In this paper, we are interested in short homologically and homotopically
independent loops based at the same point on Riemannian surfaces and metric
First, we show that for every closed Riemannian surface of genus
normalized to ,
there are at least
homotopically independent loops based at the same point of length at
a universal constant. On the one hand, this result substantially improves Theorem
5.4.A of M Gromov in [J. Differential Geom. 18 (1983) 1–147]. On the other hand, it
recaptures the result of S Sabourau on the separating systole in [Comment. Math.
Helv. 83 (2008) 35–54] and refines his proof.
Second, we show that for any two integers
, every connected
metric graph of
first Betti number
and of length
contains at least
homologically independent loops based at the same point and of length at most
In particular, this result extends Bollobàs, Szemerédi and Thomason’s
bound on the homological
systole to at least
homologically independent loops based at the same point. Moreover, we give
examples of graphs where this result is optimal.