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We introduce a norm on the real 1-cohomology of finite 2-complexes determinedby the Euler characteristics of graphs on these complexes. We also introducetwisted Alexander-Fox polynomials of groups and show that they give rise tonorms on the real 1-cohomology of groups. Our main theorem states that for afinite 2-complex X, the norm on H^1(X; R) determined by graphs on X majoratesthe Alexander-Fox norms derived from \pi_1(X).

A norm for the cohomology of 2–complexes