cohomology class. We write $\mathfrak Z(\Sigma)$ for the set of weakly Zoll pairs and $\mathfrak Z_C(\Sigma)$ for the subset of those pairs $(\mathfrak p,c)$ such that $C=\mathfrak p^*c\in H_{{\mathrm{dR}}}^2(\Sigma)$. We denote by $\mathfrak Z_C^0(\Sigma)$ the set of those $(\mathfrak p,c)\in\mathfrak Z_C(\Sigma)$ such that $\mathfrak p\in\mathfrak P^0(\Sigma)$.

If $\Omega\in\mathcal S_C(\Sigma)$ is Zoll, then $(\mathfrak p_\Omega,[{\mathrm{\omega}}_\Omega])\in\mathfrak Z_C(\Sigma)$ is a weakly Zoll pair, where ${\mathrm{\omega}}_\Omega$ is the closed two-form on $M_\Omega$ such that $\Omega=\mathfrak p_\Omega^*{\mathrm{\omega}}_\Omega$. Conversely, if $(\mathfrak p,c)\in\mathfrak Z_C(\Sigma)$ is a weakly Zoll pair, we can consider any closed two-form ${\mathrm{\omega}}$ on $M$ such that $c=[{\mathrm{\omega}}]$ and build the two-form $\mathfrak p^*{\mathrm{\omega}}\in\Xi^2_C(\Sigma)$, which is Zoll exactly when ${\mathrm{\omega}}$ is symplectic.

Let us now fix a [**reference**]{} weakly Zoll pair $$(\mathfrak p_0,c_0)\in\mathfrak Z_C^0(\Sigma),\qquad \mathfrak p_0:\Sigma\to M_0.$$ We write $e_0\in H^2_{{{\mathrm{dR}}}}(M_0)$ for minus the real Euler class of $\mathfrak p_0$ and we have the equivalence $$C^n\neq0\qquad \Longleftrightarrow\qquad e_0=0.$$ Let $\Lambda_{\mathfrak h}(\Sigma)$ be the space of one-periodic curves in the class $\mathfrak h$, and let $\widetilde{\Lambda}_{\mathfrak h}(\Sigma)$ be the space of homotopies of paths $\{\gamma_r\}_{r\in[0,1]}$ inside $\Lambda_{\mathfrak h}(\Sigma)$ such that $\gamma_0$ is some oriented $\mathfrak p_0$-fibre. The admissible homotopies are allowed to move $\gamma_0$ inside the set of oriented $\mathfrak p_0$-fibres but have to fix 