Relative systoles of relative-essential 2–complexes

Katz, Karin Usadi¹ ; Katz, Mikhail G¹ ; Sabourau, Stéphane² ; Shnider, Steven¹ ; Weinberger, Shmuel³

¹Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel
²Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France
³Department of Mathematics, University of Chicago, 5734 S University Avenue, Chicago IL 60637-1514, United States

Algebraic and Geometric Topology, Tome 11 (2011) no. 1, pp. 197-217

Résumé

We prove a systolic inequality for a ϕ–relative systole of a ϕ–essential 2–complex X, where ϕ: π1(X) → G is a homomorphism to a finitely presented group G. Thus, we show that universally for any ϕ–essential Riemannian 2–complex X, and any G, the following inequality is satisfied: sys(X,ϕ)2 ≤ 8Area(X). Combining our results with a method of L Guth, we obtain new quantitative results for certain 3–manifolds: in particular for the Poincaré homology sphere Σ, we have sys(Σ)3 ≤ 24Vol(Σ).

Détail

DOI : 10.2140/agt.2011.11.197

Keywords: coarea formula, cohomology of cyclic groups, essential complex, Grushko's theorem, Poincaré duality, systole, systolic ratio

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