resulting manifold \[\hat X := X\cup_{\partial X} ([0, \infty)\times \partial X)\] is called the symplectic completion of \(X\).

example 2.0.2

Let \(Q\) be a manifold. The cotangent bundle \(X=T^*Q\) is an exact symplectic manifold; call the primitive \(\lambda_{can}=\sum_{i=1}^np_idq_i\). In local coordinates, the Liouville vector field is \(Z= \sum_{i=1}^n p_i \partial p_i\). The flow of this vector field acts by scalar multiplication on the fibers of \(T^*Q\), \begin{align*}\phi^t: T^*Q\to T^*Q && (q, p) \mapsto (q, tp)\end{align*} which inflates the symplectic form. Now equip \(Q\) with a metric \(g\). The metric determines a unit sphere bundle \(S^*Q\subset T^*Q\), which is transverse to \(V\). Letting \(\alpha=\lambda|_{S^*Q}\) makes \((S^*Q, \alpha)\) a contact manifold. The time \(t\)-flow on \((S^*Q, \alpha)\) corresponds to the time \(t\) geodesic flow on the base (exercise 6.0.2). It is known ([Fet52]) that every closed manifold has at least 1 closed geodesic, so all of these examples of contact manifolds have a Reeb orbit.

example 2.0.3

A Stein manifold is a complex manifold \(X\) which admits a proper embedding into \(\CC^N\) with \(N\) sufficiently large; therefore \(X\) is a symplectic manifold with symplectic form \(\omega_{\CC^N}|_X\). On \(\CC^n\) we have the function \(\phi=|z|^2:\CC^n\to \RR\) which is exhausting and strictly plurisubharmonic. The symplectic form on \(\CC^n\) can be written as \(dd^c\phi\). By restricting \(\phi\) to \(X\), we obtain a primitive \(d^c\phi|_X\) for \(\omega|_X\), which makes \(X\) a Liouville manifold. In particular, every affine variety is an example of a Liouville manifold.

3: constructing \(\SH(G)\)

We now give two different constructions of the symplectic cohomology. In both constructions, it is necessary to deal with the non-compactness of the space \(\check X\). The first defines the symplectic cohomology as the Floer cohomology of a Hamiltonian whose slope increases along the symplectization coordinate to infinity. The second defines the symplectic cohomology as the limit of Floer cohomology of linear Hamiltonians, where the limit is taken over increasing slopes.

3.1: maximum principles

Given \(H_t: X\times \RR\to \RR\) a time dependent Hamiltonian, we obtain an action on the free loop space \begin{equation} A_{H_t}(\gamma)=-\int_\gamma \lambda + \int_\gamma H_t dt \end{equation}whose critical points are the time one periodic orbits of \(V_{H_t}\). Given a \(\omega\)-compatible almost complex structure, we observe that cylinders \(u: \RR_s\times S^1_t\to X\) which satisfy the \(H_t\) perturbed Floer equation \begin{equation} \partial_s u + J(\partial_t u - V_{H_t})=0 \end{equation}parameterize the negative gradient flow lines of \(A_{H_t}\). For curves \(\gamma_+, \gamma_-\), let \(\mathcal M(\gamma_+, \gamma_-)\) denote the moduli space of solutions to Floer's equation with ends limiting to \(\gamma_+, \gamma_-\). Supposing that \(X\) is an exact symplectic manifold, and that our time-dependent Hamiltonian is chosen in a generically, the Floer cochains \(\CF(X, H_t)\) are the graded vector space generated on the time one orbits of \(H_t\). We take a slightly different convention (following Equation 5.2 of [Abo10]) and give each orbit the grading \(\deg(\gamma)=n-CZ(\gamma)\), where \(CZ(\gamma)\) is the Conley-Zehnder index. The structure coefficients of the differential are given by counts of solutions to Floer's equation. The theory becomes powerful when it satisfies the following properties:

