A norm for the cohomology of 2–complexes

Abstract

We introduce a norm on the real 1–cohomology of finite 2–complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander–Fox polynomials of groups and show that they give rise to norms on the real 1–cohomology of groups. Our main theorem states that for a finite 2–complex $X$, the norm on $H^1\left(X;ℝ\right)$ determined by graphs on $X$ majorates the Alexander–Fox norms derived from ${\pi }_1\left(X\right)$.