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

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exercise 0.0.1

Consider the space \(T^*S^1\), with coordinates \((q, p)\).
  1. Determine all of the time-1 periodic orbits of the Hamiltonian \(H=|p|^2\).
  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.
  3. 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).
  4. Compute \(H_*(LS^1)\), the homology of the loop space of \(S^1\).
  5. Justify why the generators of \(H_*(LS^1)\) appear in pairs (just like they do \(\SH(T^*S^1)\)).

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exercise 0.0.2

Let \((Q, g)\) be a Riemannian manifold.
  1. 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 \(H(q, p):=g_q^*(p, p): T^*_qQ\to \RR\).
  2. 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.
  3. 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 \[H_n(S^n)\to \pi_n(S^n)\to \pi_{n-1}(\Omega S^n)\to H_{n-1}(\Omega S^n).\]
  4. 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 \(L\).
  5. 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

[BO09]Frédéric Bourgeois and Alexandru Oancea. Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. Duke mathematical journal, 146(1):71--174, 2009.