On the Berstein-Svarc Theorem in dimension 2

We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$ with $\dim X=\text{cat}X=n$, the $n$th power of the Berstein class of $X$ is nontrivial. Previously it was known for $n\ge 3$. We also prove that, for every map $f:M\to N$ of degree $\pm 1$ of closed orientable manifolds, the fundamental group of $N$ is free provided that the fundamental group of $M$ is.