On the Berstein–Svarc theorem in dimension 2

Abstract

AbstractWe prove that for any group π with cohomological dimension at least n the nth power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n:Theorem. For a connected complex X with dim X = cat X = n, we have$\ber_X^n$ ≠ 0 where$\ber_X$is the Berstein class of X.Previously it was known for n ≥ 3.We also prove that, for every map f: M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.