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 $⌈ log (2 g) + 1 ⌉$ 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 [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 $b \geq 2$ 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 and 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.

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Keywords

Riemannian surfaces, homologically independent loops, systole

Mathematical Subject Classification 2010

Primary: 30F10

References

Publication

Received: 21 October 2013

Accepted: 15 November 2013

Published: 29 May 2014

Authors

Steve Karam
Laboratoire de Mathématiques et de Physique Théorique Université de Tours UFR Sciences et Technologie Parc de Grandmont 37200 Tours France

Volume 14, issue 3 (2014)

Steve Karam

Abstract