\(\CF(X, H_t)\) is a chain complex. The key step is to show that when \(\dim(\mathcal M(\gamma_+, \gamma_-))=1\), there exists a compactification of this moduli space by including broken cylinders. A compactification of the space comes from applying Gromov compactness, while additional requirements on \(X\) are sometimes required ensure that the only configurations which appear in the compactification are broken cylinders. In our setting, the only breaking configurations which may occur are broken cylinders, as \(X\) is exact (so \(\omega(\pi_2(X))=0)\).
Given \(H_{t, 0}\) and \(H_{t, 1}\) two time-dependent Hamiltonians, there exists a continuation map \(\CF(X, H_{t, 0})\to \CF(X, H_{t, 1})\). Furthermore, this map is a homotopy equivalence.
Finally, we need some way to compute \(\CF(X, H_{t, 1})\). One way to do this is to observe that for \(C^2\) small Hamiltonians the Floer cochains agree with the Morse cochains (and only consist of constant orbits). By either using the PSS-isomorphism or by analyzing Floer trajectories, the Floer cohomology can be compared to the Morse cohomology of \(X\).

The major difference in defining the Hamiltonian Floer cohomology for Liouville domains \(X\) (as opposed to compact symplectic manifolds) comes from the proof of Gromov-compactness. The first step in the proof of Gromov-compactness is to apply Arzel\'a-Ascoli to out sequence of pseudoholomorphic maps. Because \(\hat X\) is not compact, we cannot apply the Arzel\'a-Ascoli theorem to a sequence of pseudoholomorphic cylinders \(u: S^1\times \RR \to \hat X\). We now give an example of where we can solve the issue of non-compactness.

example 3.1.1

The maximum modulus principle states that if \(\phi: D^2\to \CC\) is a holomorphic function from the disk to \(\CC\), that the maximum of \(|\phi|: D^2\to \RR_{\geq 0}\) is achieved on \(\partial D^2\). Let \(\hat X\) be a non-compact symplectic manifold with compatible almost complex structure \(J\), along with a \(J-\jmath\)-holomorphic projection \(W: \hat X\to \CC\). Suppose that the fibers of \(W\) are compact. Pick two loops \(\gamma_-, \gamma_+\subset \hat X\) and \(r_0\in \RR\) large enough so that \(U:=W^{-1}(\{z \st |z|\leq r\})\) contains \(\gamma_-, \gamma_+\). We will prove that every pseudoholomorphic cylinder \(u: S^1\times \RR\to \hat X\) with ends limiting to \(\gamma_-, \gamma_+\) has image contained within the compact subset \(U\). The composition \(W\circ u: S^1\times \RR\to \CC\) is a holomorphic map, with ends limiting to \(W(\gamma_\pm)\), and therefore satisfies the maximum modulus principle. Since the boundary is sent to \(W(\gamma_\pm)\), we obtain that \(|W|\) achieves a value no greater than \(r_0\) on \(u\); therefore \(\Im(u)\subset U\). It follows that the image of \(u\) is contained within a compact set. In order to extend example 3.1.1 to the setting of \(\hat X\), we will use the maximum principle. First, we will need to assume that we have chosen our almost complex structure for \(\hat X\) so that the sub-bundle spanned by the vector fields \(\partial_r, R\) form an almost complex subspace.

definition 3.1.2

Let \(\hat X\) be the completion of a Liouville domain. A choice of almost complex structure for \(\hat X\) is of contact type if \[d(\exp(r))\circ J = -\alpha.\] We will also need to assume that we have chosen our Hamiltonian so that over the symplectization it only depends on the \(r\)-coordinate. For such a contact type almost complex structure and Hamiltonian there exists a version of example 3.1.1.

