  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 $\dim X=\cat X=n$, 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.
