The proof of Lemma \[L:Riede\] involves a discrete curve shortening process $\Psi$ on $\mathcal{C}_\Sigma$ which satisfies properties (1) - (3) as described in the introduction. Since any curve $c \in \mathcal{C}_\Sigma$ can be continuously deformed into the unique chord joining the same end points, we can restrict $\mathcal{C}_\Sigma$ to the subset $\mathcal{C}^0_\Sigma$ consisting of all the chords with end points on $\Sigma$. The chord shortening flow is then a curve shortening process on $\mathcal{C}^0_\Sigma$ which satisfies all the required properties. Moreover, the space of chords $\mathcal{C}^0_\Sigma$ can also be described as the orbit space of $\Sigma \times \Sigma$ under the ${\mathbb{Z}}_2$-action $(p,q) \mapsto (q,p)$. As before, if we let $\Delta \subset \Sigma \times \Sigma$ be the fixed point set of the ${\mathbb{Z}}_2$-action, and $H^{{\mathbb{Z}}_2}_*(\Sigma \times \Sigma, \Delta)$, $H^*_{{\mathbb{Z}}_2}(\Sigma \times \Sigma)$ be the ${\mathbb{Z}}_2$-equivariant homology and cohomology respectively, we have by naturality $$\label{E:isomorphism}
H^{{\mathbb{Z}}_2}_*(\Sigma \times \Sigma,\Delta) \cong H^{{\mathbb{Z}}_2}_*(\mathcal{C}_\Sigma,\Delta') \qquad \text{and} \qquad H^*_{{\mathbb{Z}}_2}(\Sigma \times \Sigma) \cong H^*_{{\mathbb{Z}}_2}(\mathcal{C}_\Sigma).$$ In [@Hayashi82], Hayashi studied the equivariant (co)homology of $\Sigma \times \Sigma$ and obtained the following result in [@Hayashi82 Theorem 2].

\[L:Hayashi\] There exists $\beta \in H^{{\mathbb{Z}}_2}_{2k}(\Sigma \times \Sigma,\Delta)$ and $\alpha \in H^1_{{\mathbb{Z}}_2}(\Sigma \times \Sigma)$ such that $\alpha^k \cap \beta \neq 0$ in $H^{{\mathbb{Z}}_2}_k(\Sigma \times \Sigma, 