Relative systoles of relative-essential 2-complexes

TL;DR

This paper establishes a new systolic inequality relating the phi-relative 1-systole and area of phi-essential 2-complexes, with implications for 3-manifolds like the Poincare homology sphere.

Contribution

It proves a universal lower bound on the area of phi-essential 2-complexes in terms of their phi-relative systole, extending systolic inequalities to a broader class of complexes.

Findings

Area of phi-essential 2-complexes is at least 1/8 of the square of their phi-relative systole.

New bounds on systole and volume for certain 3-manifolds, including the Poincare homology sphere.

Application of Guth's method to derive quantitative geometric inequalities.

Abstract

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24 vol(Sigma).