proposition 3.1.3

Let \(H: \hat X\to \RR\) be a Hamiltonian which on the symplectization takes the form of \(h(\exp(r))\). Let \(\gamma_+, \gamma_-\) be time 1 orbits of \(V_{H_{t}}\). For a contact type almost complex structure, every solution \(u: \RR\times S^1\to \hat X\) of the Floer equation with ends limiting to \(\gamma_+, \gamma_-\) has image contained in the subset \(\hat X|_{\exp(r)\leq C}\), where \(C\) is the maximum value of \(\exp(r)\) on the orbits \(\gamma_+, \gamma_-\). Let \(u: \RR\times S^1\to \RR\) be a solution to the Floer equation (). Let \(\rho=\exp(r\circ u)\). By applying \(d(\exp(r))\) to the Floer equation, and using definition 3.1.2 we obtain : \begin{align*} 0=d(\exp(r))\circ \left(\partial_s u + J(\partial_t u - V_{H})\right)=& \partial_s(\rho) - \alpha(\partial_t u)+ \alpha(V_H) \end{align*} Because \(H= h(\rho)\), the Hamiltonian vector field associated to \(H\) is \(h'(\rho) V_\alpha\), where \(V_\alpha\) is the Reeb flow. From , we see that \(\alpha(V_H)=h'(\rho).\) \begin{align*} =& \partial_s(\rho) - \alpha(\partial_t u)+ h'(\rho) . \end{align*} Similarly, applying \(\alpha\) to the Floer equation: \begin{align*} 0=\alpha \left(\partial_s u + J(\partial_t u - V_{H})\right) =& \alpha(\partial_su)+ \partial_t(\exp(\rho))+ V_{H}(\exp(\rho)) \end{align*}Because \(H= h(\rho)\) has the same level sets as \(\rho\), \(V_H(\rho)=0\).\begin{align*} =& \alpha(\partial_s u) + \partial_t(\rho) \end{align*} Differentiating the first line with respect to \(s\), the second line with respect to \(t\), and summing the lines together we obtain \begin{align*} 0=& (\partial_s^2 + \partial_t^2)\circ \rho- \partial_s\alpha(\partial_t u)+\partial_s \rho h'(\rho) +\partial_t\alpha(\partial_s u)\\ \end{align*}As \([\partial_s, \partial_t]=0\), we can substitue \(-\partial_t\alpha(\partial_s u)+ \partial_s\alpha (\partial_t u)= -u^*\omega(\partial_s, \partial_t)\)\begin{align*} =& \Delta \rho- u^*\omega(\partial_s, \partial_t)+ \rho h'(\rho)\partial_s\rho + \rho h''(\rho) \partial_s\rho\\ \end{align*}By again applying Floer's equation, and using the compatibility of almost complex structure with \(J\), we may substitue \(u^*\omega(\partial_s, \partial_t)= u^*\omega(\partial_s, J\partial_t-X_H)=|\partial_s u^2|-dh'(\rho)\partial_s\)\begin{align*} =& \Delta \rho-|\partial_s|^2+\rho h''(\rho)\partial_s\rho \end{align*} We therefore obtain that \(\Delta\rho+\rho\cdot h''(\rho) \partial_s\rho\geq 0\). Observe now that where \(z\in S^1\times \RR\) is a proposed maximum for \(\rho\) that \(\partial_s\rho=0\), allowing us to write \((\partial^2_s \rho + \partial^2_t \rho)|_z \geq 0\). This implies that at least one of \(\partial^2_s, \partial^2_s\) has to be non-negative --- in particular, the second derivative test does not detect the maximum. A more general argument --- the maximum principle --- states that \(\rho\) achieves no local maxima; therefore \(\sup_{S^1\times \RR} \rho \leq \max_{t\in S^1} \exp(r\circ \gamma_\pm)=:C.\) It follows that the image of \(u\) is contained in \(\hat X|_{\rho< C}\), which is a compact set.

