Algebraic & Geometric Topology 11 (2011) 197–217

DOI: 10.2140/agt.2011.11.197

Abstract

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).

Keywords

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

Mathematical Subject Classification

Primary: 53C23, 57M20

Secondary: 57N65

References

Publication

Received: 27 October 2009
Revised: 12 July 2010
Accepted: 2 October 2010
Published: 7 January 2011

Authors