On the Berstein–Svarc theorem in dimension 2

Abstract

Abstract We prove that for any group π with cohomological dimension at least n the n th 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.