3.2: \(\SH(X)\) via quadratic \(H\)

Let \((X, \lambda)\) be a Liouville domain, and let \(\hat X\) be its completion. The goal is to construct a Floer cohomology which witnesses all of the Reeb orbits of \(\partial X\). With that in mind, it is natural to study the Hamiltonian Floer cohomology of Hamiltonians \(H_t\) with the property that over the symplectization they are of the form \[H|_{\RR\times \partial X}=h(\exp(r))\] where \(\lim_{r\to\infty} h'(\exp(r))=\infty\). This Hamiltonian witnesses every Reeb orbit with sufficiently large period.

figure 3.2.1:An increasing Hamiltonian over the symplectization witnesses all Reeb orbits

A problem with this definition is that the Hamiltonian we have chosen is time independent, and so whenever \((\partial X, \alpha)\) has any Reeb orbits, the time-1 orbits of \(H\) will necessarily be degenerate (all orbits come in \(S^1\) families from reparameterization). The standard work-around is to introduce a small time-dependent perturbation to \(H\) in such a way that we can still apply our maximum principle argument. We therefore look at time-dependent Hamiltonians \(H_t\) which, outside of a compact set, are of the form \(h_t(\exp(r))\), with \(\lim_{r\to\infty} h_t(\exp(r))=\infty\) for all \(t\). The maximum principle argument from proposition 3.1.3 can be made to hold in this setting (Proposition 4.1 [Wen]). One then takes the definition of the symplectic cohomology to be \[\SH(X):=\CF(\hat X, H_t)=\bigoplus_{\gamma\st \dot \gamma= V_{H_t}} \ZZ\langle \gamma\rangle.\] with differential given by structure coefficients \(\langle d(\gamma_+), \gamma_\rangle\) counting \(V_{H_t}\)-perturbed pseudoholomorphic cylinders with ends limiting to \(\gamma_\pm\).

3.3: \(\SH(X)\) as a limit

As before, let \((X, \lambda)\) be a Liouville domain. For \(m\not\in \ell(\Gamma)\) not a period of a Reeb orbit, define \[\SH(X)^{< m}:=\HF(\hat X, H^m_t)\] where \(H^m_t\) is a Hamiltonian which on the symplectization agrees with \(H^m\), the linear Hamiltonian of slope \(m\). Over the symplectization \(\RR\times \partial X\) there are no Hamiltonian orbits, as \(H^m\) has no Hamiltonian orbits. The \(< m\) signifies that this version of symplectic cohomology is only supposed to detect those Reeb orbits of period less than \(m\). In order to recover the symplectic cohomology, we would like to understand the limit of the groups \(SH(X)^{< m}\) as we take \(m\to\infty\). Making sense of a limit algebraically requires constructing maps between these groups. When \(m^+< m^-\), the maximum principle arguments applied to families of Hamiltonians dependent on the \(s\)-parameter hold, allowing us to construct chain maps \[\CF(\hat X, H^{m^+}_t)\to \CF(\hat X, H^{m^-}_t)\] The \(\pm\) index on the slope are meant to represent whether they are the incoming or outgoing side of a Floer trajectory, not the relative sizes of the slopes. From the perspective of 3.2, the set of Hamiltonian orbits corresponding to Reeb orbits of period less than \(m^+\) is a subcomplex of the set of Reeb orbits of period less than \(m^-\). Intuitively, the Floer trajectory should decrease the action associated to the Reeb vector field, which is the period of the Reeb orbit.

figure 3.3.1:A limit of Hamiltonians of increasing slopes eventually sees all Reeb orbits

Consider now an increasing sequence of slopes \(m_0< m_1< \cdots \) which are not the periods of any Reeb orbits of \(\partial X\). One can form the telescope complex

could not find file dig_symplenicticCohomologyTelescope

where the vertical maps are the identity, and the diagonal maps are continuations.

proposition 3.3.2

The cohomology of the telescope complex \(\bigoplus_{i=0}^\infty C^\bullet_i \oplus C^\bullet_{i-1}\) is \(\lim_{i\to\infty} H(C^\bullet_i)\). We could therefore also define \[SH(X):=\lim_{i\to\infty} \SH(X)^{< m^i}.\]

3.4: ring structure on \(\SH(X)\)

Without proof, we describe some of the algebraic structures on the symplectic cohomology. First, the construction of the pair of pants product on Hamiltonian Floer cohomology can be extended to give a pair-of-pants product on the symplectic cohomology. This makes \(\SH(X)\) into a graded ring. Furthermore, the symplectic cohomology is a ring with unit.

