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Short Homotopically independent loops on surfaces

In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs. First, we show that for every closed Riemannian surface of genus $g \geq 2$ and area normalized to $g$, there are at least $\ceil{\log(2g)+1}$ homotopically independent loops based at the same point of length at most $C\log(g)$, where $C$ is a universal constant. On the one hand, this result substantially improves Theorem $5.4.A$ of M. Gromov in \cite{G1}. On the other hand, it recaptures the result of S. Sabourau on the separating systole in \cite{SS} and refines his proof. Second, we show that for any two integers $b\geq 2$ with $1\leq n\leq b$, every connected metric graph $Γ$ of first Betti number $b$ and of length $b$ contains at least $n$ homologically independent loops based at the same point and of length at most $24(\log(b)+n)$. In particular, this result extends Bollobàs-Szemerédi-Thomason's $\log(b)$ bound on the homological systole to at least $\log(b)$ homologically independent loops based at the same point. Moreover, we give examples of graphs where this result is optimal.