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Short homotopically independent loops on surfaces

Steve Karam

Algebraic & Geometric Topology 14 (2014) 1825–1844

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 2 and area normalized to g, there are at least log(2g) + 1 homotopically independent loops based at the same point of length at most Clog(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 2 with 1 n 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.

Riemannian surfaces, homologically independent loops, systole
Mathematical Subject Classification 2010
Primary: 30F10
Received: 21 October 2013
Accepted: 15 November 2013
Published: 29 May 2014
Steve Karam
Laboratoire de Mathématiques et de Physique Théorique
Université de Tours
UFR Sciences et Technologie
Parc de Grandmont
37200 Tours