Algebraic & Geometric Topology

Relative systoles of relative-essential $2$–complexes

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

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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)28Area(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)324Vol(Σ).

Article information

Algebr. Geom. Topol., Volume 11, Number 1 (2011), 197-217.

Received: 27 October 2009
Revised: 12 July 2010
Accepted: 2 October 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57M20: Two-dimensional complexes
Secondary: 57N65: Algebraic topology of manifolds

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


Katz, Karin Usadi; Katz, Mikhail G; Sabourau, Stéphane; Shnider, Steven; Weinberger, Shmuel. Relative systoles of relative-essential $2$–complexes. Algebr. Geom. Topol. 11 (2011), no. 1, 197--217. doi:10.2140/agt.2011.11.197.

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