## Algebraic & Geometric Topology

### Relative systoles of relative-essential $2$–complexes

#### 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 $sys(Σ)3≤24Vol(Σ)$.

#### Article information

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

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

https://projecteuclid.org/euclid.agt/1513715184

Digital Object Identifier
doi:10.2140/agt.2011.11.197

Mathematical Reviews number (MathSciNet)
MR2764040

Zentralblatt MATH identifier
1228.53056

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715184

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