4: applications of \(\SH(X)\)

We look at some applications and theorems from symplectic cohomology.

proposition 4.0.1

Let \(X\) be a Liouville domain. There is a map \[H^\bullet(X)\to \SH(X).\] Let \(m_0\) be the minimal period of a Reeb orbit in \((\partial X, \alpha)\). Pick a slope \(0< m< m_0\), and consider a linear Hamiltonian \(H^m\) which is \(C^2\) small on \(X\). Then the only orbits for \(H^m\) will be the constant orbits, and there is a quasi-isomorphism of chain complexes between \(\CF(\hat X, H^m_t)\) and the Morse complex \(CM^\bullet(\hat X, H^m_t)\) sending each constant orbit to its associated critical point of \(H^m_t\). Since \(H^m\) has gradient which points outward along the boundary of \(X\), this is a valid Morse function for computing the Morse cohomology of \(X\).

application 4.0.2

We look at an application of proposition 4.0.1. Let \(X\) be a Liouville domain, and suppose that \((\partial X, \alpha)\) has no Reeb orbits. Then the map from proposition 4.0.1 is an isomorphism \[H^\bullet(X)\to \SH(X).\] Therefore, we can compute \(\SH(X)\) to show that \((\partial X)\) has a Reeb orbit.

example 4.0.3

A key example is the standard contact structure on the sphere. One can compute that the standard symplectic ball \(B^{2n}:=\{(z_1, \ldots, z_n)\st \sum_{i=1}^n |z_i|^2=1 \}\subset \CC^n\) is a Liouville domain, and that \(\SH(B^{2n})=0\). We can conclude that \((S^{2n-1}, \alpha)\) has a Reeb orbit. A generalization of this result (due to [Oan03]) states that whenever \(X\) is a subcritical Stein domain (so the Morse indexes of the critical points of \(\phi: X\to \RR\) are all less than \(n\)) then \(\SH(X)=0\).

definition 4.0.4

Suppose that \((X, \lambda)\) is a Liouville domain. A Liouville subdomain is a compact submanifold with boundary \(X_0\subset X\setminus \partial X\) such that the Liouville vector \(Z\) points outwards along \(\partial X_0\).

theorem 4.0.5 [Vit99]

Let \(X_0\subset X\) be a Liouville subdomain. Then there is a restriction map \(\SH(X)\to \SH(X_0)\), which is a unital ring homomorphism. Furthermore, we have a commutative diagram

where the horizontal maps are given by proposition 4.0.1. In the setting of subcritical Stein domains ( application 4.0.2 ), the sublevel sets \(X|_{\phi< t}\) form a nested sequence of Liouville subdomains. One way to prove the vanishing of \(\SH(X)\) is to show that the map \(\SH(X|_{\phi< t_{i+1}})\to \SH(X|_{\phi< t_{i}})\) is a an isomorphism for all \(t_{i} < t_{i+1}\). When the only critical points of \(\phi\) contained in \(\SH(X|_{\phi < t_0})\) are minima, then \(X|_{t_0}\) is a ball, which has vanishing symplectic cohomology.

5: Viterbo's theorem

One of the most striking computations for symplectic cohomology comes from Viterbo's theorem, which relates the symplectic cohomology of \(T^*Q\) with the homology of the free loop space \(\mathcal L Q\).

