#### Volume 11, issue 1 (2011)

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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
##### Abstract

We prove a systolic inequality for a $\varphi$–relative systole of a $\varphi$–essential $2$–complex $X$, where $\varphi :{\pi }_{1}\left(X\right)\to G$ is a homomorphism to a finitely presented group $G$. Thus, we show that universally for any $\varphi$–essential Riemannian $2$–complex $X$, and any $G$, the following inequality is satisfied: $sys{\left(X,\varphi \right)}^{2}\le 8Area\left(X\right)$. 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 $\Sigma$, we have $sys{\left(\Sigma \right)}^{3}\le 24Vol\left(\Sigma \right)$.

##### Keywords
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