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
and area
normalized to ,
there are at least
homotopically independent loops based at the same point of length at
most ,
where is
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
with
, 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.