definition 5.0.1

Let \(Q\) be a manifold. The free loop space of \(Q\), denote by \(\mathcal L Q\), consists of all continuous maps \[\gamma:S^1\to Q.\] The projection to base point is the map \begin{align*} \ev_0: \mathcal L Q\to& Q\\ \gamma\mapsto &\gamma(0). \end{align*} We usually regard the Hamiltonian Floer cohomology of \(X\) as being ``inspired'' by doing Morse theory on the free loop space of \(\Omega X\). However, it is not the Morse theory on the loop space, which can actually be done! From the data of \(g\) a metric for \(Q\), we can define an energy functional on the space of loops of Sobelov class \(W^{1,2}\), \begin{align*} E: \mathcal LQ\to \RR\\ \gamma\mapsto \int_{\gamma}|\gamma'|^2_gdt, \end{align*} whose critical points are either constant loops or \(g\)-geodesics on \(Q\). This functional is only Morse-Bott for generic metrics (observe that every geodesic can be reparameterized) but still satisfies the property that the critical submanifolds have finite index. Therefore, the downward flow spaces of each critical submanifold is a finite dimensional submanifold, and the union of all such manifolds recovers the homotopy type of \(\mathcal LQ\). Additionally, there is an honest gradient flow of \(E\) --- that is, from a given loop \(\gamma\) we can construct a unique \(E\)-gradient flow-line starting at \(\gamma\). Since the gradient of \(E\) points inwards along its level sets, \(CM_\bullet(\mathcal LQ, E)\) computes the homology of the loop space. Given \((X, \omega, J)\) a symplectic manifold, the Floer action functional does not have critical points of finite index or co-index --- so there are no well defined downward flow spaces, and no absolute index. Even worse, the gradient ``flow'' of the Floer action functional is not well defined (in that we don't have unique 1-dimensional families of solutions given an initial starting parameter). However, the space of gradient flow-lines between two critical points is still well-behaved, which is enough to get a ``Morse-like'' cohomology theory from \(A_{H_t}:\mathcal L X \to \RR\). Despite these apparent differences, there is a scenario where we can translate Floer theory to study the homology of the loop space. Let \((Q, g)\) be a Riemannian manifold. Let \(B^*Q\) be the unit cotangent ball. This is a Liouville domain (depending on the choice of metric \(g\)), whose contact boundary \((S^*Q, \alpha)\) has Reeb orbits corresponding to the \(g\)-geodesics of \(Q\) (exercise 6.0.2). So, it seems reasonable to conjecture a relation between \[\text{Floer functional \(A_{H}:\mathcal L(T^*Q)\to \RR\)}\Leftrightarrow\text{Energy functional \(E:\mathcal Q\to \RR\)}\] We also have the suggestive relations between cohomology theories which we know how to compute:

where the bottom arrow is inclusion of \(Q\into \mathcal LQ\) by the constant loops, and the top right arrow is given by proposition 4.0.1. From these pieces of evidence one might expect the following theorem.

theorem 5.0.2 [Vit99]

Let \((Q, g)\) be a compact Riemannian manifold. Let \(X=B^*Q\) be the unit cotangent ball. Then \[\SH(X)= H_\bullet(\mathcal L Q).\]

application 5.0.3

We now look at an application of Viterbo's theorem. Let \(X\) be a Liouville domain, and suppose that there exists \(L\subset X\) an exact Lagrangian submanifold. Then a Weinstein neighborhood \(B^*L\subset X\) provides an example of a Liouville subdomain. We therefore have a unital ring homomorphism \(\SH(X)\to \SH(B^*L)\). Since the latter is isomorphic (as a vector space) to \(H_\bullet(\mathcal L)\), it is non-vanishing. Since a unital ring homomorphism to a non-trivial target cannot have trivial domain, we conclude that \(\SH(X)\) is non-vanishing. This application is more striking in the reverse direction. Let \(X\) be a Liouville domain with vanishing symplectic cohomology (for instance, a subcritical Stein domain). Then \(X\) contains no exact Lagrangian submanifolds.

6: exercises

exercise 6.0.1

Consider the space \(T^*S^1\), with coordinates \((q, p)\).

