Let $(M,g)$ be a Riemannian manifold and ‘$c$’ be some homology class of $M$. The systole of $c$ is the minimum of the $k$-volume over all possible representatives of $c$. We will use combine recent works of Gromov and Zhu to show an upper bound for the systole of $S^2 \times \{*\}$ under the assumption that $S^2 \times \{*\}$ contains two representatives which are far enough from each other.

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Last updated: 06 Nov 2024