Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose shortest representatives have length at most $L$ is infinite? For surfaces ($n=2$) with constant curvature metrics, this is impossible. Thanks!
This cardinality is always finite, for any compact locally simply connected metric space. If there were infinitely many non-homotopic curves of length $\le L$, they would have a converging subsequence (by Arzela-Ascoli). In a locally simply connected space, any two sufficiently close curves are homotopic, so curves in the sequence are eventually homotopic to their limit, a contradiction.