\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

symplectic cohomology of \(T^*S^1\)

  1. A straighfoward computation shows that \(\varphi_t(q,p)=(q-2pt,p)\). Therefore the flow is 1-periodic if and only if \(2p=2k\pi, k\in \ZZ\), i.e. \(p=k\pi\).
  2. At the zero-section \(S^n\subset T^*S^n\) we have \(X_H\restr{S^n}=0\). Consider therefore a {\it perturbed} Hamiltonian \(\tilde H=H+\epsilon h\), where \(h(q,p):=\eta(p)\sin q\) with \(\eta:\RR\to \RR\) a cut-off function centered at \(p=0\). Note that \(\tilde H=H\) whenever \(\eta=0\), so outside a neighborhood of the zero-section we have a bijection between orbits of \(\tilde H\) and orbits of \(H\), all of which are non-constant. Furthermore, we have \(X_{\tilde H}=(-2p+\epsilon\eta(p)\cos q)\partial_q +\epsilon\eta'(p)\sin q\partial_p\) from which we deduce that, for \(\epsilon\) small enough, the only constant orbits occur when \(p=0\) and \(\cos q=0\), a total of two points. It remains to argue that there cannot be other non-constant orbits in this region. This follows from the fact that we can make \(|X_{\tilde H}|\) as small as we want --- again, by choosing suitably small \(\epsilon\) and cut-off width --- so that any periodic orbit must necessarily have period \(T>>1\).
  3. We will use the time-dependent Hamiltonian \(\tilde H_t\) to compute the symplectic cohomology \(SH^\bullet(T^*S^1)\). Note \(\tilde H_t\) has:
    • Two constant orbits ocurrying at \((q,p)=\left(\pm \frac{\pi}{2},0\right)\), call them \(\gamma^0_{max,min}\);
    • Two non-constant orbits \(\gamma_{max,min}^k\) for each \(k\in \ZZ\setminus\set{0}\) ocurrying near \(p_k=k\pi\).
    These are by definition the generators of \(FC^\bullet(\tilde H_t)\). Now, for topological reasons (homology/homotopy class of the orbits) we can exclude any pseudoholomorphic cylinders between \(\gamma^k_*\) and \(\gamma^l_*\) for \(k\neq l\). Furthermore, by assumption there is also no differential between \(\gamma^k_+\) and \(\gamma^k_-\) for any \(k\). Putting both things together we conclude the differential vanishes, and thus each orbit defines a generator of \(SH^\bullet(T^*S^1)\). In conclusion, we have \[ SH^\bullet(T^*S^1)=\bigoplus_{\substack{k\in\ZZ \\ *=max,min}}\RR\cdot \gamma^k_*\cong \RR[x]\oplus\RR[x]\]
  4. We first note that \(\pi_0(LS^1)=\ZZ\), where \(k\in \ZZ\) distinguishes homotopy classes. Let \(LS^1(k)\subset LS^1\) be the connected component consisting of loops of homotopy class equal to \(k\in\pi_1(S^1)\cong\ZZ\). Then we have a splitting \[ H_*(LS^1)\cong\oplus_{k\in \ZZ}H_*(LS^1(k))\] There is an obvious inclusion \(S^1\into LS^1(k), p\mapsto (t\mapsto pe^{i2\pi k t})\), and one can see the map \(LS^1(k)\to S^1\) that records the base-point of the loop is a homotopy inverse. Therefore \(S^1\simeq LS^1(k)\) have the same homology, and we conclude \[ H_*(LS^1)=\oplus_{k\in \ZZ} \RR^2\cong \RR[x]\oplus\RR[x]\]
  5. Each component \(LS^1(k)\) contributes two generators to \(H_*(LS^1)\). Consider a generator of \(H_*(LS^1)\) coming from \(H_0(LS^1(k))\), which is just a point \(\gamma\in LS^1(k)\). Then there is an obvious 1-cycle \(\Gamma\in C_1(LS^1(k)), \partial\Gamma=0\) that we can build, namely \(\Gamma(s)=\gamma(\cdot+s)\), the 'rotation' of the loop \(\gamma\). Note that recording the basepoint \(\Gamma(s)(0)=\gamma(s)\) of each rotated loop we obtain the fundamental cycle of \(S^1\), so according to our identification \(S^1\simeq LS^1(k)\) the 1-cycle \(\Gamma\) is precisely the degree one generator of \(H_*(LS^1(k))\).