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.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 \((M, \alpha)\).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.
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.definition 1.0.8
Let \((M, \alpha)\) be a contact manifold. The symplectization of \((M, \alpha)\) is the symplectic manifold \[(\RR\times M, d(\exp(r)\cdot \alpha)).\]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 \(H^m(r, x)= m\cdot \exp(r)\).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 \(X\).
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 \(X\).
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.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 \[d(\exp(r))\circ J = -\alpha.\]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 \(\gamma_+, \gamma_-\).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. 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. 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 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 \(\lim_{i\to\infty} H(C^\bullet_i)\).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 \[H^\bullet(X)\to \SH(X).\]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.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 4.0.1.5: 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*}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 \(\{\gamma_{\min}\}_{\gamma\in TI(\tilde H)}\cup \{\gamma_{\max}\}_{\gamma\in TI(\tilde H)}\).
- 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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[Wen] | Chris Wendl. A beginnerâ€™s overview of symplectic homology. Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf. |