Relative systoles of relative-essential $2$–complexes

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 $sys{\left(\Sigma \right)}^3\le 24Vol\left(\Sigma \right)$.