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Relative systoles of relative-essential $2$–complexes

Karin Usadi Katz, Mikhail G Katz, Stéphane Sabourau, Steven Shnider and Shmuel Weinberger

Algebraic & Geometric Topology 11 (2011) 197–217

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).

coarea formula, cohomology of cyclic groups, essential complex, Grushko's theorem, Poincaré duality, systole, systolic ratio
Mathematical Subject Classification 2000
Primary: 53C23, 57M20
Secondary: 57N65
Received: 27 October 2009
Revised: 12 July 2010
Accepted: 2 October 2010
Published: 7 January 2011
Karin Usadi Katz
Department of Mathematics
Bar Ilan University
52900 Ramat Gan
Mikhail G Katz
Department of Mathematics
Bar Ilan University
52900 Ramat Gan
Stéphane Sabourau
Laboratoire de Mathématiques et Physique Théorique
Université de Tours
Parc de Grandmont
37200 Tours
Steven Shnider
Department of Mathematics
Bar Ilan University
52900 Ramat Gan
Shmuel Weinberger
Department of Mathematics
University of Chicago
5734 S University Avenue
Chicago IL 60637-1514
United States