Determine all of the time-1 periodic orbits of the Hamiltonian \(H=|p|^2\).
Construct a Hamiltonian \(\tilde H\) with the property that
- The non-constant orbits of \(H\) are in bijection with the non-constant orbits of \(\tilde H\) and;
- \(\tilde H\) has only 2 constant orbits.
Let \(TI(\tilde H)/S^1\) denote the set of time independent orbits of \(\tilde H\) up to reparameterization. It is shown in [BO09] that there exists a time dependent Hamiltonian \(\tilde H_t: T^*S^1\to \RR\) whose
- constant orbits are in bijection with the constant orbits of \(\tilde H\) and;
- whose non-constant orbits are of the form \(\{\gamma_{\min}\}_{\gamma\in TI(\tilde H)}\cup \{\gamma_{\max}\}_{\gamma\in TI(\tilde H)}\).
Furthermore, the homology class of \(\gamma_{\min}, \gamma_{\max}\) agree with \(\gamma\in TI(\tilde H)\), and there is (in this example) no differential between \(\gamma_{\min}, \gamma_{\max}\). Using this, compute \(\SH(T^*S^1)\) (as an ungraded vector space).
Compute \(H_*(LS^1)\), the homology of the loop space of \(S^1\).
Justify why the generators of \(H_*(LS^1)\) appear in pairs (just like they do \(\SH(T^*S^1)\)).

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exercise 6.0.2

Let \((Q, g)\) be a Riemannian manifold.

Describe a metric on \( g^*:T^*_qQ\to T^*_qQ\) on the cotangent bundle. In local coordinates \((q, p)\) for \(T^*Q\), write down the Hamiltonian vector field \(V_g\) associated to the Hamiltonian \(H(q, p):=g_q^*(p, p): T^*_qQ\to \RR\).
Recall that a geodesic on \(Q\) is a curve \(\gamma:\RR\to Q\) which is locally length minimizing. In particular, it is a minimizer for the action \[E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\dot \gamma(t), \dot \gamma(t)) dt\] for every \(a,b\in \RR\). Prove that if \(\hat \gamma: \RR\to T^*Q\) is a flow-line of \(V_g\), that \(\gamma:=\pi_Q(\hat \gamma)\) is a geodesic.
Let \(n>1\). Use the Serre spectral sequence for the fibration \[\Omega S^n\to PS^n\to S^n\] to conclude that \(H_*(\Omega S^n)\simeq \ZZ[x]\). Here, \(\Omega M\) is the based loop space of \(M\), and \(PM\) is the based path space of \(M\). The element \(x\) is of degree \(n-1\) and is determined by \[H_n(S^n)\to \pi_n(S^n)\to \pi_{n-1}(\Omega S^n)\to H_{n-1}(\Omega S^n).\]
Let \(n>1\). Using the Serre spectral sequence for the fibration \[\Omega S^n\to LS^n\to S^n\] show that as a vector space \(H_*(LS^n)\neq 0, \ZZ/n\ZZ\) or \(\ZZ^2\). Here, \(LM\) is the free loop space of \(L\).
Show that if there exists a metric \(g\) for \(S^n\) which has no closed geodesics, that there exists a Hamiltonian \(H: S^n\to \RR\) which has no non-constant orbits, and two constant orbits. Conclude that every metric on \(S^n\) has at least one closed geodesic.

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References

[Abo10]	Mohammed Abouzaid. A geometric criterion for generating the Fukaya category. Publications Mathématiques de l'IHÉS, 112:191--240, 2010.
[BO09]	Frédéric Bourgeois and Alexandru Oancea. Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. Duke mathematical journal, 146(1):71--174, 2009.
[Fet52]	Abram Il'ich Fet. Variational problems on closed manifolds. Matematicheskii Sbornik, 72(2):271--316, 1952.
[Oan03]	Alexandru Oancea. La suite spectrale de Leray-Serre en homologie de Floer des variétés symplectiques compactes à bord de type contact. PhD thesis, Université Paris Sud-Paris XI, 2003.
[Sei06]	Paul Seidel. A biased view of symplectic cohomology. Current developments in mathematics, 2006(1):211--254, 2006.
[Vit99]	Claude Viterbo. Functors and computations in Floer homology with applications, i. Geometric & Functional Analysis GAFA, 9(5):985--1033, 1999.
[Wen]	Chris Wendl. A beginner’s overview of symplectic homology. Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf.